Models.bfc.Sofc Unit Cell

Solved with COMSOL Multiphysics 4.2a

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Cu r r e n t Den s i t y D i s t r i b u t i o n i n a S o l i d Ox i d e Fu e l C e l l

Introduction

This model studies the current density distribution in a solid oxide fuel cell (SOFC). It includes the full coupling between the mass balances at the anode and cathode, the momentum balances in the gas channels, the gas flow in the porous electrodes, the balance of the ionic current carried by the oxide ion, and an electronic current balance.

Model Definition

An SOFC is constructed with two porous gas diffusion electrodes (GDEs) with an electrolyte sandwiched in the middle; see Figure 1.

Cathode outlet

Cathode inlet

Flow channel

Flow channel

Porous gas diffusion electrode

Porous gas diffusion electrode

Electrolyte

ANODE SIDE

CATHODE SIDE

Anode inlet

Anode outlet

Figure 1: Geometry of the unit cell, with anode at the bottom and cathode at the top.

The fuel feed in the cathode and anode is counterflow, with hydrogen-rich anode gas entering from the left.

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The electrochemical reactions in the cell are given below.

• Anode

• Cathode:

The model includes the following processes:

• Electronic charge balance (Ohm’s law)

• Ionic charge balance (Ohm’s law)

• Butler-Volmer charge transfer kinetics

• Flow distribution in gas channels (Navier-Stokes)

• Flow in the porous GDEs (Brinkman equations)

• Mass balances in gas phase in both gas channels and porous electrodes (Maxwell-Stefan Diffusion and Convection)

C H A R G E B A L A N C E S

The electronic and ionic charge balance in the anode and cathode current feeders, the electrolyte and GDEs are solved for using a Secondary Current Distribution interface.

Assume that Butler-Volmer charge transfer kinetics describe the charge transfer current density. At the anode, hydrogen is reduced to form water, and assuming the first electron transfer to be the rate determining step, the following charge transfer kinetics equation applies:

Here i0,a is the anode exchange current density (A/m2), ch2 is the molar concentration of hydrogen, ch2o is the molar concentration of water, ct the total concentration of species (mol/m3). ch2,ref and ch2,ref is the reference concentrations (mol/m3). Furthermore, F is Faraday’s constant (C/mol), R the gas constant (J/(mol·K)), T the temperature (K), and η the overvoltage (V).

For the cathode, use the relation

H2 O2- H2O 2e-+→+

12---O2 2e-

+ O2-→

ia ct, i0 a,ch2

ch2 ref,----------------- 0.5F

RT------------η⎝ ⎠

⎛ ⎞ ch2och2o, ref------------------ 1.5– F

RT----------------η⎝ ⎠

⎛ ⎞exp–exp⎝ ⎠⎛ ⎞=

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where i0,c is the cathode exchange current density (A/m2), and xo2 is the molar fraction of oxygen.

The overvoltage is defined as

where Δφeq is the equilibrium potential difference (V).

At the anode’s inlet boundary, the potential is fixed at a reference potential of zero. At the cathode’s inlet boundary, set the potential to the cell voltage, Vcell. The latter is given by

(1)

where Vpol is the polarization. In this model, φeq,aΔ 0 V= and φeq,cΔ 1 V= , and you simulate the fuel cell over the range 0,2 V Vcell 0,95 V≤ ≤ by using Vpol in the range 0.05 V through 0.8 V as the parameter for the parametric solver.

For the ionic charge balance equations, apply insulating boundary conditions at all external boundaries.

At the interior boundaries, continuity in current and potential applies by default.

M U L T I C O M P O N E N T TR A N S P O R T

SOFCs can be operated on many different fuels. This model describes a unit running on hydrogen and air. At the anode, a humidified hydrogen gas is supplied as fuel, meaning that the gas consists of two components: hydrogen and water vapor. In the cathode, humidified air is supplied, consisting of three components: oxygen, water vapor, and nitrogen.

The material transport is described by the Maxwell-Stefan’s diffusion and convection equations, solved for by using a Transport of Concentrated Species physics interface for each electrode flow compartment.

