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Current Density Distribution in a Solid Oxide Fuel Cell
Introduction
This example 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.
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.
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
•
Mass balances in gas phase in both gas channels and GDEs (Maxwell-Stefan Diffusion and Convection)
•
Flow distribution in gas channels (Navier-Stokes equations)
•
Flow in the porous GDEs (Brinkman equations)
Charge Balances
The electronic and ionic charge balance in the anode and cathode current feeders, the electrolyte and GDEs are solved for using a Hydrogen Fuel Cell interface.
Assume that Butler-Volmer charge transfer kinetics describe the charge transfer current density. At the anode, hydrogen is oxidized 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 (SI unit: A/m2), ph2 is the partial pressure of hydrogen, ph2o is the partial pressure of water, ph2,ref and ph2o,ref is the reference pressures (SI unit: Pa). Furthermore, F is Faraday’s constant (SI unit: C/mol), R the gas constant (SI unit: J/(mol·K)), T the temperature (SI unit: K), and η the overvoltage (SI unit: V).
For the cathode, use the relation
where i0,c is the cathode exchange current density (SI unit: A/m2), and po2 is the partial pressure of oxygen.
The overvoltage is defined as
where is the equilibrium potential difference (SI unit: 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
where Vpol is the polarization. In this model, you simulate the fuel cell for a range of cell voltage (ranging from around 0.2 V to 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.
Multicomponent Mass Transport
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 the Hydrogen Fuel Cell interface.
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.
Continuity in composition and flux apply for all mass balances at the interfaces between the GDEs and the channels.
Gas-Flow Equations
The Free and Porous Media Flow interface is used for solving for the velocity field and pressure. The 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).
Couplings for the density, dynamic viscosity, velocity, pressure and net mass sources and sinks are made between the Hydrogen Fuel Cell and Free and Porous Media Flow interface by using Reacting Flow, H2 Gas Phase and Reacting Flow, O2 Gas Phase multiphysics nodes.
Results and Discussion
Figure 2 shows the oxygen mole fraction in the cathode at a cell polarization of 0.8 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 0.8 V.
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 0.8 V cell voltage.
A consequence of the concentration distribution is that the current density is nonuniform in the GDEs. Figure 4 depicts the current density distribution at the cathode side of the ionic conductor.
Figure 4: The electrolyte current density in the unit cell operating at 0.8 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.
Figure 5 shows the voltage as a function of the total current (polarization curve).
Figure 5: Polarization curve.
Figure 6 shows the power output as a function of the cell voltage. The model predicts a maximum power-output of about 1100 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, P.A., 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.
Application Library path: Fuel_Cell_and_Electrolyzer_Module/Fuel_Cells/sofc_unit_cell
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  3D.
2
In the Select Physics tree, select Electrochemistry>Hydrogen Fuel Cells>Solid Oxide (fc).
3
Click Add.
4
In the Select Physics tree, select Fluid Flow>Porous Media and Subsurface Flow>Free and Porous Media Flow (fp).
5
Click Add.
6
In the Velocity field text field, type u_c.
7
In the Velocity field components table, enter the following settings:
u_c
v_c
w_c
8
In the Pressure text field, type p_c.
9
In the Select Physics tree, select Fluid Flow>Porous Media and Subsurface Flow>Free and Porous Media Flow (fp).
10
Click Add.
11
In the Velocity field text field, type u_a.
12
In the Velocity field components table, enter the following settings:
u_a
v_a
w_a
13
In the Pressure text field, type p_a.
14
Click  Study.
15
In the Select Study tree, select Preset Studies for Selected Physics Interfaces>Hydrogen Fuel Cell>Stationary with Initialization.
16
Click  Done.
Global Definitions
Define the parameters using the text file provided.
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file sofc_unit_cell_parameters.txt.
Geometry 1
Create the geometry by first defining the 2D cross section of the device, then extrude it to create the 3D model geometry.
Work Plane 1 (wp1)
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, locate the Plane Definition section.
3
From the Plane list, choose yz-plane.
Add rectangles as described below.
