Introduction to the Batteries & Fuel Cells Module · 2014-04-10 · 2 | Introduction The Batteries...

VERSION 4.4

Introduction toBatteries & Fuel Cells Module

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Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Battery Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

The Battery Modeling Physics Interfaces . . . . . . . . . . . . . . . . . . . . . . 6

Fuel Cell Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

The Fuel Cell Modeling Physics Interfaces . . . . . . . . . . . . . . . . . . . . 10

The Physics Interface List by Space Dimension and Study

Type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

The Model Libraries Window . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Tutorial Model of a Lithium-Ion Battery . . . . . . . . . . . . . . . . . . . 15

Model Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Tutorial Model of a Fuel Cell Cathode . . . . . . . . . . . . . . . . . . . . 30

Model Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Solving Electrochemical Models . . . . . . . . . . . . . . . . . . . . . . . . . . 61

| i

Introduction

The Batteries & Fuel Cells Module models and simulates the fundamental processes in the electrodes and electrolytes of batteries and fuel cells. These simulations may involve the transport of charged and neutral species, current conduction, fluid flow, heat transfer, and electrochemical reactions in porous electrodes.You can use this module to investigate the performance of batteries and fuel cells at different operating conditions for different electrode configurations, separators, current collectors and feeders, materials, and chemistry. The description of the involved processes and phenomena is rather detailed and you can therefore apply different hypotheses to gain an understanding of the investigated systems. You can study the influence of different electrocatalysts, pore distribution, electrolyte composition, and other fundamental parameters directly in the physics interface.The figure below shows the Batteries & Fuel Cells physics interfaces and other physics interfaces in COMSOL Multiphysics that are modified by the module, for example the Heat Transfer interfaces.

Figure 1: The 3D model physics list for the Batteries & Fuel Cells Module as shown in the Model Wizard.

Introduction | 1

The Batteries & Fuel Cells physics interfaces are based on the conservation of current, charge, chemical species, and energy.The Battery with Binary Electrolyte ( ), the Lead-Acid Battery ( ), and the Lithium-Ion Battery ( ) interfaces form the basis for battery modeling. In addition, the Primary Current Distribution ( ), Secondary Current Distribution ( ), and Tertiary Current Distribution, Nernst-Planck ( ) interfaces are general physics interfaces to model any electrochemical cell. The Chemical Species Transport ( ), the Fluid Flow ( ) and the Heat Transfer ( ) interfaces are also extended with functionality for battery modeling, as discussed in “Battery Modeling” on page 3.The Primary, Secondary, and Tertiary Current Distribution interfaces, in combination with the interfaces for Chemical Species Transport, Fluid Flow, and Heat Transfer, form the basis for fuel cell modeling, as discussed in “Fuel Cell Modeling” on page 8.

2 | Introduction

Battery Modeling

The Batteries & Fuel Cells Module has a number of physics interfaces to model battery unit cells that consist of:• Current collectors and current feeders• Porous electrodes• The electrolyte that separates the anode and cathode

The module treats both rechargeable batteries (secondary cells) and non-rechargeable batteries (primary cells).The Batteries & Fuel Cells Module can be used to study the charge and discharge processes in a rechargeable battery as described below.

Figure 2: Direction of the current and charge transfer current during discharge in a battery with porous electrodes.

During discharge, chemical energy is transferred to electrical energy in the charge transfer reactions at the anode and cathode. The conversion of chemical to electrical energy during discharge may involve electrochemical reactions, transport of electric current, transport of ions in the electrolyte, neutral chemical species transport, fluid flow, and the release of heat in irreversible losses, such as ohmic losses and losses due to activation energies.

Cathodic charge transfer reaction

Anodic charge transfer reactionCurrent in positive electrodeCurrent in the electrolyteCurrent in negative electrode

negative electrode positive electrode

Anode Cathode

Negative electrode Positive electrode

Ecell

iloc

iloc, a

iloc, c

E

Battery Modeling | 3

Figure 2 shows a schematic picture of the discharge process. The current enters the cell from the current feeder at the negative electrode. The charge transfer reaction occurs at the interface between the electrode material and electrolyte contained in the porous electrode, also called the pore electrolyte. Here, an oxidation of the electrode material may take place through an anodic charge transfer reaction, denoted iloc, a in Figure 2. The shapes of the two curves in the graph are described by the electrode kinetics for the specific materials. The reaction may also involve the transport of chemical species from the pore electrolyte and also from the electrode particles.From the pore electrolyte, the current is conducted by the transport of ions through the electrolyte that separates the positive and negative electrode to the pore electrolyte in the positive electrode.At the interface between the pore electrolyte and the surface of the particles in the porous electrode, the charge transfer reaction transfers the electrolyte current to current conducted by electrons in the positive electrode. At this interface, a reduction of the electrode material takes place through a cathodic charge transfer reaction, denoted iloc, c in Figure 3. Also here, the charge transfer reaction may involve the transport of chemical species in the electrolyte and in the electrode particles.

Figure 3: Electrode polarization during discharge.

The current leaves the cell through the current collector. The conduction of current and the electrochemical charge transfer reactions also releases heat due to ohmic losses, activation losses, and other irreversible processes.The graph in Figure 3 plots the charge transfer current density, iloc, as a function of the electrode potential, E. These curves describe the polarization of the electrodes during discharge.

Positive electrodeNegative electrodeiloc

iloc, a

iloc, c

Anode Cathode

Ecell

Eocv

E

4 | Battery Modeling

The negative electrode is polarized anodically during discharge, positive current as indicated by the arrow in Figure 3. The potential of the negative electrode increases. The positive electrode is polarized cathodically, a negative current as indicated by the arrow. The potential of the positive electrode decreases.Consequently, Figure 3 also shows that the potential difference between the electrodes, here denoted Ecell, decreases during discharge compared to the open cell voltage, here denoted Eocv. The value of Ecell is the cell voltage at a given current iloc, if the ohmic losses in the cell are negligible. This is usually not the case in most batteries. This implies that the cell voltage in most cases is slightly smaller than that shown in Figure 3.During recharge, the processes are reversed; see Figure 4. Electrical energy is transformed to chemical energy that is stored in the battery.

Figure 4: During recharge, the positive electrode acts as the anode while the negative one acts as the cathode. The cell voltage increases, at a give current, compared to the open cell voltage.

The current enters the cell at the positive electrode. Here, during recharge, an oxidation of the products from the discharge takes place through an anodic charge transfer reaction. The positive electrode is polarized anodically, with a positive current, and the electrode potential increases.The current is then conducted from the pore electrolyte, through the electrolyte that separates the electrodes, to the negative electrode.

Anodic charge transfer reaction

Cathodic charge transfer reactionCurrent in positive electrodeCurrent in the electrolyteCurrent in negative electrode


Cathode Anode

iloc

iloc, a

iloc, c

EcellE



In the negative electrode, a reduction of the reduced products from the previous discharge reaction takes place through a cathodic charge transfer reaction. The negative electrode is polarized cathodically and the electrode potential decreases.

Figure 5: Electrode polarization during recharge.