The boundary conditions at the walls of the gas channel and GDE are zero mass flux (insulating condition). At the inlet, the composition is specified, while the outlet condition is convective flux. This assumption means that the convective term dominates the transport perpendicular to this boundary.

ic ct, i0,c3.5FRT------------η⎝ ⎠

⎛ ⎞ xo2

ct

co2,ref--------------- 0.5– F

RT----------------η⎝ ⎠

⎛ ⎞exp–exp=

η φelectronic φionic– Δφeq–=

Vcell Δφeq,c Δφeq,a– Vpol–=



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Continuity in composition and flux apply for all mass balances at the interfaces between the GDEs and the channels.

G A S - F L O W E Q U A T I O N S

A Free and Porous Media Flow physics interface is used for solving for the velocity field and pressure. The weakly compressible Navier-Stokes equations govern the flow in the open channels and the Brinkman equations describe the flow velocity in the porous GDEs.

At the inlet and outlet, you set the pressure, specifying a slight overpressure at the inlet to drive the flow (2 Pa at the anode, and 6 Pa at the cathode).

Results and Discussion

Figure 2 shows the oxygen mole fraction in the cathode at a cell polarization of 0.5 V. The oxygen depletion is substantial, which has implications on the reaction distribution at the cathode.

Figure 2: Oxygen mole fraction in the gas channel and in the gas diffusion cathode while operating at a cell voltage of 0.5 V.



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The mole fraction of hydrogen in the anode also decreases along the channel. Figure 3 below shows the distribution of hydrogen. It shows that the depletion is not as pronounced as for the cathode.

Figure 3: Hydrogen distribution in the anode at 0.5 V cell voltage.

A consequence of the concentration distribution is that the current density will be nonuniform in the GDEs. Figure 4 depicts the current density distribution at the cathode side of the ionic conductor.



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Figure 4: The current density in the unit cell operating at 0.5 V. The cathode inlet is to the right.

As a consequence of oxygen depletion, the current density distribution is poor, with most of the current produced close to the cathode inlet. One way to improve the operating conditions is to increase the cathode flow rate, thus improving the oxygen mass transport.



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Figure 5 shows the voltage as a function of the total current (polarization curve).

Figure 5: Polarization curve.



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Figure 6 shows the power output as a function of the cell voltage. The model predicts a maximum power-output of 940 W/m2 for the unit cell.

Figure 6: Power output as a function of cell voltage.

References

1. J. Hartvigsen, S. Elangovan, and A. Khandkar, Science and Technology of Zirconia V, S.P.S. Badwal, M.J. Bannister, and R.H.J. Hannink (eds), p. 682, Technomic Publishing Company Inc., Lancaster, PA, 1993.

2. R. Herbin, J.M. Fiard, and J.R. Ferguson, First European Solid Oxide Fuel Cell Forum Proceedings, U. Bossel (ed.), p. 317, Lucerne, Switzerland, 1994.

Model Library path: Batteries_and_Fuel_Cells_Module/SOFC/sofc_unit_cell



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Modeling Instructions

M O D E L W I Z A R D

1 Go to the Model Wizard window.

2 Click Next.

3 In the Add physics tree, select Electrochemistry>Secondary Current Distribution (siec).

4 Click Add Selected.

5 In the Add physics tree, select Chemical Species Transport>Transport of Concentrated

Species (chcs).


7 Click Add Mass Fraction.

8 In the Mass fractions table, enter the following settings:

9 In the Add physics tree, select Chemical Species Transport>Transport of Concentrated

Species (chcs).


11 In the Mass fractions table, enter the following settings:

12 In the Add physics tree, select Fluid Flow>Porous Media and Subsurface Flow>Free and

Porous Media Flow (fp).


14 Click Next.

15 Find the Studies subsection. In the tree, select Preset Studies for Selected

Physics>Stationary.