4
Click  Show Work Plane.
Work Plane 1 (wp1)>Rectangle 1 (r1)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type W_channel+W_rib.
4
In the Height text field, type H_gde.
5
Click  Build Selected.
6
Click the  Zoom Extents button in the Graphics toolbar.
Work Plane 1 (wp1)>Rectangle 2 (r2)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type W_channel+W_rib.
4
In the Height text field, type H_electrolyte.
5
Locate the Position section. In the yw text field, type -H_electrolyte.
6
Click  Build Selected.
Work Plane 1 (wp1)>Rectangle 3 (r3)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type W_channel+W_rib.
4
In the Height text field, type H_gde.
5
Locate the Position section. In the yw text field, type -H_electrolyte-H_gde.
6
Click  Build Selected.
Work Plane 1 (wp1)>Rectangle 4 (r4)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type W_channel.
4
In the Height text field, type H_channel.
5
Locate the Position section. In the xw text field, type W_rib/2.
6
In the yw text field, type H_gde.
7
Click  Build Selected.
Work Plane 1 (wp1)>Rectangle 5 (r5)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type W_channel.
4
In the Height text field, type H_channel.
5
Locate the Position section. In the xw text field, type W_rib/2.
6
In the yw text field, type -H_gde-H_electrolyte-H_channel.
7
Click  Build Selected.
8
Click the  Zoom Extents button in the Graphics toolbar.
Extrude 1 (ext1)
1
In the Model Builder window, under Component 1 (comp1)>Geometry 1 right-click Work Plane 1 (wp1) and choose Extrude.
2
In the Settings window for Extrude, locate the Distances section.
3
In the table, enter the following settings:
Distances (m)
L
4
Click  Build All Objects.
5
Click the  Zoom Extents button in the Graphics toolbar.
The model geometry is now complete, and it should look like that in the figure below.
Definitions
Now make a number of selections to facilitate choosing different parts of the geometry when setting up the model.
Anode Flow Channel
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Anode Flow Channel in the Label text field.
3
Select Domain 4 only.
Anode Gas Diffusion Electrode
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Anode Gas Diffusion Electrode in the Label text field.
3
Select Domain 1 only.
Cathode Gas Diffusion Electrode
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Cathode Gas Diffusion Electrode in the Label text field.
3
Select Domain 3 only.
Cathode Flow Channel
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Cathode Flow Channel in the Label text field.
3
Select Domain 5 only.
Membrane
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Membrane in the Label text field.
3
Select Domain 2 only.
Free and Porous Media Flow - Cathode
Set up the domains applicable for the flow interfaces.
1
In the Model Builder window, under Component 1 (comp1) click Free and Porous Media Flow (fp).
2
In the Settings window for Free and Porous Media Flow, type Free and Porous Media Flow - Cathode in the Label text field.
3
Locate the Domain Selection section. In the list, choose 1, 2, and 4.
4
Click  Remove from Selection.
5
Select Domains 3 and 5 only.
Free and Porous Media Flow - Anode
1
In the Model Builder window, under Component 1 (comp1) click Free and Porous Media Flow 2 (fp2).
2
In the Settings window for Free and Porous Media Flow, type Free and Porous Media Flow - Anode in the Label text field.
3
Locate the Domain Selection section. In the list, choose 2, 3, and 5.
4
Click  Remove from Selection.
5
Select Domains 1 and 4 only.
Multiphysics
Next, couple the interfaces appropriately using the reacting flow multiphysics coupling nodes. Note that currently, the multiphysics nodes may not be applicable to any domain selections, but the selections will be automatically updated when the Hydrogen Fuel Cell interface is set up.
Reacting Flow, H2 Gas Phase 1 (rfh1)
1
In the Physics toolbar, click  Multiphysics Couplings and choose Domain>Reacting Flow, H2 Gas Phase.
2
In the Settings window for Reacting Flow, H2 Gas Phase, locate the Coupled Interfaces section.
3
From the Fluid flow list, choose Free and Porous Media Flow - Anode (fp2).