The difference in potential between the electrodes, here denoted Ecell, at a given iloc, increases during recharge, compared to the open cell voltage, here denoted Eocv. The value of Ecell is equal to the cell voltage when ohmic losses are neglected. In most cells, these losses are not negligible and they would add to the cell voltage.The battery processes and phenomena described in the figures above can all be investigated using the Batteries & Fuel Cells Module. The physics included in the module allow you to investigate the influence on battery performance and thermal management of parameters such as the:• Choice of materials and chemistry• Dimensions and geometry of the current collectors and feeders• Dimension and geometry of the electrodes• Size of the particles that the porous electrodes are made of• Porosity and specific surface area of the porous electrode• Configuration of the battery components

The Battery Modeling Physics InterfacesThe Battery with Binary Electrolyte interface ( ) describes the conduction of electric current in the electrodes, the charge transfer reactions in the porous electrodes, the mass transport of ions in the pore electrolyte and in the electrolyte that separates the electrodes, and the intercalation of species in the particles that form the porous electrodes. The descriptions are available for cells with basic

iloc

iloc, a

iloc, c

Positive electrodeNegative electrode

Cathode AnodeEocv

Ecell

E

6 | Battery Modeling

binary electrolyte, which covers the nickel-metal hydride and the nickel-cadmium batteries.The Lead-Acid Battery interface ( ) is tailored for this type of battery and includes functionality that describes the transport of charged species, charge transfer reactions, the variation of porosity due to charge and discharge, and the average superficial velocity of the electrolyte caused by the change in porosity.The Lithium-Ion Battery interface ( ) is tailored for this type of battery and includes functionality that describes the transport of charged species, charge transfer reactions, internal particle diffusion, and the solid electrolyte interface (SEI).The Tertiary Current Distribution, Nernst-Planck, interface ( ) describes the transport of charged species in electrolytes through diffusion, migration, and convection. In addition, it also includes ready-made formulations for porous and non-porous electrodes, including charge transfer reactions and current conduction in the electronic conductors.The Chemical Species Transport interfaces ( ) describe the transport of ions in the pore electrolyte and in the electrolyte that separates the anode and cathode. Other reactions can be added other than pure electrochemical reactions to, for example, describe the degradation of materials. In combination with the Secondary Current Distribution interface ( ), the Transport of Concentrated Species interface ( ), and the Species Transport in Porous Media interface ( ) can be used to model the transport of charged species and the electrochemical reactions in most battery systems.The Fluid Flow interfaces ( ) describe the fluid flow in the porous electrodes and in free media if this is relevant for a specific type of battery, for example, certain types of lead-acid batteries.The Heat Transfer in Porous Media interface ( ) describes heat transfer in the cells. This includes the effects of Joule heating in the electrode material and in the electrolyte, heating due to activation losses in the electrochemical reactions, and of the net change of entropy. The heat of reactions from other reactions than the electrochemical reactions can also be described by these interfaces.


Fuel Cell Modeling

This module includes functionality to model fuel cell unit cells that consist of:• Current collectors and current feeders• Gas channels usually formed by grooves in the current collectors and feeders• Porous gas diffusion electrodes (GDEs)• An electrolyte that separates the anode and cathode

Figure 6 shows a schematic drawing of a fuel cell unit cell and the structure of one of the GDEs. It represents a fuel cell unit cell and a magnified section of the cathode GDE and its contact with the electrolyte.

Figure 6: Fuel cell unit cell and a magnified section of the cathode GDE and its contact with the electrolyte.

Oxygen and hydrogen are supplied to the cell through the gas channels in the current collector and current feeder, respectively. The current collector and the current feeder are usually made of electronic conducting materials and are equipped with grooves that form the gas channels. These grooves are open channels with the open side facing the surface of the GDEs.The current collectors and feeders also conduct the current to the wires connected to the load. They can also supply cooling required during operation and heating required during start-up of the cell.

Cathode

AnodeElectrolyte

Electrolyte

LoadPore electrolyte GDE

Gas

Current collector

Current feeder

Direction of the current

Gas channels in current collectors and current feeders

8 | Fuel Cell Modeling

The GDE magnified in Figure 6 is an oxygen-reducing cathode in a fuel cell with acidic electrolyte, for example the PEMFC. In the PEMFC, the active GDE is confined to a thin active layer supported by a pure gas diffusion layer (GDL).

Figure 7: Transport of oxygen, water, protons, and electrons to and from the reaction site in an oxygen reducing GDE.

Figure 7 shows the principle of the oxygen reduction process in the electrode. From the free electrolyte, current enters the electrolyte contained in the GDE (also called pore electrolyte) as protons and is transferred to electron current in the charge transfer reaction at the reaction sites. These reaction sites are situated at the interface between the electrocatalyst in the electrode material and the pore electrolyte.Figure 7 also describes the schematic path of the current in the electrode. The current in the pore electrolyte decreases as a function of the distance from the free electrolyte as it is transferred to electron current in the electrode. The direction of the current in the electrode is opposite to that of the electrons, by definition.The supply of oxygen takes place in conjunction with the charge transfer reaction and can be subject to mass transport resistance both in the gas phase and in the thin layer of pore electrolyte that covers the reaction site.

H+

2H++1/2O2+2e- = H2O

e-

O2

H2O

Current by ion transport

Current by electron transport

Charge transfer reaction

Oxygen transport

Water transport

Charge transfer reaction:

Electrolyte

Electrode

Gas

Fuel Cell Modeling | 9

The water balance in the electrode is maintained through evaporation and transport through the gas pores.The pore electrolyte has to form an unbroken path from the free electrolyte, between the anode and the cathode, to the reaction site. Also, the electrode material and the gas pores must each form an unbroken path to the reaction site or to the pore electrolyte covering the reaction site.The processes described above including fluid flow, chemical species transport, heat transfer, current conduction in the collectors, feeders, electrodes and electrolytes, and the electrochemical reactions are all coupled together, and determine the characteristics of a unit cell.Several important design parameters can be investigated by modeling these processes. Among these parameters are:• Porosity, active surface area, and pore electrolyte content of the GDEs• Geometry of the GDEs (active layer and GDL for the PEMFC) and

electrolyte in relation to the gas channels, the current collectors, and feeders• Geometry of the grooves that form the gas channels and dimensions of the

current collectors and feeders

The Fuel Cell Modeling Physics InterfacesThe fluid flow in the gas channels and in the GDEs are addressed by the Fluid Flow interfaces—the Laminar Flow ( ), Free and Porous Media Flow ( ), and Darcy’s Law ( ) interfaces.The transport of gaseous species and the mass transport resistance in the pore electrolyte are handled by the Chemical Species Transport interfaces ( ), which all have nodes that couple the transport in the gas phase to the electrochemical reactions. The Chemical Species Transport interfaces are also coupled to the Fluid Flow interfaces ( ) through the gas density, which is influenced by the gas composition.The Heat Transfer interfaces ( ) handle the effects of Joule heating in the electrolyte, in the pore electrolyte, and in the electrodes. They include the contribution to the thermal balance from the electrochemical reactions due to the activation overpotential and the net change of entropy.The current transport by ions in the free electrolyte and in the pore electrolyte, the current transport by electrons, and the charge transfer reactions are all treated in the Primary Current Distribution ( ), Secondary Current Distribution ( ), and the Tertiary Current Distribution, Nernst-Planck ( ) interfaces. The Primary Current Distribution interface neglects the variations in composition in the electrolyte and the activation losses for the charge transfer reactions. It may only be used for electrolytes with fixed charges or well mixed electrolytes, and in

10 | Fuel Cell Modeling

the cases where the activation losses are substantially smaller than the activation losses. In the Secondary Current Distribution interface, the variations in composition in the electrolyte are also neglected, while the activation losses for the charge transfer reactions are taken into account. In the Tertiary Current Distribution, Nernst-Planck interface, also the contribution of diffusion to the transport of ions, and thus the contribution to the current in the electrolyte, is taken into account.The Electrode, Shell interface ( ) models electric current conduction in the tangential direction on a boundary. The interface is suitable to use for thin electrodes where the potential variation in the normal direction to the electrode is negligible. This assumption allows for the thin electrode domain to be replaced by a partial differential equation formulation on the boundary. In this way the problem size can be reduced, and potential problems with mesh anisotropy in the thin layer can be avoided.