16 Click Finish.

G L O B A L D E F I N I T I O N S

Define the parameters using the text file provided.

wO2

wH2Oc

wN2

wH2

wH2Oa



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Parameters1 In the Model Builder window, right-click Global Definitions and choose Parameters.

2 Go to the Settings window for Parameters.

3 Locate the Parameters section. Click Load from File.

4 Browse to the model’s Model Library folder and double-click the file sofc_unit_cell_parameters.txt.

Here, R_const and F_const are predefined constants representing the gas constant and the Faraday constant, respectively.

G E O M E T R Y 1

In this model you will first create the 2D cross section of the device. Then you will extrude it to create your 3D model geometry. Then you creat a mapped mesh on the faces at one end of the model. This mapped mesh will then be swept along the length. This technique is particularly useful if you have a stretched-out object where you want to spend a few mesh layers along the object length.

Work Plane 11 In the Model Builder window, right-click Model 1>Geometry 1 and choose Work Plane.

2 Go to the Settings window for Work Plane.

3 Locate the Work Plane section. From the Plane list, choose yz-plane.

Add rectangles as described below.

Rectangle 11 In the Model Builder window, right-click Geometry and choose Rectangle.

2 Go to the Settings window for Rectangle.

3 Locate the Size section. In the Width edit field, type W_channel+W_rib.

4 In the Height edit field, type H_gde.

5 Click the Build Selected button.

6 Click the Zoom Extents button on the Graphics toolbar.

Rectangle 21 In the Model Builder window, right-click Work Plane 1>Geometry and choose

Rectangle.



4 In the Height edit field, type H_electrolyte.

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5 Locate the Position section. In the y edit field, type -H_electrolyte.



Rectangle.



4 In the Height edit field, type H_gde.

5 Locate the Position section. In the y edit field, type -H_electrolyte-H_gde.



Rectangle.


3 Locate the Size section. In the Width edit field, type W_channel.

4 In the Height edit field, type H_channel.

5 Locate the Position section. In the x edit field, type W_rib/2.

6 In the y edit field, type H_gde.



Rectangle.


3 Locate the Size section. In the Width edit field, type W_channel.

4 In the Height edit field, type H_channel.

5 Locate the Position section. In the x edit field, type W_rib/2.

6 In the y edit field, type -H_gde-H_electrolyte-H_channel.



Extrude 11 In the Model Builder window, right-click Work Plane 1 and choose Extrude.

2 Go to the Settings window for Extrude.



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3 Locate the Distances from Work Plane section. In the table, enter the following settings:

4 Click the Build All button.


The model geometry is now complete, and it should look like that in Figure 1.

D E F I N I T I O N S

Explicit 11 In the Model Builder window, right-click Model 1>Definitions and choose

Selections>Explicit.

2 Select Domain 4 only.

3 Right-click Explicit 1 and choose Rename.

4 Go to the Rename Explicit dialog box and type Anode Flow Channel in the New

name edit field.

5 Click OK.

Explicit 21 Right-click Definitions and choose Explicit.



4 Go to the Rename Explicit dialog box and type Anode Electrode in the New name edit field.

5 Click OK.




4 Go to the Rename Explicit dialog box and type Cathode Electrode in the New name edit field.

5 Click OK.

DISTANCES (M)

L



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4 Go to the Rename Explicit dialog box and type Cathode Flow Channel in the New

name edit field.

5 Click OK.

Variables 11 Right-click Definitions and choose Variables.

2 Go to the Settings window for Variables.

3 Locate the Geometric Entity Selection section. From the Geometric entity level list, choose Domain.

4 From the Selection list, choose Anode Electrode.

5 Locate the Variables section. In the Variables table, enter the following settings:

Variables 21 In the Model Builder window, right-click Definitions and choose Variables.

