mass transport analysis of a high temperature pem fuel cell

Created in COMSOL Multiphysics 5.5

Ma s s T r a n s po r t Ana l y s i s o f a H i g h T empe r a t u r ePEM Fu e l C e l l

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Introduction

A high-temperature PEM based on phosphoric acid doped polymer membranes typically operates in a temperature window between 100°C and 200°C.

Water is produced on the cathode side due to the electrochemical reactions, but water may also be present on the anode side if the hydrogen fuel stream contains water. The latter may be the case if the hydrogen fuel stems from reforming of hydrocarbon fuels such as natural gas, alcohols or diesel.

Water build-up in the cell results in lower partial pressures of hydrogen and oxygen on the anode and cathode, respectively, and thereby also lowers the rate of the cell reactions.

The local water concentration in the cell may have a beneficial impact on the ion conductivity of the membrane, but if water condenses in the cell it may result in leaching of phosphoric acid out from the polymer matrix, irreversibly reducing the ion conducting capabilities of the membrane and the performance of the whole cell.

During normal operation above 100°C, water normally does not condensate in the cell, but knowledge about the water content in the porous electrodes during operation may also be desired in order to optimize shut-down procedures, since water may condensate when the temperature is lowered below 100°C.

This model example investigates the steady-state transport of reactants and water in a cell including both anode and cathode mass and momentum transport phenomena in the flow channels, gas diffusion layers (GDLs) and porous electrodes, as well as electrochemical currents in the GDLs, the porous electrodes and the polymer membrane.

2 | M A S S T R A N S P O R T A N A L Y S I S O F A H I G H T E M P E R A T U R E P E M F U E L C E L L

Model Definition

The geometry is shown in Figure 1.

Figure 1: Modeled geometry.

The following unknown variables are solved for in the model:

• — the electronic potential

• — the ionic potential

• wH2 — the hydrogen mass fraction (in the anode compartment)

• wH2Oa — the water mass fraction (in the anode compartment)

• wO2 — the oxygen mass fraction (in the cathode compartment)

• wH2Oc — the water mass fraction (in the cathode compartment)

• wN2 — the nitrogen mass fraction (in the cathode compartment)

• ua and uc — the velocity field vectors in the anode and cathode compartments

• pa and pc — the pressure in the anode and cathode compartments

This is done by coupling two Reacting Flow in Porous Media, Concentrated Species interfaces to one Secondary Current Distribution interface.

CATHODE SIDE

ANODE SIDEPorous electrode

Flow channel

GDL

GDL

Porous electrode

Membrane

OutletsFlow channel

Inlets

φs

φl


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

The Secondary Current Distribution interface is used to model the electrochemical currents using Ohm’s law and solving for in the GDLs, and in the porous electrodes, and in the electrolyte membrane. In the porous electrodes the local current densities depend on the ionic and electronic potentials, but also on local reactant concentrations.

On the anode, hydrogen oxidation occurs according to

(no water molecules are assumed to be involved in the proton transport), and the following local current density expression is used for the hydrogen oxidation reaction:

where pH2 is the local hydrogen partial pressure and pref the reference pressure of 1 atm. is the overpotential with respect to the equilibrium potential of the reaction at

atmospheric pressures of the reacting gases.

At the cathode, oxygen reacts together with the protons to form water according to

and the following current density expression is used for the oxygen reduction reaction:

where pO2 and pH2O are the local partial pressures of oxygen and water vapor, respectively.

The anode GDL boundaries facing the flow pattern ribs are set to zero electronic potential, and the corresponding boundaries at the cathode side are set to the cell potential. All other external boundaries are electrically isolating.

M A S S T R A N S F E R

wH2, wH2Oa, wO2, wH2Oc, and wH2Oc are solved for in the flow channels, GDLs and porous electrode using in the Maxwell-Stefan equations in two different Transport of Concentrated Species interfaces. One interface is used for each electrode compartment;

φs φs φlφl

H2 2H+

2e-+

ia i0 ref a, ,pH2

pref---------

αa a,RT-----------Fηa αc c,

RT----------Fηa–

exp–exp =

ηa

O2 4H+

4e-+ + 2H2O

ic i0 ref c, ,pH2O

pref-------------

2 αa c,RT-----------Fηc pO2

pref---------

αc c,RT----------Fηc–

exp–exp =


hydrogen and water species are present on the anode whereas oxygen, water and nitrogen are present on the anode.