Fuel Cell Modeling | 11

The Physics Interface List by Space Dimension and Study Type

The table lists the physics interfaces available in the Batteries & Fuel Cells Module in addition to those included with the COMSOL basic license.

INTERFACE ICON TAG SPACE DIMENSION

PRESET STUDY TYPE

Chemical Species Transport

Surface Reactions chsr all dimensions stationary; time dependent

Transport of Concentrated Species

chcs all dimensions stationary; time dependent

Species Transport in Porous Media

chpm all dimensions stationary; time dependent

Reacting Flow, Concentrated Species

rfcs 3D, 2D, 2D axisymmetric

stationary; time dependent

Reacting Flow, Diluted Species rfds 3D, 2D, 2D axisymmetric


Electrochemistry

Primary Current Distribution siec all dimensions stationary; time dependent; AC impedance stationary; AC impedance time dependent

Secondary Current Distribution

siec all dimensions stationary; time dependent; AC impedance stationary; AC impedance time dependent

Tertiary Current Distribution, Nernst-Planck

tcdee all dimensions stationary; time dependent; AC impedance stationary; AC impedance time dependent

Electroanalysis elan all dimensions stationary; time dependent; AC impedance stationary; AC impedance time dependent

Electrode, Shell els 3D, 2D, 2D axisymmetric


12 | The Physics Interface List by Space Dimension and Study Type

Battery Interfaces

Lithium-Ion Battery liion all dimensions stationary; time dependent; AC impedance stationary; AC impedance time dependent

Battery with Binary Electrolyte batbe all dimensions stationary; time dependent; AC impedance stationary; AC impedance time dependent

Lead Acid Battery leadbat all dimensions stationary; time dependent; AC impedance stationary; AC impedance time dependent

Fluid Flow

Porous Media and Subsurface Flow

Brinkman Equations br 3D, 2D, 2D axisymmetric


Darcy’s Law dl all dimensions stationary; time dependent

Free and Porous Media Flow fp 3D, 2D, 2D axisymmetric


Heat Transfer

Heat Transfer in Porous Media ht all dimensions stationary; time dependent

INTERFACE ICON TAG SPACE DIMENSION

PRESET STUDY TYPE

The Physics Interface List by Space Dimension and Study Type | 13

The Model Libraries Window

To open a Batteries & Fuel Cells Module model library model, click Blank Model in the New screen. Then on the Home or Main toolbar click Model Libraries . In the Model Libraries window that opens, expand the Batteries & Fuel Cells Module folder and browse or search the contents. Click Open Model to open the model in COMSOL Multiphysics or click Open PDF Document to read background about the model including the step-by-step instructions to build it. The MPH-files in the COMSOL model library can have two formats—Full MPH-files or Compact MPH-files.• Full MPH-files, including all meshes and solutions. In the Model Libraries

window these models appear with the icon. If the MPH-file’s size exceeds 25MB, a tip with the text “Large file” and the file size appears when you position the cursor at the model’s node in the Model Libraries tree.

• Compact MPH-files with all settings for the model but without built meshes and solution data to save space on the DVD (a few MPH-files have no solutions for other reasons). You can open these models to study the settings and to mesh and re-solve the models. It is also possible to download the full versions—with meshes and solutions—of most of these models when you update your model library. These models appear in the Model Libraries window with the icon. If you position the cursor at a compact model in the Model Libraries window, a No solutions stored message appears. If a full MPH-file is available for download, the corresponding node’s context menu includes a Download Full Model item ( ).

To check all available Model Libraries updates, select Update COMSOL Model Library ( ) from the File>Help menu (Windows users) or from the Help menu (Mac and Linux users).

Two models from the model library are used as tutorials in this guide. See “Tutorial Model of a Lithium-Ion Battery” starting on page 15 and the “Tutorial Model of a Fuel Cell Cathode” starting on page 30. The next two sections introduce you to the battery and fuel cell modeling basics.

14 | The Model Libraries Window

Tutorial Model of a Lithium-Ion Battery

The following is a two-dimensional model of a lithium-ion battery. The cell geometry could be a small part of an experimental cell but here it is only meant to demonstrate a 2D model setup. A realistic 2D geometry is shown in the model Edge Effects in a Spirally Wound Li-Ion Battery available in the Batteries & Fuel Cells Module model library (see “The Model Libraries Window” on page 14 to access this model).

Model DefinitionThe cell geometry is shown in the figure below. Due to symmetry along the height of the battery, the 3D geometry can be modeled using a 2D cross section. The figure shows the position of the positive and negative electrode, and the position of the collector and feeder contacts during discharge.

During discharge, the current collector is in contact with the outer face of the battery (red face, middle) while the current feeder runs inside of the folded structure (blue faces, middle). The modeled 2D cross section is shown in light blue (right).

Positive electrode

Electrolyte Negative electrode

Cross-sectionCurrent collector (discharge)

Current feeder (discharge)

Tutorial Model of a Lithium-Ion Battery | 15

The resulting 2D cell geometry is shown in the next figure. During discharge, the positive electrode acts as the cathode and the contact of the metallic tab acts as a current collector. The negative electrode then acts as the anode and the contact of the metallic tab acts as the current feeder.

The model defines and solves the current and material balances in the lithium-ion battery. The intercalation of lithium inside the particles in the positive and negative electrode is solved using a fourth independent variable for the particle radius (x, y, and t are the other three). The reaction kinetics and the intercalation are coupled to the material and current balances at the surface of the particles. The model equations are found in the Batteries & Fuel Cells Module User’s Guide. The model was originally formulated for 1D simulations by John Newman and his co-workers at the University of California at Berkeley.

Results and DiscussionThe purpose of the 2D simulation is to reveal the distribution of the depth of discharge, as a function of discharge time, in the different parts of the electrodes. This distribution is caused by the position of the current collector and feeder and the thickness of the electrodes and electrolyte in combination with the electrode kinetics and transport properties.The figure below shows the concentration of lithium at the surface of the particles in the electrodes after 100 s of discharge at 200 A/m2, with respect to the current collector at the positive electrode.The high concentration at the positive electrodes is proportional to the local depth of discharge of these parts of the electrode. Conversely, the low concentration of lithium at the negative electrodes is proportional to the local depth of discharge

Current collector

Current feeder

Positive electrode

Negative electrode

1.3 mm

16 | Tutorial Model of a Lithium-Ion Battery

for this electrode. The figure shows that the back side of the electrodes, with respect to the position of the current collector, are less utilized during discharge. As the discharge process continues, also these parts will be discharged. However, for repeated cycling of the cell (charge and discharge), the different parts of the electrodes will age in a nonuniform way if the electrodes only are discharged to a moderate degree during cycling

Despite the simplicity of this model, it shows on a problem that may arise in realistic battery geometry if the shape and configuration of the electrodes, current collectors, and current feeders are not investigated thoroughly using modeling and simulations.The following instructions show how to formulate, solve, and reproduce this model.

Model Wizard

Note: These instructions are for the user interface on Windows but apply, with minor differences, also to Linux and Mac.