2 Go to the Settings window for Variables.

3 Locate the Geometric Entity Selection section. From the Geometric entity level list, choose Domain.

4 From the Selection list, choose Cathode Electrode.

NAME EXPRESSION DESCRIPTION

eta_a phis-phil-dphieq_a Anode activation overpotential

ict_a io_a*(chcs2.c_wH2/c_h2ref*exp(0.5*F_const*eta_a/(R_const*T))-chcs2.c_wH2Oa/c_h2oref*exp(-1.5*F_const*eta_a/(R_const*T)))

Anode charge transfer current density



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5 Locate the Variables section. In the Variables table, enter the following settings:

6 In the Model Builder window, right-click Definitions and choose Probes>Boundary

Probe.

7 Go to the Settings window for Boundary Probe.

8 Locate the Probe Settings section. From the Type list, choose Average.

9 Locate the Source Selection section. Click Clear Selection.

10 Select Boundary 6 only.

11 In the upper-right corner of the Expression section, click Replace Expression.

12 From the menu, choose Secondary Current Distribution>Electrolyte current density

vector>Electrolyte current density vector, z component (siec.Ilz).

S E C O N D A R Y C U R R E N T D I S T R I B U T I O N

1 In the Model Builder window, click Model 1>Secondary Current Distribution.

2 Select Domains 1–3 only.

Porous Electrode 11 Right-click Model 1>Secondary Current Distribution and choose Porous Electrode.

2 Go to the Settings window for Porous Electrode.

3 Locate the Domain Selection section. From the Selection list, choose Anode Electrode.

4 Locate the Electrolyte Current Conduction section. From the σl list, choose User

defined. In the associated edit field, type kleff_a.

5 From the Effective conductivity correction list, choose No correction.

6 Locate the Electrode Current Conduction section. From the σs list, choose User

defined. In the associated edit field, type kseff_a.


NAME EXPRESSION DESCRIPTION

eta_c phis-phil-dphieq_c Cathode activation overpotential

ict_c io_c*(exp(3.5*F_const*eta_c/(R_const*T))-chcs.c_wO2/c_o2ref*exp(-0.5*F_const*eta_c/(R_const*T)))

Cathode charge transfer current



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Porous Electrode Reaction 11 In the Model Builder window, expand the Porous Electrode 1 node, then click Porous

Electrode Reaction 1.

2 Go to the Settings window for Porous Electrode Reaction.

3 Locate the Electrode Kinetics section. From the Kinetics expression type list, choose User defined. In the iloc edit field, type ict_a.

4 Locate the Active Specific Surface Area section. In the av edit field, type Sa_a.

Porous Electrode 21 In the Model Builder window, right-click Secondary Current Distribution and choose

Porous Electrode.

2 Go to the Settings window for Porous Electrode.

3 Locate the Domain Selection section. From the Selection list, choose Cathode

Electrode.

4 Locate the Electrolyte Current Conduction section. From the σl list, choose User

defined. In the associated edit field, type kleff_c.


6 Locate the Electrode Current Conduction section. From the σs list, choose User

defined. In the associated edit field, type kseff_c.


Porous Electrode Reaction 11 In the Model Builder window, expand the Porous Electrode 2 node, then click Porous

Electrode Reaction 1.

2 Go to the Settings window for Porous Electrode Reaction.

3 Locate the Electrode Kinetics section. From the Kinetics expression type list, choose User defined. In the iloc edit field, type ict_c.

4 Locate the Active Specific Surface Area section. In the av edit field, type Sa_c.

Electrolyte 11 In the Model Builder window, click Electrolyte 1.

2 Go to the Settings window for Electrolyte.

3 Locate the Electrolyte section. From the σl list, choose User defined. In the associated edit field, type kl.



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Electric Ground 11 In the Model Builder window, right-click Secondary Current Distribution and choose

Electrode>Electric Ground.

2 Select Boundaries 3 and 20 only.

Electric Potential 11 In the Model Builder window, right-click Secondary Current Distribution and choose

Electrode>Electric Potential.


3 Go to the Settings window for Electric Potential.

4 Locate the Electric Potential section. In the φs,bnd edit field, type V_cell.

Initial Values 21 In the Model Builder window, right-click Secondary Current Distribution and choose

Initial Values.