Reacting Flow, O2 Gas Phase 1 (rfo1)
In the Physics toolbar, click  Multiphysics Couplings and choose Domain>Reacting Flow, O2 Gas Phase.
Hydrogen Fuel Cell (fc)
Set up the current distribution and mass transport model. The default gas species are hydrogen and water on the anode side, and oxygen and nitrogen on the cathode side. Additionally, include water on the cathode side.
1
In the Model Builder window, under Component 1 (comp1) click Hydrogen Fuel Cell (fc).
2
In the Settings window for Hydrogen Fuel Cell, locate the O2 Gas Mixture section.
3
Select the H2O check box.
Add the relevant domain nodes.
Membrane 1
1
In the Physics toolbar, click  Domains and choose Membrane.
2
In the Settings window for Membrane, locate the Domain Selection section.
3
From the Selection list, choose Membrane.
H2 Gas Diffusion Electrode 1
1
In the Physics toolbar, click  Domains and choose H2 Gas Diffusion Electrode.
2
In the Settings window for H2 Gas Diffusion Electrode, locate the Domain Selection section.
3
From the Selection list, choose Anode Gas Diffusion Electrode.
H2 Gas Flow Channel 1
1
In the Physics toolbar, click  Domains and choose H2 Gas Flow Channel.
2
In the Settings window for H2 Gas Flow Channel, locate the Domain Selection section.
3
From the Selection list, choose Anode Flow Channel.
O2 Gas Diffusion Electrode 1
1
In the Physics toolbar, click  Domains and choose O2 Gas Diffusion Electrode.
2
In the Settings window for O2 Gas Diffusion Electrode, locate the Domain Selection section.
3
From the Selection list, choose Cathode Gas Diffusion Electrode.
O2 Gas Flow Channel 1
1
In the Physics toolbar, click  Domains and choose O2 Gas Flow Channel.
2
In the Settings window for O2 Gas Flow Channel, locate the Domain Selection section.
3
From the Selection list, choose Cathode Flow Channel.
Add Material
1
In the Home toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Fuel Cell and Electrolyzer>Solid Oxides>Yttria-Stabilized Zirconia, 8YSZ, (ZrO2)0.92-(Y2O3)0.08.
4
Right-click and choose Add to Component 1 (comp1).
5
In the Home toolbar, click  Add Material to close the Add Material window.
In the Electrolyte Phase node, the electrolyte conductivity is set to be taken from the Materials node. In the H2 Gas Phase and O2 Gas Phase nodes, the settings are either the default option or automatically set by the multiphysics coupling nodes.
Hydrogen Fuel Cell (fc)
H2 Gas Diffusion Electrode 1
Set up the properties of the H2 Gas Diffusion Electrode node. The details of electrode kinetics are set in the child node. Note that the reference equilibrium potential is calculated automatically when the default Built in option is used.
1
In the Settings window for H2 Gas Diffusion Electrode, locate the Electrode Charge Transport section.
2
In the σs text field, type kseff_a.
3
Locate the Effective Electrolyte Charge Transport section. From the Effective conductivity correction list, choose User defined. In the fl text field, type fl_a.
4
Locate the Gas Transport section. In the εg text field, type e_por.
H2 Gas Diffusion Electrode Reaction 1
1
In the Model Builder window, click H2 Gas Diffusion Electrode Reaction 1.
2
In the Settings window for H2 Gas Diffusion Electrode Reaction, click to expand the Reference Pressures section.
3
In the pH2,ref text field, type p_h2ref.
4
In the pH2O,ref text field, type p_h2oref_a.
5
Locate the Electrode Kinetics section. In the i0,ref(T) text field, type i0_a.
6
Locate the Active Specific Surface Area section. In the av text field, type Sa_a.
O2 Gas Diffusion Electrode 1
Similarly, set up the properties of the O2 Gas Diffusion Electrode node. The details of electrode kinetics are set in the child node. Note that the reference equilibrium potential is calculated automatically when the default Built in option is used.
1
In the Model Builder window, under Component 1 (comp1)>Hydrogen Fuel Cell (fc) click O2 Gas Diffusion Electrode 1.