1 To start the software, double-click the COMSOL icon on the desktop. When the software opens, you can choose to use the Model Wizard to create a new COMSOL model or Blank Model to create one manually. For this tutorial, click the Model Wizard button.If COMSOL is already open, you can start the Model Wizard by selecting New from the File menu and then click Model Wizard .

The Model Wizard guides you through the first steps of setting up a model. The next window lets you select the dimension of the modeling space.

2 In the Space Dimension window click the 2D button .3 On the Add Physics window, under Electrochemistry>Battery Interfaces ,

double-click Lithium-Ion Battery (liion) to add it to the Selected physics list.

4 Click Study .5 On the Studies window under Preset Studies, click Time Dependent .6 Click Done .

Global Definit ions

Load the parameter values to be used in the model from a parameter file.

Parameters1 On the Home toolbar click Parameters .Note: On Linux and Mac, the Home toolbar refers to the specific set of controls near the top of the Desktop.

2 In the Model Builder under Global Definitions, click Parameters.3 In the Parameters settings window locate the Parameters section.4 In the settings window under Parameters, click Load from File . 5 Browse to the file li_battery_tutorial_2d_parameters.txt in the model

library folder on your computer, models\Batteries_and_Fuel_Cells_Module\Tutorial Models. Double-click to add or click Open.

Note: The location of the file used in this exercise varies based on the installation of COMSOL Multiphysics. For example, if the installation is on your hard drive, the file path might be similar to C:\Program Files\COMSOL44\models\.


The parameters are added to the table as in this figure.

Geometry 1

Also import the geometry from a file.1 On the Home toolbar click Import .2 In the settings window under Import, click Browse.3 Browse to the file li_battery_tutorial_2d.mphbin in the model library

folder on your computer, Batteries_and_Fuel_Cells_Module\Tutorial Models. Double-click to add or click Open.

Note: The location of the file used in this exercise varies based on the installation of COMSOL Multiphysics. For example, if the installation is on your hard drive, the file path might be similar to C:\Program Files\COMSOL44\models\.


4 Click Import and then the Build All Objects button .

Materials

Use the Batteries and Fuel Cells material library to set up the material properties for the electrolyte and electrode materials. By adding the electrolyte material to the model first, this material becomes the default material for all domains.

1:2 EC:DMC / LiPF6 (Li-ion Battery)1 On the Home toolbar click Add Material .


2 Go to the Add Material window. In the tree, select Batteries and Fuel Cells>1:2 EC:DMC / LiPF6 (Li-ion Battery). Right-click and select Add to Component.

LixC6 Electrode (Negative, Li-ion Battery)1 Go to the Add Material window. 2 In the tree, select Batteries and Fuel Cells>LixC6 Electrode (Negative, Li-ion

Battery). In the Add Material window, click Add to Component.

LixMn2O4 Electrode (Positive, Li-ion Battery)3 Go to the Add Material window.4 In the tree, select Batteries and Fuel Cells>LixMn2O4 Electrode (Positive,

Li-ion Battery). In the Add Material window, click Add to Component.

The node sequence in the Model Builder under the Materials node should match this figure.


Lithium-Ion Battery Interface

An Electrolyte node has already been added to the model by default. The electrolyte parameters, that depend on the electrolyte concentration, are set by default to be taken from the Materials node. Modify the Model Inputs section so that the material parameters use the electrolyte concentration as input.

Electrolyte 11 In the Model Builder under Component 1>Lithium-Ion Battery, click

Electrolyte 1 . The ‘D’ in the upper left corner of a node means it is a default node. Locate the Model Inputs section and click the Make All Model Inputs Editable button in the upper right corner.

2 From the Concentration c list, choose Electrolyte salt concentration (liion/ice1).

Porous Electrode 11 On the Physics toolbar click Domains . Choose Porous Electrode to add it

to the model.

2 Locate the Domain Selection section of the Porous Electrode node and select Domain 3 only.

Note: There are many ways to select geometric entities. When you know the geometric entity to add, such as in these exercises, you can click the Paste Selection button and enter the information in the Selection field. For more information about selecting geometric entities in the Graphics window, see the COMSOL Multiphysics Reference Manual.

3 Go to the Porous Electrode settings window. Under Model Inputs, from the c list, select Electrolyte salt concentration (liion/liion).

4 In the same window, under Electrode Properties, select LixC6 Electrode (Negative, Li-ion Battery) from the Electrode material list.


5 Under Particle Intercalation from the Particle material list, select LixC6 Electrode (Negative, Li-ion Battery).

6 Enter cs0_neg as the Initial species concentration (cs,init) and rp_neg in the Particle mean center-surface distance (rp) field.

7 Under Volume Fractions, in the Electrode volume fraction (s) field, enter epss_neg. In the Electrolyte volume fraction (l) field enter epsl_neg.

8 Under Effective Transport Parameter Correction, choose Bruggeman from all the lists—Electrolyte conductivity, Electrical conductivity and Diffusion.

Porous Electrode Reaction 11 In the Model Builder, expand the Porous

Electrode 1 node and click the Porous Electrode Reaction 1 node.


2 Under Model Inputs from the c list, select Insertion particle concentration, surface (liion/pce1).

3 Under Equilibrium Potential, select From material.

4 Under Materials select LixC6 Electrode (Negative, Li-ion Battery) from the list.

Now set up the positive porous electrode in a similar way.

Porous Electrode 21 On the Physics toolbar, click Domains. Choose Porous Electrode to add a

second node of this type to the model.2 Select Domain 1 only.3 In the Porous Electrode settings window, under Model Inputs, select

Electrolyte salt concentration (liion/liion) from the c list.4 Under Electrode Properties, select LixMn2O4 Electrode (Positive, Li-ion

Battery) from the Electrode material list.5 Under Particle Intercalation, select LixMn2O4 Electrode (Positive, Li-ion

Battery) from the Particle material list. Enter cs0_pos in the Initial species concentration (cs,init) field, and enter rp_pos in the Particle mean center-surface distance (rp) field.

6 Under Volume Fractions, enter epss_pos in the Electrode volume fraction (s) field. In the Electrolyte volume fraction (l) field, enter epsl_pos.

7 Under Effective Transport Parameter Correction, choose Bruggeman from all the lists—Electrolyte conductivity, Electrical conductivity, and Diffusion.


Porous Electrode Reaction 11 In the Model Builder, expand the Porous

Electrode 2 node and click Porous Electrode Reaction 1.

2 Under Model Inputs from the c list, select Insertion particle concentration, surface (liion/liion).

3 Under Equilibrium Potential, select From material.

4 Under Materials select LixMn2O4 Electrode (Positive, Li-ion Battery) from the list.

Electrolyte is the default domain node . This default node is now applied to Domain 2 only. No additional settings are needed for this node.

Initial Values 2Initial values are needed for the solver to converge. Start with the positive electrode.1 On the Physics toolbar click Domains and choose Initial Values.2 In the Model Builder under Component 1>Lithium-Ion Battery, click Initial

Values 2. Select Domain 1 only.3 In the Initial Values settings window

under Initial Values, enter -0.2 in the Electrolyte potential (phil) field and 4.1 the Electric potential (phis) field.

Initial Values 1 (the Default Node)The default Initial Values node is now applied to the negative electrode and the electrolyte domains only.


1 In the Model Builder click Initial Values 1 . 2 Go to the Initial Values settings window. Under

Initial Values enter -0.2 in the Electrolyte potential (phil) field.