Couplings are made to the Secondary Current Distribution interface for the mass sources and sinks.

At the channel inlets the mass fractions are specified, and outflow conditions are used at the channel outlets. All other external boundaries use zero flux conditions.

M O M E N T U M T R A N S F E R

u and p are modeled by the Navier-Stokes equations in the flow channels and the Brinkman equations for the porous gas diffusion layers (GDLs) and electrodes, using the Brinkman Equations interface.

Couplings for the density, velocity, pressure and net mass sources and sinks are made to the Transport of Concentrated Species interfaces by using Reacting Flow multiphysics nodes.

At the flow channel inlet boundaries, laminar inlet flow velocity profiles are specified, whereas a pressure is specified at the flow channel outlet boundaries. To model a multiple parallel channel configuration, symmetry boundary conditions are applied along the long sides of the GDLs and the porous electrodes. All other wall boundaries use no slip conditions.


Results and Discussion

Figure 2 shows the polarization plot of the cell.

Figure 2: Polarization plot.

Figure 3 shows the ionic current in the z direction at the center of the membrane for 0.4 V. In the y direction the current density is lower towards the outlet (due to lower reactant concentrations).In the x direction the currents density is highest in the region close to the channel, where the reactant concentrations are higher, but the current density


is reduced towards the very center of the channel. This is due to Ohmic drops in the GDLs.

Figure 3: Ionic current in the polymer membrane at 0.4 V.

Figure 4 shows the hydrogen and oxygen concentrations for the same voltage level. The oxygen concentration is significantly lower in the porous electrode and towards the end of the flow channel compared to the inlet level. For the anode the trend is the same but the hydrogen concentration level is more uniform.


Figure 4: Hydrogen concentration at the anode (top) and oxygen concentration at the cathode (bottom) in the cell at 0.4 V.

Figure 5 shows the water concentration in the cell for the same voltage level. The concentration increase due to water production at the cathode is much larger than the effect of removing hydrogen from the gas stream at the anode for these flow and current levels.


Figure 5: Water concentration on the anode (top) and cathode (bottom) in the cell at 0.4 V.

Reference

1. E.U. Ubong, Z. Shi, and X. Wang, “Three-Dimensional Modeling and Experimental Study of a High Temperature PBI-Based PEM Fuel Cell,” J. Electrochemical Soc., vol. 156, no. 10, pp. B1276–B1282, 2009.


Application Library path: Batteries_and_Fuel_Cells_Module/Fuel_Cells/ht_pem

Modeling Instructions

From the File menu, choose New.

N E W

In the New window, click Model Wizard.

M O D E L W I Z A R D

1 In the Model Wizard window, click 3D.

2 In the Select Physics tree, select Electrochemistry>

Primary and Secondary Current Distribution>Secondary Current Distribution (cd).

3 Click Add.

4 In the Select Physics tree, select Chemical Species Transport>

Reacting Flow in Porous Media>Transport of Concentrated Species.

5 Click Add.

6 In the Velocity field text field, type ua.

7 In the Velocity field components table, enter the following settings:

8 In the Pressure text field, type pa.

9 In the Added physics interfaces tree, select Transport of Concentrated Species (tcs).

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

11 In the Select Physics tree, select Chemical Species Transport>

Reacting Flow in Porous Media>Transport of Concentrated Species.

12 Click Add.

ua

va

wa

wH2

wH2Oa


13 In the Velocity field text field, type uc.

14 In the Velocity field components table, enter the following settings:

15 In the Pressure text field, type pc.

16 In the Added physics interfaces tree, select Transport of Concentrated Species (tcs2).

17 In the Number of species text field, type 3.

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

19 Click Study.

20 In the Select Study tree, select General Studies>Stationary.

21 Click Done.

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

Load the model parameters from a text file.

Parameters 11 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 ht_pem_parameters.txt.

G E O M E T R Y 1

Draw the channels, gas diffusion layers, porous electrodes and the membrane using rectangular blocks.

Block 1 (blk1)1 In the Geometry toolbar, click Block.