2
In the Settings window for O2 Gas Diffusion Electrode, locate the Electrode Charge Transport section.
3
In the σs text field, type kseff_c.
4
Locate the Effective Electrolyte Charge Transport section. From the Effective conductivity correction list, choose User defined. In the fl text field, type fl_c.
5
Locate the Gas Transport section. In the εg text field, type e_por.
O2 Gas Diffusion Electrode Reaction 1
1
In the Model Builder window, click O2 Gas Diffusion Electrode Reaction 1.
2
In the Settings window for O2 Gas Diffusion Electrode Reaction, click to expand the Reference Pressures section.
3
In the pO2,ref text field, type p_o2ref.
4
Locate the Electrode Kinetics section. In the i0,ref(T) text field, type i0_c.
5
Locate the Active Specific Surface Area section. In the av text field, type Sa_c.
There are no settings required on the flow channel nodes, other than the domain selection. Next, set up the boundary conditions and initial values.
Electronic Conducting Phase 1
In the Model Builder window, under Component 1 (comp1)>Hydrogen Fuel Cell (fc) click Electronic Conducting Phase 1.
Electric Ground 1
1
In the Physics toolbar, click  Attributes and choose Electric Ground.
2
Select Boundaries 3 and 20 only.
Electronic Conducting Phase 1
In the Model Builder window, click Electronic Conducting Phase 1.
Electric Potential 1
1
In the Physics toolbar, click  Attributes and choose Electric Potential.
2
Select Boundaries 10 and 22 only.
3
In the Settings window for Electric Potential, locate the Electric Potential section.
4
In the φs,bnd text field, type V_cell.
Initial Values 1
1
In the Model Builder window, expand the Component 1 (comp1)>Hydrogen Fuel Cell (fc)>H2 Gas Phase 1 node, then click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Composition section.
3
In the x0,H2O text field, type x_h2oref_a.
H2 Gas Phase 1
In the Model Builder window, click H2 Gas Phase 1.
H2 Inlet 1
1
In the Physics toolbar, click  Attributes and choose H2 Inlet.
2
Select Boundary 11 only.
3
In the Settings window for H2 Inlet, locate the Mixture Specification section.
4
From the list, choose Mass fractions.
5
In the ω0,H2O text field, type w_h2oref_a.
H2 Gas Phase 1
In the Model Builder window, click H2 Gas Phase 1.
H2 Outlet 1
1
In the Physics toolbar, click  Attributes and choose H2 Outlet.
2
Select Boundary 29 only.
Initial Values 1
1
In the Model Builder window, under Component 1 (comp1)>Hydrogen Fuel Cell (fc)>O2 Gas Phase 1 click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Composition section.
3
In the x0,H2O text field, type x_h2oref_c.
4
In the x0,N2 text field, type x_n2ref.
O2 Gas Phase 1
In the Model Builder window, click O2 Gas Phase 1.
O2 Inlet 1
1
In the Physics toolbar, click  Attributes and choose O2 Inlet.
2
Select Boundary 30 only.
3
In the Settings window for O2 Inlet, locate the Mixture Specification section.
4
From the list, choose Mass fractions.
5
In the ω0,H2O text field, type w_h2oref_c.
6
In the ω0,N2 text field, type w_n2ref.
O2 Gas Phase 1
In the Model Builder window, click O2 Gas Phase 1.
O2 Outlet 1
1
In the Physics toolbar, click  Attributes and choose O2 Outlet.
2
Select Boundary 15 only.
Free and Porous Media Flow - Cathode (fp)
Next, set up the fluid flow model on the cathode side. Note that the flow is compressible.
1
In the Model Builder window, under Component 1 (comp1) click Free and Porous Media Flow - Cathode (fp).
2
In the Settings window for Free and Porous Media Flow, locate the Physical Model section.
3
From the Compressibility list, choose Compressible flow (Ma<0.3).
Define the pressure reference level in the interface properties.
4
In the pref text field, type p_atm.