Finish the model by setting up the boundary conditions. Ground the negative electrode current feeder, and apply a current density condition at the positive current collector.

Electric Ground 11 On the Physics toolbar click Boundaries and choose Electric Ground.

2 Select boundaries 7, 9, and 15 only (or click the Paste Selection button and enter 7, 9, 15 in the Paste Selection window).

Electrode Current Density 11 On the Physics toolbar click Boundaries and

choose Electrode Current Density. 2 Select Boundary 14 only.3 In the settings window under Electrode

Current Density, enter -200[A/m^2] in the Inward electrode current density (in,s) field.


Mesh 1

In this example the automatically generated mesh is used. No manual mesh settings are required.

Study 1

Set up a 100 second time-dependent solver to store the solution at 1 second intervals during the first 10 seconds, and 10 second intervals during the last 90 seconds. Then solve the problem.

Step 1: Time Dependent1 In the Model Builder expand the Study 1 node and click Step 1: Time

Dependent .2 In the settings window under

Study Settings, in the Times field enter (or copy and paste) range(0,1,10), range(20,10,100).

3 Click to select the Relative tolerance check box and enter 0.001.

4 On the Home toolbar click Compute .

Results

2D Plot Group 3The following steps create a plot of the solid lithium concentration at the surface of the electrode particles at 100 seconds.1 On the Home toolbar click Add Plot Group and choose 2D Plot Group .2 In the 2D Plot Group settings window under Data, choose 100 from the Time

list.


3 On the 2D Plot Group toolbar click Surface .4 In the Surface settings window click Replace Expression (it is in the

upper-right corner of the Expression section). Choose Insertion particle concentration, surface (liion.cs_surface). On the 2D Plot Group toolbar click Plot .

5 On the Graphics toolbar click the Zoom Extents button .

1D Plot Group 4The following steps create a plot of the cell potential during the simulation.1 On the Home toolbar click Add Plot Group and choose 1D Plot Group .2 On the 1D Plot Group toolbar click Point Graph .3 In the Model Builder under Results>1D Plot Group 4, click Point Graph 1.4 Select Point 11 only.5 In the Point Graph settings window under y-Axis Data, enter phis in the

Expression field.


6 On the 1D Plot Group toolbar, click Plot .


Tutorial Model of a Fuel Cell Cathode

One of the most important—and most difficult—parameters to model in a fuel cell is the mass transport through gas diffusion and reactive layers. Gas concentrations are often quite large and are significantly affected by the reactions that take place. This makes Fickian diffusion an inappropriate assumption to model the mass transport.Figure 8 shows an example 3D geometry of a cathode from a fuel cell with perforated current collectors. It is often seen in self-breathing cathodes or in small experimental cells. Due to the perforation layout, a 3D model is needed in the study of the mass transport, current, and reaction distributions.

Figure 8: A fuel cell cathode with a perforated current collector.

This model investigates such a geometry and the mass transport that occurs through Maxwell-Stefan diffusion. It couples this mass transport to a generic, Tafel-like electrochemical kinetics in the reaction term at a cathode.

Model DefinitionFigure 9 shows details for a unit cell from Figure 8. The circular hole in the collector is where the gas enters the modeling domain, where the composition is known. The upper rectangular domain is the reaction-zone electrode. It is a porous structure that contains the feed-gas mixture, an electronically conducting material covered with an electrocatalyst, and an electrolyte. The lower domain

Gas inlet hole

Unit cell

Reactive laye

r

Electrolyte layer

30 | Tutorial Model of a Fuel Cell Cathode

corresponds to a free electrolyte ionically interconnecting the two electrodes of the fuel cell. No reaction takes place in this domain. Neither are there pores to allow gas to flow or material for electronic current—current is conducted ionically.The reaction zone is 75 mm thick, as is the electrolyte layer. The unit cell is 1.5-by-1.5 mm in surface, and the gas inlet hole has a radius of 1.0 mm.

Figure 9: The modeled fuel cell cathode unit cell. The marked zone is the surface of the cathode that is open to the feed gas inlet, while the rest of the top surface sits flush against the current collector. In the unit cell the top domain is the porous cathode, while the bottom domain is the free electrolyte.

The electronic and ionic current balances are modeled using a Secondary Current Distribution interface.The species (mass) transport is modeled by the Maxwell-Stefan equations for oxygen (Species 1) and water (Species 2) in the gas phase using a Transport of Concentrated Species interface. Mass transport is solved for in the electrode domain only. The velocity vector is solved for using a Darcy’s law interface.

Results and DiscussionFigure 10 shows the oxygen concentration at a total potential drop over the modeled domain of 190 mV. The figure shows that concentration variations are

Tutorial Model of a Fuel Cell Cathode | 31

small along the thickness of the cathode for this relatively small current density, while they are substantially larger along the electrode’s width.

Figure 10: Isosurfaces of the weight fraction of oxygen at a total potential drop over the modeled domain of 190 mV.

Figure 11 shows the gas velocity in the porous cathode. There is a significant velocity peak at the edge of the inlet orifice. This is caused by the contributions of the reactive layer underneath the current collector because in this region the convective flux dominates the mass transport. Thus it is important to model the velocity field properly. In this case, the combination of a circular orifice and square unit cell eliminates the possibility to approximate the geometry with a rotationally symmetric model.The electrochemical reaction rate, represented by the local current density, is related to both the local overvoltage and oxygen concentration. Figure 12 depicts the local overvoltage, which is rather even throughout the cathode. This is caused by the high electronic conductivity in the porous material. Another observation is that the maximum overvoltage is -180 mV. This means that there is a voltage loss of 10 mV in the electrolyte layer.


Figure 11: Velocity field for the gas phase in the cathode’s porous reactive layer.

Figure 12: Local overvoltage in the cathode reactive layer.


Although the local overvoltage distribution is rather even, the concentration of oxygen is not. This means that the reaction rate is nonuniform in the reactive layer. One way to study the distribution of the reaction rate is to plot the ionic current density at the bottom boundary of the free electrolyte. Figure 13 shows such a plot.

Figure 13: Current density perpendicular to the lower, free electrolyte boundary.

The current-density distribution shows that the variations are rather large. The reaction rate and the current production are higher beneath the orifice and decrease as the distance to the gas inlet increases. This means that the mass transport of reactant dictates the electrode’s efficiency for this design at these particular conditions.The following instructions show how to formulate, solve, and reproduce this model.

Model Wizard

Start by adding a Secondary Current Distribution interface and a Stationary study to set up and solve for a current distribution model of the cell. Later you will add more physics for modeling mass transport and convection.


1 If COMSOL is already open, you can start the Model Wizard by selecting New from the File menu and then click Model Wizard .

2 In the Space Dimension window click the 3D button .3 In the Select physics tree, select

Electrochemistry>Secondary Current Distribution (siec) .

4 Click the Add button.5 Click the Study button.

6 Select Preset Studies>Stationary . 7 Click Done .

Geometry 1

Now draw the model geometry. Use blocks to define the electrolyte and the porous electrode domains. Then use a workplane to draw the inlet hole at the top of the porous electrode. Facilitate geometry selection later (when setting up the physics) by enabling Create Selections and renaming the geometry objects.Begin by setting the default length unit to millimeters.1 In the Model Builder under Component 1, click Geometry 1 .2 In the Geometry settings window locate the Units section.3 From the Length unit list, choose mm.

Create the first Block1 On the Geometry toolbar click Block .2 In the Model Builder under Component 1

(comp1)>Geometry 1, click Block 1.