2 In the Settings window for Block, type Anode Channel in the Label text field.

3 Locate the Size and Shape section. In the Width text field, type W_ch.

uc

vc

wc

wO2

wH2Oc

wN2


4 In the Depth text field, type L.

5 In the Height text field, type H_ch.

6 Locate the Position section. In the x text field, type W_rib/2.

7 Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.

8 Click Build Selected.

Anode Channel 1 (blk2)1 Right-click Anode Channel and choose Duplicate.

2 In the Settings window for Block, type Anode GDL in the Label text field.

3 Locate the Size and Shape section. In the Width text field, type (W_ch+W_rib).

4 In the Height text field, type H_gdl.

5 Locate the Position section. In the x text field, type 0.

6 In the z text field, type H_ch.


Anode GDL 1 (blk3)1 Right-click Anode GDL and choose Duplicate.

2 In the Settings window for Block, type Anode Electrode in the Label text field.

3 Locate the Size and Shape section. In the Height text field, type H_electrode.

4 Locate the Position section. In the z text field, type H_ch+H_gdl.


Anode Electrode 1 (blk4)1 Right-click Anode Electrode and choose Duplicate.

2 In the Settings window for Block, type Membrane in the Label text field.

3 Locate the Size and Shape section. In the Height text field, type H_membrane.

4 Locate the Position section. In the z text field, type H_ch+H_gdl+H_electrode.


Membrane 1 (blk5)1 Right-click Membrane and choose Duplicate.

2 In the Settings window for Block, type Cathode Electrode in the Label text field.

3 Locate the Size and Shape section. In the Width text field, type W_ch+W_rib.

4 In the Height text field, type H_electrode.


5 Locate the Position section. In the z text field, type H_ch+H_gdl+H_electrode+H_membrane.


Cathode Electrode 1 (blk6)1 Right-click Cathode Electrode and choose Duplicate.

2 In the Settings window for Block, type Cathode GDL in the Label text field.

3 Locate the Size and Shape section. In the Height text field, type H_gdl.

4 Locate the Position section. In the z text field, type H_ch+H_gdl+H_electrode+H_membrane+H_electrode.


Cathode GDL 1 (blk7)1 Right-click Cathode GDL and choose Duplicate.

2 In the Settings window for Block, type Cathode Channel in the Label text field.

3 Locate the Size and Shape section. In the Width text field, type W_ch.

4 In the Height text field, type H_ch.

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

6 In the z text field, type H_ch+H_gdl+H_electrode+H_membrane+H_electrode+H_gdl.

7 Click Build All Objects.

D E F I N I T I O N S

Now create some additional selections of certain parts of the geometry to facilitate setting up the physics later on.

Explicit 11 In the Definitions toolbar, click Explicit.

2 In the Settings window for Explicit, type Anode Inlet in the Label text field.

3 Locate the Input Entities section. From the Geometric entity level list, choose Boundary.

4 Select Boundary 23 only.


2 In the Settings window for Explicit, type Cathode Inlet in the Label text field.





2 In the Settings window for Explicit, type Anode Outlet in the Label text field.




2 In the Settings window for Explicit, type Cathode Outlet in the Label text field.



Union 11 In the Definitions toolbar, click Union.

2 In the Settings window for Union, type Anode Compartment in the Label text field.

3 Locate the Input Entities section. Under Selections to add, click Add.

4 In the Add dialog box, in the Selections to add list, choose Anode Channel, Anode GDL, and Anode Electrode.

5 Click OK.

Union 21 In the Definitions toolbar, click Union.

2 In the Settings window for Union, type Cathode Compartment in the Label text field.

3 Locate the Input Entities section. Under Selections to add, click Add.

4 In the Add dialog box, in the Selections to add list, choose Cathode Electrode, Cathode GDL, and Cathode Channel.

5 Click OK.

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 ( C D )

Set up the model for the current distribution. There are no currents flowing in the gas channels.

Porous Electrode 11 In the Model Builder window, under Component 1 (comp1) right-click

Secondary Current Distribution (cd) and choose Porous Electrode.


2 In the Settings window for Porous Electrode, locate the Domain Selection section.

3 From the Selection list, choose Anode Electrode.

4 Locate the Electrolyte Current Conduction section. From the σl list, choose User defined. In the associated text field, type sigma_m.