3 Go to the Settings window for Initial Values.

4 Locate the Initial Values section. In the phis edit field, type V_cell.

TR A N S P O R T O F C O N C E N T R A T E D S P E C I E S

1 In the Model Builder window, right-click Model 1>Transport of Concentrated Species and choose Rename.

2 Go to the Rename Transport of Concentrated Species dialog box and type Transport of Concentrated Species - Cathode in the New name edit field.

3 Click OK.

4 Select Domains 3 and 5 only.

5 Go to the Settings window for Transport of Concentrated Species.

6 Locate the Transport Mechanisms section. From the Diffusion model list, choose Maxwell-Stefan.

7 Locate the Species section. From the From mass constraint list, choose wN2.

TR A N S P O R T O F C O N C E N T R A T E D S P E C I E S - C A T H O D E

Convection and Diffusion 11 In the Model Builder window, expand the - Cathode node, then click Convection and

Diffusion 1.

2 Go to the Settings window for Convection and Diffusion.



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3 Locate the Density section. From the Mixture density list, choose Ideal gas.

4 In the MwO2 edit field, type Mo2.

5 In the MwH2Oc edit field, type Mh2o.

6 In the MwN2 edit field, type Mn2.

7 Locate the Diffusion section. In the Dik table, enter the following settings:

8 Locate the Model Inputs section. From the u list, choose Velocity field (fp/fp1).

9 In the T edit field, type T.

10 From the p list, choose Pressure (fp/fp1).

11 In the pref edit field, type p_atm.

Initial Values 11 In the Model Builder window, click Initial Values 1.


3 Locate the Initial Values section. In the wO2 edit field, type w_o2ref.

4 In the wH2Oc edit field, type w_h2oref.

Convection and Diffusion 21 In the Model Builder window, right-click Transport of Concentrated Species - Cathode

and choose Convection and Diffusion.


3 Locate the Domain Selection section. From the Selection list, choose Cathode Flow

Channel.


5 In the MwO2 edit field, type Mo2.

6 In the MwH2Oc edit field, type Mh2o.

7 In the MwN2 edit field, type Mn2.


1 Do2h2oeff Do2n2eff

Do2h2oeff 1 Dn2h2oeff

Do2n2eff Dn2h2oeff 1

1 Do2h2o Do2n2



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Porous Electrode Coupling 11 In the Model Builder window, right-click Transport of Concentrated Species - Cathode

and choose Porous Electrode Coupling.

2 Go to the Settings window for Porous Electrode Coupling.


Electrode.

Reaction Coefficients 11 In the Model Builder window, expand the Porous Electrode Coupling 1 node, then click

Reaction Coefficients 1.

2 Go to the Settings window for Reaction Coefficients.

3 Locate the Model Inputs section. From the iv list, choose Local current source (siec/

per1).

4 Locate the Stoichiometric Coefficients section. In the nm edit field, type 4.

5 In the νwO2 edit field, type -1.

Inflow 11 In the Model Builder window, right-click Transport of Concentrated Species - Cathode

and choose Inflow.


3 Go to the Settings window for Inflow.

4 Locate the Inflow section. In the w0,wO2 edit field, type w_o2ref.

5 In the w0,wH2Oc edit field, type w_h2oref.

Outflow 11 In the Model Builder window, right-click Transport of Concentrated Species - Cathode

and choose Outflow.


Do2h2o 1 Dn2h2o

Do2n2 Dn2h2o 1



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TR A N S P O R T O F C O N C E N T R A T E D S P E C I E S 2

1 In the Model Builder window, right-click Model 1>Transport of Concentrated Species

2 and choose Rename.

2 Go to the Rename Transport of Concentrated Species dialog box and type Transport of Concentrated Species 2 - Anode in the New name edit field.

3 Click OK.


5 Go to the Settings window for Transport of Concentrated Species.

6 Locate the Transport Mechanisms section. From the Diffusion model list, choose Maxwell-Stefan.

7 Locate the Species section. From the From mass constraint list, choose wH2Oa.

TR A N S P O R T O F C O N C E N T R A T E D S P E C I E S 2 - A N O D E

Convection and Diffusion 11 In the Model Builder window, expand the - Anode node, then click Convection and

Diffusion 1.