3 In the Block settings window, locate the Size section.- In the Width edit field, type 1.5.- In the Depth edit field, type 1.5.- In the Height edit field, type 0.075.

4 Locate the Selections of Resulting Entities section. Select the Create selections check box.

5 Right-click Component 1 >Geometry 1>Block 1 and choose Rename (or press F2).

6 Go to the Rename Block dialog box and type Electrolyte in the New name edit field.

7 Click OK.


8 Click the Build Selected button .

Duplicate the Block and change z Position1 Right-click Component 1 >Geometry 1>Block 1 and

choose Duplicate.

2 In the Block settings window, locate the Position section.

3 In the z edit field, type 0.075.4 Click the Electrolyte 1 node and press F2.

5 Go to the Rename Block dialog box and type Porous Electrode in the New name edit field.

6 Click OK.

Create a Work Plane for drawing the inlet hole

1 On the Geometry toolbar click Work Plane .

2 In the Model Builder under Component 1 >Geometry 1, click Work Plane 1.

3 In the Work Plane settings window locate the Plane Definition section.

4 In the z-coordinate field, type 0.15.5 Locate the Selections of Resulting Entities section. Select the Create selections

check box.


6 Right-click Component 1 >Geometry 1>Work Plane 1 and choose Rename (or click the node and press F2).

7 Go to the Rename Work Plane dialog box and type Inlet in the New name edit field.

8 Click OK.

Draw the inlet hole1 In the Model Builder under

Component 1>Geometry 1>Inlet, right-click Plane Geometry and choose Circle .

2 In the Circle settings window, locate the Size and Shape section.

- In the Radius edit field, type 1.- In the Sector angle edit field, type 90.

3 Locate the Position section.- In the xw edit field, type 1.5.- In the yw edit field, type 1.5.

4 Locate the Rotation Angle section. In the Rotation edit field, type 180.

5 Click the Build Selected button .

Form a Union to Finalize the Geometry1 In the Model Builder click the Form Union node and then click the Build

Selected button . 2 Click the Zoom Extents button on the Graphics toolbar.


Your finished geometry should now look like this:

Global Definit ions

Load the model parameters and variables from text files.

Parameters1 On the Home toolbar click

Parameters .Note: On Linux and Mac, the Home toolbar refers to the specific set of controls near the top of the Desktop.

2 In the Model Builder under Global Definitions, click Parameters.


3 In the Parameters settings window locate the Parameters section. Click Load from File .

4 In the model library folder on your computer, models\Batteries_and_Fuel_Cells_Module\Tutorial Models, double-click the file fuel_cell_cathode_parameters.txt to import it to the Parameters table. Note that the location of the files used in this exercise may vary depending on the installation. For example, if the installation is on your hard drive, the file path might be similar to C:\Program Files\COMSOL44\models\.

Variables1 On the Home toolbar click Variables and choose

Local Variables .2 In the Model Builder under

Component 1>Definitions, click Variables 1.3 In the Variables settings window locate the

Variables section. Click Load from File .

4 In the model library folder on your computer, models\Batteries_and_Fuel_Cells_Module\Tutorial Models, double-click the file fuel_cell_cathode_variables.txt.

Note: The v_in variable is marked in orange because the p (pressure) variable is not yet defined. A physics containing the pressure variable is added later.

SelectionsNow manually add selections for the bottom electrolyte and top current collector boundaries.


1 On the Definitions toolbar in the Selections section, click Explicit .

2 In the Model Builder under Component 1>Definitions, click Explicit 1.3 In the Explicit settings window, locate the Input Entities section.4 From the Geometric entity level list, choose Boundary.5 Select Boundary 3 only.6 Click Explicit 1 and press F2. 7 Go to the Rename Explicit dialog box and type Electrolyte Boundary in the

New name edit field.8 Click OK.

Add a selection for the Current Collector in a similar way:1 On the Definitions toolbar in the Selections section click Explicit .2 In the Model Builder, under Component 1 >Definitions click Explicit 2.3 In the Explicit settings window, locate the Input Entities section.4 From the Geometric entity level list, choose Boundary. Select Boundary 7 only.5 Click Explicit 2 and press F2.6 Go to the Rename Explicit dialog box and type Current Collector in the New name edit field.

7 Click OK.


Now start defining the physics for the current distribution model. Add a porous electrode and specify the electrode reaction parameters, then add potential boundary nodes for both the electrolyte and the electrode phase. Note that an Electrolyte node already has been added automatically by default.

Porous Electrode 11 On the Physics toolbar click Domains and choose Porous Electrode .


2 In the Model Builder under Component 1>Secondary Current Distribution, click Porous Electrode 1. Keep the default From material setting for the conductivities in this node (the Materials node is specified later.)

3 In the Porous Electrode settings window, locate the Domain Selection section and select Porous Electrode.

4 From the Effective conductivity correction list, choose No correction in both the Electrolyte Current Conduction and the Electrode Current Conduction sections.

Porous Electrode Reaction 1

1 In the Model Builder expand the Porous Electrode 1 node, then click Porous Electrode Reaction 1 .

2 In the Porous Electrode Reaction settings window, locate the Model Inputs section. In the T edit field, type T.

3 Locate the Electrode Kinetics section. From the Kinetics expression type list, choose Butler-Volmer.

4 In the i0 edit field, type i0.5 Locate the Active Specific Surface Area section.

In the av edit field, type S.

Electrolyte Potential 11 On the Physics toolbar click Boundaries

and choose Electrolyte Potential.2 In the Model Builder under Component 1>Secondary Current Distribution,

click Electrolyte Potential 1.


3 Locate the Boundary Selection section. From the Selection list choose Electrolyte Boundary.

Electric Potential 11 On the Physics toolbar click Boundaries and choose Electric Potential.2 In the Model Builder under

Component 1>Secondary Current Distribution, click Electric Potential 1.

3 In the Electric Potential settings window, locate the Boundary Selection section. Choose Current Collector from the Selection list.

4 Locate the Electric Potential section. In the s,bnd edit field, type E_pol.

Initial Values 1Also provide initial values for the potentials. This reduces the computational time.1 In the Model Builder under Component 1>Secondary Current Distribution

(siec), click Initial Values 1 .2 In the Initial Values settings window, locate the Initial Values section. In the

phis edit field, type E_pol.

Materials

The Materials node is marked in red. This indicates that there are material parameters missing in the model. Add two different material nodes for the electrolyte and the porous electrode, and specify the conductivity values.

Material 1 (Electrolyte)1 On the Home toolbar click New Material .2 In the Material settings window locate the Geometric Entity Selection section.3 From the Selection list choose Electrolyte.4 Locate the Material Contents section. In the table enter the following settings:

PROPERTY NAME VALUE

Electrolyte conductivity sigmal 5[S/m]


Material 2 (Porous Electrode)1 On the Home toolbar click New Material .2 In the Material settings window locate the

Geometric Entity Selection section.3 From the Selection list choose Porous Electrode.4 Locate the Material Contents section. In the table enter the following settings:

Note: You can provide the conductivities in the table either directly by typing in the values or by using parameters defined in the Parameters node.

Mesh 1

Use the default mesh sequence that is induced by the physics but change to a finer size.1 In the Model Builder under Component 1,

click Mesh 1 .2 In the Mesh settings window locate the Mesh

Settings section.3 From the Element size list choose Fine.4 Click the Build All button .