5 In the εl text field, type eps_l.

6 Locate the Electrode Current Conduction section. From the σs list, choose User defined. In the associated text field, type sigma_gdl.

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

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

Porous Electrode Reaction 1.

2 In the Settings window for Porous Electrode Reaction, locate the Equilibrium Potential section.

3 From the Eeq list, choose Nernst equation.

4 In the Eeq, ref(T) text field, type Eeq_ref_H2.

The tcs.p_wH2 variable represents the partial pressure of hydrogen.

5 In the CR text field, type tcs.p_wH2/p_ref.

6 Locate the Stoichiometric Coefficients section. In the n text field, type 2.

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

8 From the Exchange current density type list, choose From Nernst Equation.

9 In the i0,ref(T) text field, type i0_ref_a.

10 Locate the Active Specific Surface Area section. In the av text field, type Av.

Use the Default Model Inputs node to set the Temperature at this stage for rest of the model. This node may be accessed by multiple physics nodes.

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

Default Model Inputs1 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.


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 ( C D )

Porous Electrode 21 In the Physics toolbar, click Domains and choose Porous Electrode.

2 In the Settings window for Porous Electrode, locate the Domain Selection section.

3 From the Selection list, choose Cathode Electrode.

4 Locate the Electrolyte Current Conduction section. From the σl list, choose User defined. In the associated text field, type sigma_m.

5 In the εl text field, type eps_l.

6 Locate the Electrode Current Conduction section. From the σs list, choose User defined. In the associated text field, type sigma_gdl.

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

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

Porous Electrode Reaction 1.

2 In the Settings window for Porous Electrode Reaction, locate the Equilibrium Potential section.

3 From the Eeq list, choose Nernst equation.

4 In the Eeq, ref(T) text field, type Eeq_ref_O2.

5 In the CR text field, type (tcs2.p_wH2Oc/p_ref)^2.

6 In the CO text field, type tcs2.p_wO2/p_ref.


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

9 From the Exchange current density type list, choose From Nernst Equation.

10 In the i0,ref(T) text field, type i0_ref_c.

11 In the αa text field, type 4-alpha_c.

12 Locate the Active Specific Surface Area section. In the av text field, type Av.

Electrode 1In the gas diffusion layers there are no electrochemical reactions occurring, and there is no electrolyte present. The current conduction in these domains are modeled using Electrode nodes.

1 In the Physics toolbar, click Domains and choose Electrode.


2 Select Domains 1 and 5 only.

3 In the Settings window for Electrode, locate the Electrode section.

4 From the σs list, choose User defined. In the associated text field, type sigma_gdl.

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

2 In the Settings window for Electrolyte, locate the Electrolyte section.

3 From the σl list, choose User defined. In the associated text field, type sigma_m.

Electric Ground 1The domain settings describing the physics of the electrochemical reactions and currents are now complete. Now ground the anode side of the cell and set the cathode side to the cell potential.

1 In the Physics toolbar, click Boundaries and choose Electric Ground.

2 Select Boundaries 3 and 33 only.

Electric Potential 11 In the Physics toolbar, click Boundaries and choose Electric Potential.

2 Select Boundaries 16 and 35 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 21 In the Physics toolbar, click Domains and choose Initial Values.


3 In the Settings window for Initial Values, locate the Initial Values section.

4 In the phis text field, type V_cell.

B R I N K M A N E Q U A T I O N S ( B R )

1 In the Model Builder window, under Component 1 (comp1) click Brinkman Equations (br).

2 In the Settings window for Brinkman Equations, type Brinkman Equations - Anode in the Label text field.

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

4 Locate the Physical Model section. Clear the Neglect inertial term (Stokes flow) check box.

Define the pressure reference level in the interface properties.


5 In the pref text field, type p_ref.

Fluid and Matrix Properties 11 In the Model Builder window, under Component 1 (comp1)>Brinkman Equations -

Anode (br) click Fluid and Matrix Properties 1.

2 In the Settings window for Fluid and Matrix Properties, locate the Fluid Properties section.

3 From the μ list, choose User defined. In the associated text field, type mu_anode.

4 Locate the Porous Matrix Properties section. From the εp list, choose User defined. In the associated text field, type eps_gdl.