4 In the MwH2 edit field, type Mh2.

5 In the MwH2Oa edit field, type Mh2o.








3 Locate the Initial Values section. In the wH2 edit field, type w_h2ref.

1 Dh2h2oeff

Dh2h2oeff 1



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Convection and Diffusion 21 In the Model Builder window, right-click Transport of Concentrated Species 2 - Anode

and choose Convection and Diffusion.


3 Locate the Domain Selection section. From the Selection list, choose Anode Flow

Channel.

4 In the Model Builder window, click Convection and Diffusion 2.



7 In the Model Builder window, click Convection and Diffusion 2.


9 Locate the Density section. In the MwH2 edit field, type Mh2.

10 In the MwH2Oa edit field, type Mh2o.






Porous Electrode Coupling 11 In the Model Builder window, right-click Transport of Concentrated Species 2 - Anode

and choose Porous Electrode Coupling.



Reaction Coefficients 11 In the Model Builder window, click Reaction Coefficients 1.



per1).


1 Dh2h2o

Dh2h2o 1


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5 In the νwH2 edit field, type 1.

6 In the νwH2Oa edit field, type -1.

Inflow 11 In the Model Builder window, right-click Transport of Concentrated Species 2 - Anode

and choose Inflow.


3 Go to the Settings window for Inflow.

4 Locate the Inflow section. In the w0,wH2 edit field, type w_h2ref.

Outflow 11 In the Model Builder window, right-click Transport of Concentrated Species 2 - Anode

and choose Outflow.


F R E E A N D PO R O U S M E D I A F L O W

1 In the Model Builder window, click Model 1>Free and Porous Media Flow.

2 Select Domains 1 and 3–5 only.

3 Go to the Settings window for Free and Porous Media Flow.

4 Locate the Physical Model section. From the Compressibility list, choose Compressible

flow (Ma<0.3).

Fluid Properties 11 In the Model Builder window, expand the Free and Porous Media Flow node, then click

Fluid Properties 1.

2 Go to the Settings window for Fluid Properties.

3 Locate the Fluid Properties section. From the ρ list, choose Density (chcs2/chcs2).

4 From the μ list, choose User defined. In the associated edit field, type mu.

Wall 21 In the Model Builder window, right-click Free and Porous Media Flow and choose Wall.

2 Select Boundaries 2, 8, 23, and 25 only.

3 Go to the Settings window for Wall.

4 Locate the Boundary Condition section. From the Boundary condition list, choose Slip.


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Fluid Properties 21 In the Model Builder window, right-click Free and Porous Media Flow and choose Fluid

Properties.


3 Go to the Settings window for Fluid Properties.

4 Locate the Fluid Properties section. From the ρ list, choose Density (chcs/chcs).

5 From the μ list, choose User defined. In the associated edit field, type mu.

Porous Matrix Properties 11 In the Model Builder window, right-click Free and Porous Media Flow and choose

Porous Matrix Properties.

2 Go to the Settings window for Porous Matrix Properties.



5 Locate the Porous Matrix Properties section. From the εp list, choose User defined. In the associated edit field, type e_por.

6 From the κbr list, choose User defined. In the associated edit field, type perm_a.

Porous Electrode Coupling 11 In the Model Builder window, right-click Free and Porous Media Flow and choose

Porous Electrode Coupling.



4 Locate the Species section. Click Add.

5 In the Species table, enter the following settings:





per1).

MOLAR MASS (KG/MOL)

Mh2

Mh2o



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5 In the ν1 edit field, type 1.

6 In the ν2 edit field, type -1.

Porous Matrix Properties 21 In the Model Builder window, right-click Free and Porous Media Flow and choose

Porous Matrix Properties.



Electrode.


5 Locate the Porous Matrix Properties section. From the εp list, choose User defined. In the associated edit field, type e_por.