PROPERTY NAME VALUE

Electrolyte conductivity sigmal 1[S/m]

Electrical conductivity sigma sigma_s


The finished mesh should now look like this:

Study 1

The current distribution model is now ready for solving.1 On the Home toolbar click Compute .

Results

Electrolyte potential plots are created by default. Now create a plot of the local overpotential by first adding a plot group followed by a volume plot.

3D Plot Group 31 On the Home toolbar click Add Plot Group and choose 3D Plot Group .


2 On the 3D Plot Group 3 toolbar click Volume .

3 In the Model Builder, under Results>3D Plot Group 3 click Volume 1.

4 In the Volume settings window click Replace Expression (it is in the upper-right corner of the Expression section). Choose Secondary Current Distribution>Electrode kinteics>Overpotential (siec.eta_per1).

5 Click the Plot button .

The overpotential plot should now look like this:

1 In the Model Builder right-click 3D Plot Group 3 and choose Rename.2 Go to the Rename 3D Plot Group dialog box and type Local Overpotential

in the New name edit field.3 Click OK.


Data SetsCreate a data set with a selection on the bottom electrolyte boundary only. Then use this data set to plot a surface plot of the normal electrolyte current density.1 On the Results toolbar click More Data Sets and choose Solution .

2 In the Model Builder, under Results>Data Sets, right-click Solution 2 and choose Add Selection.

3 In the Selection settings window locate the Geometric Entity Selection section. From the Geometric entity level list, choose Boundary.

4 From the Selection list, choose Electrolyte Boundary.

5 Select the Propagate to lower dimensions check box.

3D Plot Group 41 On the Home toolbar click Add Plot Group and choose 3D Plot Group .2 In the Model Builder under Results, click 3D Plot

Group 4.3 In the 3D Plot Group settings window locate the

Data section. From the Data set list, choose Solution 2.

4 On the 3D Plot Group 4 toolbar click Surface .5 In the Surface settings window click Replace

Expression and choose Normal electrolyte current density (siec.nIl).

You can use operators to modify any postprocessing variable. In this case, use the abs() function to plot the absolute value of the current density. 6 Locate the Expression section. In the Expression edit field, type abs(siec.nIl).

7 On the 3D Plot Group toolbar, click Plot .


The current density plot should now look like this:

1 In the Model Builder right-click 3D Plot Group 4 and choose Rename.2 Go to the Rename 3D Plot Group dialog box and type Electrolyte Current Density in the New name edit field.

3 Click OK.

Change Element Order and Recompute

Now change to second order elements in the finite element discretization. This will increase the accuracy of the solution and render a smoother plot. (Alternatively you could increase the resolution of the mesh).1 In the Model Builder window’s

toolbar, click the Show button and select Discretization in the menu.


2 In the Secondary Current Distribution settings window, click to expand the Discretization section.

3 From the Electrolyte potential list, choose Quadratic.4 From the Electric potential list, choose Quadratic.5 On the Home toolbar click Compute .

The current density plot should now look like this:

Add Mass Transport and Fluid Flow Physics to the Model

Now, add more physics to the model by adding a Transport of Concentrated Species interface for gas phase mass transport and a Darcy's law interface for the convective flow.1 On the Home toolbar click Add Physics .2 Go to the Add Physics window.3 In the Add physics tree under Chemical Species Transport click Transport of

Concentrated Species (chcs) .


4 Click to expand the Dependent variables section. In the Number of species edit field, type 3. In the Mass fractions table, enter the following settings:

5 In the Add physics window click Add to Component .6 Go to the Add Physics window again. In the Add physics tree under Fluid

Flow>Porous Media and Subsurface Flow click Darcy's Law (dl) .7 In the Add physics window, click Add to Component .

Transport of Concentrated Species

Now define the settings for the mass transport of nitrogen, oxygen and water.1 In the Model Builder under Component 1, click

Transport of Concentrated Species . 2 Locate the Domain Selection section. From the

Selection list, choose Porous Electrode.3 Locate the Transport Mechanisms section.

From the Diffusion model list, choose Maxwell-Stefan.

Convection and Diffusion 11 In the Model Builder expand the Transport of

Concentrated Species node, then click Convection and Diffusion 1 .

w_n2

w_o2

w_h2o


2In the Convection and Diffusion settings window, locate the Density section.-In the Mwn2 edit field, type M_n2.-In the Mwo2 edit field, type M_o2.-In the Mwh2o edit field, type M_h2o.3Locate the Diffusion section. In the Dik table, enter the following settings:

4 Both the velocity and pressure are coupled to Darcy's law. Locate the Model Inputs section. From the u list, choose Darcy's velocity field (dl/dlm1).- In the T edit field, type T.- From the p list, choose Pressure

(dl/dlm1).- In the pref edit field, type p_atm.

Couple the reaction rate of oxygen to the electrochemical currents.Use a porous electrode coupling to create a mass sink in the domain corresponding to the oxygen leaving the gas phase due to the electrochemical reactions.1 On the Physics toolbar click Domains and choose Porous Electrode

Coupling .2 In the Model Builder, under Component 1>Transport of Concentrated Species,

click Porous Electrode Coupling 1.3 In the Porous Electrode Coupling settings window locate the Domain Selection

section. From the Selection list, choose Porous Electrode.

1 D_o2n2_eff D_n2h2o_eff

D_o2n2_eff 1 D_o2h2o_eff

D_n2h2o_eff D_o2h2o_eff 1


4 In the Model Builder under Component 1>Transport of Concentrated Species>Porous Electrode Coupling 1, click Reaction Coefficients 1.

5 In the Reaction Coefficients settings window, locate the Model Inputs section. From the iv list, choose Local current source (siec/pce1/per1).

6 Locate the Stoichiometric Coefficients section. - In the nm edit field, type 4.- In the w_o2 edit field, type -1.

Define the Initial Values Node1 Click the Initial Values 1 node . 2 In the settings window under Initial Values, enter Mass fraction values w_o2_ref in the w0,o2 field and w_h2o_ref in the w0,h2o field.

Define the Inflow Node1 On the Physics toolbar click Boundaries and choose Inflow. A node is added

to the model.2 In the Model Builder under Component 1>Transport of Concentrated Species,

click Inflow 1.3 In the Inflow settings window locate the Boundary Selection section. From the

Selection list choose Inlet.4 Locate the Inflow section.

- In the w0,wo2 edit field, type w_o2_ref.- In the w0,wh2o edit field, type w_h2o_ref.

Darcy's Law

Now do the settings for Darcy's law. Also here the electrochemical currents will result in a mass sink due to the oxygen molecules leaving the domain.


1 In the Model Builder under Component 1, click Darcy's Law .

2 In the Darcy's Law settings window locate the Domain Selection section. From the Selection list, choose Porous Electrode.

Define the Fluid and Matrix Properties1 In the Model Builder expand the Darcy's

Law (dl) node then click Fluid and Matrix Properties 1. 2 Locate the Fluid Properties section.

- From the list, choose User defined. - In the associated edit field, type chcs.rho. (The density is now taken from

the Transport of Concentrated Species interface.)- From the list, choose User defined. In the associated edit field, type mu.

3 Locate the Matrix Properties section. - From the p list, choose User defined. In the associated edit field, type e_por.- From the list, choose User defined. In the associated edit field, type perm.

Porous Electrode Coupling 11 On the Physics toolbar click Domains and choose Porous Electrode

Coupling.2 In the Model Builder under Component 1>Darcy's Law, click Porous Electrode

Coupling 1. Locate the Domain Selection section and from the Selection list, choose Porous Electrode.