5 From the κ list, choose User defined. In the associated text field, type kappa_gdl.

Wall 11 Click the Show More Options button in the Model Builder toolbar.

2 In the Show More Options dialog box, in the tree, select the check box for the node Physics>Advanced Physics Options.

3 Click OK.

4 In the Model Builder window, click Wall 1.

5 In the Settings window for Wall, click to expand the Constraint Settings section.

6 From the Apply reaction terms on list, choose All physics (symmetric).

Fluid and Matrix Properties 21 In the Physics toolbar, click Domains and choose Fluid and Matrix Properties.

2 In the Settings window for Fluid and Matrix Properties, locate the Domain Selection section.


4 Locate the Fluid Properties section. From the μ list, choose User defined. In the associated text field, type mu_anode.

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

6 From the κ list, choose User defined. In the associated text field, type kappa_cl.

Fluid Properties 11 In the Physics toolbar, click Domains and choose Fluid Properties.

2 Select Domain 6 only.

3 In the Settings window for Fluid Properties, locate the Fluid Properties section.

4 From the μ list, choose User defined. In the associated text field, type mu_anode.


Inlet 1Now set up the inlet and outlet conditions on the corresponding boundaries.

1 In the Physics toolbar, click Boundaries and choose Inlet.

2 In the Settings window for Inlet, locate the Boundary Selection section.

3 From the Selection list, choose Anode Inlet.

4 Locate the Boundary Condition section. From the list, choose Fully developed flow.

5 Locate the Fully Developed Flow section. In the Uav text field, type U_in_anode.

Outlet 11 In the Physics toolbar, click Boundaries and choose Outlet.

2 In the Settings window for Outlet, locate the Boundary Selection section.

3 From the Selection list, choose Anode Outlet.

4 Locate the Pressure Conditions section. Select the Normal flow check box.

5 Click to expand the Constraint Settings section.

Symmetry 11 In the Physics toolbar, click Boundaries and choose Symmetry.

2 Select Boundaries 1, 4, 36, and 37 only.

T R 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 ( T C S )

1 In the Model Builder window, under Component 1 (comp1) click Transport of Concentrated Species (tcs).

2 In the Settings window for Transport of Concentrated Species, type Transport of Concentrated Species - Anode in the Label text field.

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

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

Stefan.

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

Transport Properties 11 In the Model Builder window, under Component 1 (comp1)>

Transport of Concentrated Species - Anode (tcs) click Transport Properties 1.

2 In the Settings window for Transport Properties, locate the Density section.

3 In the MwH2 text field, type MH2.

4 In the MwH2Oa text field, type MH2O.


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

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


3 From the Mixture specification list, choose Mole fractions.

4 In the x0,wH2 text field, type x_H2_in.

Porous Media Transport Properties 1Set up the fluid flow and mass transport for the cathode compartment in a similar way as for the anode.

1 In the Model Builder window, click Porous Media Transport Properties 1.


3 In the Settings window for Porous Media Transport Properties, locate the Density section.

4 In the MwH2 text field, type MH2.

5 In the MwH2Oa text field, type MH2O.

Set up the diffusivities. Use the values for a pure gas phase, a correction for the porous domains will be applied by the Porous Matrix nodes which will be added later.


Use a Bruggeman correction to take into account the impact of the porous matrix on gas phase mass transport.

7 From the Effective diffusivity model list, choose Bruggeman model.

Porous Electrode Coupling 1Use a Porous Electrode Coupling domain node in the porous electrodes to create a mass sink coupled to the Secondary Current Distribution interface. This will account for the hydrogen leaving the gas phase due to the electrochemical reactions.

1 In the Physics toolbar, click Domains and choose Porous Electrode Coupling.

2 In the Settings window for Porous Electrode Coupling, locate the Domain Selection section.

Species 1 Species 2 Diffusion coefficient (m^2/s)

wH2 wH2Oa D_H2_H2O


wH2 wH2Oa D_H2_H2O



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

Reaction Coefficients 1.

2 In the Settings window for Reaction Coefficients, locate the Model Inputs section.

3 From the iv list, choose Local current source, Porous Electrode Reaction 1 (cd/pce1/per1).


5 In the νwH2 text field, type 1.



Inflow 11 In the Physics toolbar, click Boundaries and choose Inflow.