Set up the properties of the porous gas diffusion electrode and the flow channel, followed by the boundary conditions. Note that the density and viscosity of the gas mixture are calculated by the Hydrogen Fuel Cell interface and automatically set by the multiphysics coupling nodes.
Porous Medium 1
1
In the Physics toolbar, click  Domains and choose Porous Medium.
2
In the Settings window for Porous Medium, locate the Domain Selection section.
3
From the Selection list, choose Cathode Gas Diffusion Electrode.
Porous Matrix 1
1
In the Model Builder window, click Porous Matrix 1.
2
In the Settings window for Porous Matrix, locate the Matrix Properties section.
3
From the εp list, choose User defined. In the associated text field, type e_por.
4
From the κ list, choose User defined. In the associated text field, type perm_c.
Inlet 1
1
In the Physics toolbar, click  Boundaries and choose Inlet.
2
Select Boundary 30 only.
3
In the Settings window for Inlet, locate the Boundary Condition section.
4
From the list, choose Pressure.
5
Locate the Pressure Conditions section. In the p0 text field, type dp_c.
Outlet 1
1
In the Physics toolbar, click  Boundaries and choose Outlet.
2
Select Boundary 15 only.
3
In the Settings window for Outlet, locate the Pressure Conditions section.
4
Select the Normal flow check box.
Wall 2
1
In the Physics toolbar, click  Boundaries and choose Wall.
2
Select Boundaries 8 and 25 only.
3
In the Settings window for Wall, locate the Boundary Condition section.
4
From the Wall condition list, choose Slip.
Free and Porous Media Flow - Anode (fp2)
Set up the fluid flow model on the anode side in the same way.
1
In the Model Builder window, under Component 1 (comp1) click Free and Porous Media Flow - Anode (fp2).
2
In the Settings window for Free and Porous Media Flow, locate the Physical Model section.
3
From the Compressibility list, choose Compressible flow (Ma<0.3).
Define the pressure reference level in the interface properties.
4
In the pref text field, type p_atm.
Porous Medium 1
1
In the Physics toolbar, click  Domains and choose Porous Medium.
2
In the Settings window for Porous Medium, locate the Domain Selection section.
3
From the Selection list, choose Anode Gas Diffusion Electrode.
Porous Matrix 1
1
In the Model Builder window, click Porous Matrix 1.
2
In the Settings window for Porous Matrix, locate the Matrix Properties section.
3
From the εp list, choose User defined. In the associated text field, type e_por.
4
From the κ list, choose User defined. In the associated text field, type perm_a.
Inlet 1
1
In the Physics toolbar, click  Boundaries and choose Inlet.
2
Select Boundary 11 only.
3
In the Settings window for Inlet, locate the Boundary Condition section.
4
From the list, choose Pressure.
5
Locate the Pressure Conditions section. In the p0 text field, type dp_a.
Outlet 1
1
In the Physics toolbar, click  Boundaries and choose Outlet.
2
Select Boundary 29 only.
3
In the Settings window for Outlet, locate the Pressure Conditions section.
4
Select the Normal flow check box.
Wall 2
1
In the Physics toolbar, click  Boundaries and choose Wall.
2
Select Boundaries 2 and 23 only.
3
In the Settings window for Wall, locate the Boundary Condition section.
4
From the Wall condition list, choose Slip.
Global Definitions
Default Model Inputs
Default Model Inputs node can be used to set the Temperature for the entire model. This node may be accessed by multiple physics nodes.
1
In the Model Builder window, under Global Definitions click Default Model Inputs.
2
In the Settings window for Default Model Inputs, locate the Browse Model Inputs section.
3
In the tree, select General>Temperature (K) - minput.T.
4
Find the Expression for remaining selection subsection. In the Temperature text field, type T.
Mesh 1
The physics settings for the model is now complete. A mapped mesh, swept in the channel direction, is suitable for this geometry. Control the size in the y direction by using an individual Edge node.
Edge 1
1
In the Mesh toolbar, click  Boundary and choose Edge.
2
Select Edges 2, 10, 15, 18, 24, and 27 only.