3 Locate the Species section. Click Add twice.4 In the Species table, enter the following settings:

SPECIES MOLAR MASS (KG/MOL)

1 M_n2

2 M_o2

3 M_h2o


Reaction Coefficients 11 In the Model Builder expand the Porous

Electrode Coupling 1 node then click Reaction Coefficients 1.

2 Under Model Inputs, from the Coupled reaction iv list, choose Local current source (siec/per1).

3 Under Stoichiometric Coefficients enter 4 in the nm field and -1 in the 2 field.

Inlet 11 On the Physics toolbar click Boundaries and

choose Inlet.2 In the Model Builder under Component 1>Darcy’s Law, click Inlet 1.3 In the Inlet settings window locate the Boundary Selection section. From the

Selection list, choose Inlet.Set an inflow condition using the v_in variable that you loaded from the text file previously. The v_in variable has been defined assuming an atmospheric pressure 1 mm above the inlet hole. In this way very steep pressure gradients at the hole edge are avoided.

4 Locate the Inlet section. In the U0 edit field, type v_in.



Make the Electrode Reaction Concentration DependentFinalize the physics settings by modifying the porous electrode reaction current density to depend on the oxygen concentration.1 In the Model Builder under Component

1>Secondary Current Distribution>Porous Electrode 1, click Porous Electrode Reaction 1.

2 In the Porous Electrode Reaction settings window, locate the Electrode Kinetics section.

3 From the Kinetics expression type list, choose Concentration dependent kinetics.

4 In the CO edit field, type chcs.c_w_o2/c_o2_ref. (chcs.c_w_o2 is the molar concentration variable, defined by the Transport of Concentrated Species interface.)

Add a New Study and Recompute

Add a second study step that solves for all physics. Modify the first study step so that it only solves for the Secondary Current Distribution interface. (This is actually not needed for this fairly simple model, but it is usually good practice to solve for the potentials in a first step, and then solve for the full problem.)

Step 1: Stationary1 On the Study toolbar click Study Steps and choose Stationary .


2 In the Model Builder under Study 1, click Step 1: Stationary.

3 In the Stationary settings window, locate the Physics and Variables Selection section.

4 In the table, enter the following settings:

5 On the Home toolbar click Compute .

PHYSICS SOLVE FOR

Transport of Concentrated Species ×

Darcy's Law ×


Results for the Concentration Dependent Problem

The overpotential plot should now look like this:


Your current density plot should now look like this:

Isosurface 1Create a plot of the mass fraction of oxygen in the gas phase as follows:1 On the Home toolbar click Add Plot Group and choose 3D Plot

Group .2 On the 3D Plot Group 5 toolbar click Isosurface .3 In the Isosurface settings window, click Replace Expression in the

upper-right corner of the Expression section and choose Mass fraction (w_o2). Locate the Levels section. In the Total levels edit field, type 10.


4 On the 3D Plot Group toolbar click Plot . The oxygen mass fraction plot should now look like this:

5 In the Model Builder right-click 3D Plot Group 5 and choose Rename.6 Go to the Rename 3D Plot Group dialog box and type Oxygen Mass Fraction in the New name edit field.

7 Click OK.

Slice 1Create a slice plot of the gas velocity magnitude as follows:1 On the Home toolbar, click Add Plot Group . Choose 3D Plot Group .2 On the 3D Plot Group 6 toolbar click Slice .3 In the Slice settings window click Replace Expression and choose Darcy’s

velocity magnitude (dl.U). 4 Right-click Results>3D Plot Group 6>Slice 1 and choose Duplicate.5 In the Slice settings window locate the Plane Data section. From the Plane list,

choose zx-planes.6 Click to expand the Inherit style section. Locate the Inherit Style section. From

the Plot list, choose Slice 1.7 On the 3D Plot Group toolbar click Plot .


The velocity plot should look like this:

1 In the Model Builder right-click 3D Plot Group 6 and choose Rename.2 Go to the Rename 3D Plot Group dialog box and type Velocity Magnitude

in the New name edit field.3 Click OK.


Solving Electrochemical Models

Due to the highly nonlinear nature of electrode kinetics, some electrochemical models can be difficult to solve. This section includes some general tips and tricks to facilitate model solving, troubleshooting, and improving solution accuracy.

General Current Distribution Problems

Start with the following suggestions if you encounter difficulty solving a problem.• Make sure that the potential levels are boot-strapped somewhere in the

model, preferably by grounding one electrode.• Review the Initial Values, especially the potentials. Suitable initial potential

values can usually be derived making a “potential walk” through the geometry, starting at the grounded boundary and assuming zero overpotentials for the main electrode reactions.

• Switch to a Linearized Butler-Volmer kinetics while troubleshooting. This may be useful to help achieve a solution for a model that does not solve with nonlinear kinetics, thereby indicating suitable initial values for the nonlinear problem.

• If you model contains porous electrodes, try refining the mesh resolution in these domains, especially towards the electrolyte boundaries.

Electrochemistry Coupled to Mass Transport

If the model involves electrochemistry coupled to mass transport, here are a few things to try to help improve model convergence.• If steep concentration gradients are expected close to electrode surfaces, use

boundary layer meshing at these boundaries.• When setting up user defined kinetics expressions, avoid evaluating negative

concentrations by using expressions such as max(c, eps^2), where eps is the machine epsilon, that is, a very small number.

Solving Electrochemical Models | 61

• Try to solve for low currents or low overpotentials first, then increase the cell load (for stationary problems this may be done using a parametric sweep).

• If a problem involving mass transport is hard to solve for high currents, but solves for low currents, it may due to mass transport limitations. In this case, review the transport parameter values and check that the current magnitudes are reasonable. If the current densities are unreasonably high, review the electrode reaction settings.

Setting Up a Study Sequence for Multiphysics Problems

For multiphysics problems, try to adjust the study sequence.• Solve certain physics in a sequence. This can in many cases reduce

computational time and improve convergence. Analyzing the results when solving a physics separately may also help when troubleshooting nonconverging models.

• A good strategy is often to solve for the potentials only (that is, disable mass transport and flow interfaces), using a stationary study step, before solving the full model in the study sequence. In this way the stationary solution is used as initial values for the following steps.

• In many models the flow profile is only slightly (or not at all) affected by changes in current density. Therefore it can be a good strategy to separately solve for the flow early in the study sequence, and then solve for the other physics in the subsequent steps. (If the flow is not affected at all by the current distribution, solving for the flow may be disabled entirely in consecutive steps).

Time-dependent Problems with Load Steps

For time dependent problems, try the following to mitigate accuracy and convergence issues with regards to sudden current or potential load steps.• Use smoothed current/potential load functions in order to avoid discrete

load steps.

62 | Solving Electrochemical Models

• Add a double layer capacitance to the model, which may improve the numerical stability.

• Reduce the Maximum step taken by the solver if you want to prevent the solver from “missing” short square load steps, or change the Steps taken by solver setting from Free to Strict, or Intermediate, to control the time steps using the Times edit field. Using the Events interface can also be an option in certain cases if the load cycle itself varies dynamically.

Solver Settings

Try adjusting the solver settings.• In rare cases, try to increase the maximum number of iterations.If you know the order of magnitude of a dependent variables beforehand, setting the manual scales for these may improve convergence.

Solving Electrochemical Models | 63

64 | Solving Electrochemical Models

Introduction to the Batteries & Fuel Cells Module · 2014-04-10 · 2 | Introduction The Batteries...

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