6 From the κbr list, choose User defined. In the associated edit field, type perm_c.

Porous Electrode Coupling 21 In the Model Builder window, right-click Free and Porous Media Flow and choose

Porous Electrode Coupling.



Electrode.

4 Locate the Species section. Click Add.

5 Click Add.

6 In the Species table, enter the following settings:





per1).

MOLAR MASS (KG/MOL)

Mo2

Mh2o

Mn2



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5 In the ν1 edit field, type -1.



3 Locate the Initial Values section. In the p edit field, type 0.

Inlet 11 In the Model Builder window, right-click Free and Porous Media Flow and choose Inlet.


3 Go to the Settings window for Inlet.

4 Locate the Boundary Condition section. From the Boundary condition list, choose Pressure, no viscous stress.

5 Locate the Pressure, No Viscous Stress section. In the p0 edit field, type dp_c.

Inlet 21 In the Model Builder window, right-click Free and Porous Media Flow and choose Inlet.


3 Go to the Settings window for Inlet.

4 Locate the Boundary Condition section. From the Boundary condition list, choose Pressure, no viscous stress.

5 Locate the Pressure, No Viscous Stress section. In the p0 edit field, type dp_a.

Outlet 11 In the Model Builder window, right-click Free and Porous Media Flow and choose

Outlet.


3 Go to the Settings window for Outlet.

4 Locate the Boundary Condition section. From the Boundary condition list, choose Laminar outflow.

5 Click the Exit pressure button.

6 In the Lexit edit field, type 1e-3.

7 Select the Constrain outer edges to zero check box.



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M E S H 1

Mapped 11 In the Model Builder window, right-click Model 1>Mesh 1 and choose More

Operations>Mapped.

2 Select Boundaries 1, 4, 7, 11, and 15 only.

Distribution 11 Right-click Mapped 1 and choose Distribution.


3 Select Edges 12, 13, 17, 20, 22, and 26 only.

4 Go to the Settings window for Distribution.

5 Locate the Distribution section. In the Number of elements edit field, type 5.

Distribution 21 In the Model Builder window, right-click Mapped 1 and choose Distribution.

2 Select Edges 2, 10, 24, and 27 only.




2 Select Edges 1 and 30 only.













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Mapped 1In the Model Builder window, right-click Mapped 1 and choose Build Selected.

Swept 1Right-click Mesh 1 and choose Swept.

Distribution 11 In the Model Builder window, right-click Swept 1 and choose Distribution.



Swept 1In the Model Builder window, right-click Swept 1 and choose Build Selected.

S T U D Y 1

1 In the Model Builder window, right-click Study 1 and choose Show Default Solver.

2 Expand the Study 1>Solver Configurations node.

Solver 11 In the Model Builder window, expand the Study 1>Solver Configurations>Solver 1

node.

This rather small problem solves more quickly using a direct solver.

2 In the Model Builder window, expand the Stationary Solver 1 node.

3 Right-click Stationary Solver 1>Direct and choose Enable.

4 Right-click Stationary Solver 1 and choose Parametric.

5 Go to the Settings window for Parametric.

6 Locate the General section. From the Defined by study step list, choose User defined.

7 In the Parameter names edit field, type V_pol.

8 In the Parameter values edit field, type 0.05 range(0.1,0.1,0.8).

9 In the Model Builder window, right-click Study 1 and choose Compute.

The computation takes about 10 minutes.

R E S U L T S

3D Plot Group 10Follow these instructions to reproduce the plot in Figure 2 of the oxygen distribution in the anode at a cell voltage of 0.5 V.



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1 In the Model Builder window, right-click Results and choose 3D Plot Group.

2 Right-click Results>3D Plot Group 10 and choose Rename.

3 Go to the Rename 3D Plot Group dialog box and type Oxygen Mole Fraction in the New name edit field.

4 Click OK.

5 Go to the Settings window for 3D Plot Group.

6 Locate the Data section. From the Parameter value (V_pol) list, choose 0.5.

7 Locate the Plot Settings section. Clear the Plot data set edges check box.

Oxygen Mole Fraction1 Right-click Results>3D Plot Group 10 and choose Slice.