Size 1
1
Right-click Edge 1 and choose Size.
2
In the Settings window for Size, locate the Element Size section.
3
Click the Custom button.
4
Locate the Element Size Parameters section. Select the Maximum element size check box.
5
In the associated text field, type W_channel/8.
6
Click  Build Selected.
Mapped 1
1
In the Mesh toolbar, click  Boundary and choose Mapped.
2
Select Boundaries 1, 4, 7, 11, and 15 only.
Distribution 1
1
Right-click Mapped 1 and choose Distribution.
2
Select Edges 12, 17, 22, and 26 only.
3
In the Settings window for Distribution, locate the Distribution section.
4
From the Distribution type list, choose Predefined.
5
In the Number of elements text field, type 10.
6
In the Element ratio text field, type 3.
7
Select the Symmetric distribution check box.
Distribution 2
1
In the Model Builder window, right-click Mapped 1 and choose Distribution.
2
Select Edges 7 and 34 only.
3
In the Settings window for Distribution, locate the Distribution section.
4
From the Distribution type list, choose Predefined.
5
In the Number of elements text field, type 8.
6
In the Element ratio text field, type 3.
Distribution 3
1
Right-click Distribution 2 and choose Duplicate.
2
In the Settings window for Distribution, locate the Edge Selection section.
3
Click  Clear Selection.
4
Select Edges 1 and 30 only.
5
Locate the Distribution section. Select the Reverse direction check box.
Distribution 4
1
In the Model Builder window, right-click Mapped 1 and choose Distribution.
2
Select Edges 4 and 32 only.
3
In the Settings window for Distribution, locate the Distribution section.
4
In the Number of elements text field, type 3.
Mapped 1
Right-click Mapped 1 and choose Build Selected.
Swept 1
In the Mesh toolbar, click  Swept.
Size 1
1
Right-click Swept 1 and choose Size.
2
In the Settings window for Size, locate the Element Size section.
3
Click the Custom button.
4
Locate the Element Size Parameters section. Select the Maximum element size check box.
5
In the associated text field, type W_channel.
6
Click  Build All.
7
Click the  Zoom Extents button in the Graphics toolbar.
The meshing is now complete, and it should look like that in the figure below.
Definitions
Before solving, add a probe for the average cell current density, it will be plotted during the solver process.
Cell Current Density Probe
1
In the Definitions toolbar, click  Probes and choose Domain Probe.
2
In the Settings window for Domain Probe, type Cell Current Density Probe in the Label text field.
3
In the Variable name text field, type I_cell.
4
Locate the Source Selection section. From the Selection list, choose Anode Gas Diffusion Electrode.
5
Click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1)>Hydrogen Fuel Cell>Electrode kinetics>fc.ivtot - Electrode reaction source - A/m³.
6
Locate the Expression section. In the Expression text field, type fc.ivtot*H_gde.
7
Select the Description check box.
8
In the associated text field, type Average cell current density.
Study 1
The problem is now ready for solving. Firstly, solve for current distribution initialization (both primary and secondary) in two study steps, followed by flow in two subsequent study steps. Finally, solve the entire model including the multiphysics couplings in the final step, along with an auxiliary sweep with continuation to solve for a range of different cell polarization voltages.
Step 1: Current Distribution Initialization
1
In the Model Builder window, under Study 1 click Step 1: Current Distribution Initialization.
2
In the Settings window for Current Distribution Initialization, locate the Physics and Variables Selection section.
3
In the table, clear the Solve for check boxes for Reacting Flow, H2 Gas Phase 1 (rfh1) and Reacting Flow, O2 Gas Phase 1 (rfo1).
Current Distribution Initialization 2
1
In the Study toolbar, click  Study Steps and choose Other>Current Distribution Initialization.
2
Drag and drop below Step 1: Current Distribution Initialization.
3
In the Settings window for Current Distribution Initialization, locate the Study Settings section.
4
From the Current distribution type list, choose Secondary.