2 Go to the Settings window for Slice.


4 From the menu, choose Transport of Concentrated Species - Cathode>Species

wO2>Mole fraction (chcs.x_wO2).

5 Locate the Plane Data section. From the Plane list, choose yz-planes.

6 In the Planes edit field, type 20.

7 Click the Plot button.

3D Plot Group 11The following steps reproduce the corresponding plot for hydrogen (compare with Figure 3).



3 Go to the Rename 3D Plot Group dialog box and type Hydrogen Mole Fraction in the New name edit field.

4 Click OK.


6 Locate the Data section. From the Parameter value (V_pol) list, choose 0.5.


Hydrogen Mole Fraction1 Right-click Results>3D Plot Group 11 and choose Slice.

2 Go to the Settings window for Slice.



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4 From the menu, choose Transport of Concentrated Species 2 - Anode>Species

wH2>Mole fraction (chcs2.x_wH2).

5 Locate the Plane Data section. From the Plane list, choose yz-planes.

6 In the Planes edit field, type 20.


1D Plot Group 12The following instructions reproduce the plot of the polarization curve for the SOFC (see Figure 5).



3 Go to the Rename 1D Plot Group dialog box and type Polarization curve in the New name edit field.

4 Click OK.


6 Locate the Title section. From the Title type list, choose Manual.

7 In the Title text area, type Polarization curve.

8 Locate the Plot Settings section. Select the x-axis label check box.

9 In the associated edit field, type Average current density (A/m2).

10 Select the y-axis label check box.

11 In the associated edit field, type Vcell (V).

Polarization curve1 Right-click Results>1D Plot Group 12 and choose Global.

2 Go to the Settings window for Global.

3 Locate the y-Axis Data section. In the table, enter the following settings:

4 Locate the x-Axis Data section. From the Parameter list, choose Expression.

5 In the Expression edit field, type bnd1.


EXPRESSION DESCRIPTION

V_cell Cell Voltage

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1D Plot Group 13Next, reproduce a plot showing the power output as a function of the cell voltage (Figure 6.



3 Go to the Rename 1D Plot Group dialog box and type Power vs Current in the New

name edit field.

4 Click OK.


6 Locate the Title section. From the Title type list, choose Manual.

7 In the Title text area, type Total output power.

8 Locate the Plot Settings section. Select the x-axis label check box.

9 In the associated edit field, type Average current density (A/m2).

10 Select the y-axis label check box.

11 In the associated edit field, type Average Cell Power (W/m2).

Power vs Current1 Right-click Results>1D Plot Group 13 and choose Global.

2 Go to the Settings window for Global.

3 Locate the y-Axis Data section. In the table, enter the following settings:

4 Locate the x-Axis Data section. From the Parameter list, choose Expression.

5 In the Expression edit field, type bnd1.


3D Plot Group 14Next reproduce the plot in Figure 4 showing the current density in the unit cell at 0.5 V.



EXPRESSION DESCRIPTION

V_cell*bnd1 Average Cell Power


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3 Go to the Rename 3D Plot Group dialog box and type Electrolyte current density in the New name edit field.

4 Click OK.

Before defining the plot, add the data set for the Boundary 9.

Data Sets1 In the Model Builder window, right-click Results>Data Sets and choose Solution.

2 Right-click Probe Solution 2 and choose Add Selection.

3 Go to the Settings window for Selection.

4 Locate the Geometric Entity Selection section. From the Geometric entity level list, choose Boundary.


Electrolyte current density1 In the Model Builder window, click Results>Electrolyte current density.


3 Locate the Data section. From the Data set list, choose Probe Solution 2.

4 From the Parameter value (V_pol) list, choose 0.5.


6 Right-click Results>Electrolyte current density and choose Surface.

7 Go to the Settings window for Surface.


9 From the menu, choose Secondary Current Distribution>Electrolyte current density

vector>Electrolyte current density vector, z component (siec.Ilz).



Models.bfc.Sofc Unit Cell

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