5
Locate the Physics and Variables Selection section. In the table, clear the Solve for check boxes for Reacting Flow, H2 Gas Phase 1 (rfh1) and Reacting Flow, O2 Gas Phase 1 (rfo1).
Step 3: Stationary
1
In the Model Builder window, click Step 3: Stationary.
2
In the Settings window for Stationary, locate the Physics and Variables Selection section.
3
In the table, clear the Solve for check boxes for Hydrogen Fuel Cell (fc) and Free and Porous Media Flow - Anode (fp2).
4
In the table, clear the Solve for check boxes for Reacting Flow, H2 Gas Phase 1 (rfh1) and Reacting Flow, O2 Gas Phase 1 (rfo1).
Stationary 2
1
In the Study toolbar, click  Study Steps and choose Stationary>Stationary.
2
In the Settings window for Stationary, locate the Physics and Variables Selection section.
3
In the table, clear the Solve for check boxes for Hydrogen Fuel Cell (fc) and Free and Porous Media Flow - Cathode (fp).
4
In the table, clear the Solve for check boxes for Reacting Flow, H2 Gas Phase 1 (rfh1) and Reacting Flow, O2 Gas Phase 1 (rfo1).
Stationary 3
1
In the Study toolbar, click  Study Steps and choose Stationary>Stationary.
2
In the Settings window for Stationary, click to expand the Study Extensions section.
3
Select the Auxiliary sweep check box.
4
Click  Add.
5
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
V_pol (Initial cell polarization)
0.05 range(0.1,0.1,0.8)
V
6
In the Study toolbar, click  Compute.
The computation takes a few minutes.
Results
Several default plots are generated. Among them are the plots seen in Figure 2 and Figure 3 that show the oxygen and hydrogen mole fraction distribution, respectively, at a cell voltage of 0.8 V.
Polarization Curve
The following instructions reproduce the plot of the polarization curve for the SOFC (see Figure 5).
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Polarization Curve in the Label text field.
3
Click to expand the Title section. From the Title type list, choose Manual.
4
In the Title text area, type Polarization curve.
5
Locate the Plot Settings section. Select the x-axis label check box.
6
In the associated text field, type Average current density (A/m<sup>2</sup>).
7
Select the y-axis label check box.
8
In the associated text field, type V<sub>cell</sub> (V).
Global 1
1
Right-click Polarization Curve and choose Global.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
V_cell
V
Cell Voltage
4
Locate the x-Axis Data section. From the Parameter list, choose Expression.
5
In the Expression text field, type I_cell.
6
In the Polarization Curve toolbar, click  Plot.
Power vs. Current
Next, reproduce a plot showing the power output as a function of the cell voltage (Figure 6).
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Power vs. Current in the Label text field.
3
Locate the Title section. From the Title type list, choose Manual.
4
In the Title text area, type Total output power.
5
Locate the Plot Settings section. Select the x-axis label check box.
6
In the associated text field, type Average current density (A/m<sup>2</sup>).
7
Select the y-axis label check box.
8
In the associated text field, type Average Cell Power (W/m<sup>2</sup>).
Global 1
1
Right-click Power vs. Current and choose Global.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
V_cell*I_cell
W/m^2
Average Cell Power
4
Locate the x-Axis Data section. From the Parameter list, choose Expression.
5
In the Expression text field, type I_cell.
6
In the Power vs. Current toolbar, click  Plot.
Electrolyte Current Density
Next reproduce the plot in Figure 4 showing the current density in the unit cell at 0.8 V.
1
In the Home toolbar, click  Add Plot Group and choose 3D Plot Group.
2
In the Settings window for 3D Plot Group, type Electrolyte Current Density in the Label text field.
Surface 1
1
Right-click Electrolyte Current Density and choose Surface.
2
In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1)>Hydrogen Fuel Cell>Electrolyte current density vector - A/m²>fc.Ilz - Electrolyte current density vector, z component.
Selection 1
1
Right-click Surface 1 and choose Selection.
2
Select Boundary 9 only.
3
In the Electrolyte Current Density toolbar, click  Plot.
4
Click the  Zoom Extents button in the Graphics toolbar.