2 In the Settings window for Inflow, locate the Boundary Selection section.

3 From the Selection list, choose Anode Inlet.

4 Locate the Inflow section. From the Mixture specification list, choose Mole fractions.

5 In the x0,wH2 text field, type x_H2_in.

6 Click to expand the Constraint Settings section. From the Apply reaction terms on list, choose All physics (symmetric).

Outflow 11 In the Physics toolbar, click Boundaries and choose Outflow.

2 In the Settings window for Outflow, locate the Boundary Selection section.

3 From the Selection list, choose Anode Outlet.

B R I N K M A N E Q U A T I O N S 2 ( B R 2 )

1 In the Model Builder window, under Component 1 (comp1) click Brinkman Equations 2 (br2).

2 In the Settings window for Brinkman Equations, type Brinkman Equations 2 - Cathode in the Label text field.

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


4 Locate the Physical Model section. Clear the Neglect inertial term (Stokes flow) check box.

Define the pressure reference level in the interface properties.

5 In the pref text field, type p_ref.

Fluid and Matrix Properties 11 In the Model Builder window, under Component 1 (comp1)>Brinkman Equations 2 -

Cathode (br2) click Fluid and Matrix Properties 1.

2 In the Settings window for Fluid and Matrix Properties, locate the Fluid Properties section.

3 From the μ list, choose User defined. In the associated text field, type mu_cathode.

4 Locate the Porous Matrix Properties section. From the εp list, choose User defined. In the associated text field, type eps_gdl.

5 From the κ list, choose User defined. In the associated text field, type kappa_gdl.

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

2 In the Settings window for Wall, locate the Constraint Settings section.

3 From the Apply reaction terms on list, choose All physics (symmetric).

Fluid Properties 11 In the Physics toolbar, click Domains and choose Fluid Properties.

2 Select Domain 7 only.

3 In the Settings window for Fluid Properties, locate the Fluid Properties section.

4 From the μ list, choose User defined. In the associated text field, type mu_cathode.

Fluid and Matrix Properties 21 In the Physics toolbar, click Domains and choose Fluid and Matrix Properties.

2 In the Settings window for Fluid and Matrix Properties, locate the Domain Selection section.


4 Locate the Fluid Properties section. From the μ list, choose User defined. In the associated text field, type mu_cathode.

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

6 From the κ list, choose User defined. In the associated text field, type kappa_cl.




Inlet 11 In the Physics toolbar, click Boundaries and choose Inlet.

2 In the Settings window for Inlet, locate the Boundary Selection section.

3 From the Selection list, choose Cathode Inlet.

4 Locate the Boundary Condition section. From the list, choose Fully developed flow.

5 Locate the Fully Developed Flow section. In the Uav text field, type U_in_cathode.

Outlet 11 In the Physics toolbar, click Boundaries and choose Outlet.

2 In the Settings window for Outlet, locate the Boundary Selection section.

3 From the Selection list, choose Cathode Outlet.

4 Locate the Pressure Conditions section. Select the Normal flow check box.

T R 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 ( T C S 2 )

1 In the Model Builder window, under Component 1 (comp1) click Transport of Concentrated Species 2 (tcs2).

2 In the Settings window for Transport of Concentrated Species, type Transport of Concentrated Species 2 - Cathode in the Label text field.


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

Stefan.

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

Transport Properties 11 In the Model Builder window, under Component 1 (comp1)>

Transport of Concentrated Species 2 - Cathode (tcs2) click Transport Properties 1.

2 In the Settings window for Transport Properties, locate the Density section.

3 In the MwO2 text field, type MO2.

4 In the MwH2Oc text field, type MH2O.

5 In the MwN2 text field, type MN2.



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


3 From the Mixture specification list, choose Mole fractions.

4 In the x0,wO2 text field, type x_O2_in.

5 In the x0,wH2Oc text field, type x_H2O_in.

Porous Media Transport Properties 11 In the Model Builder window, click Porous Media Transport Properties 1.

2 In the Settings window for Porous Media Transport Properties, locate the Domain Selection section.

3 From the Selection list, choose Cathode GDL.

4 Locate the Density section. In the MwO2 text field, type MO2.

5 In the MwH2Oc text field, type MH2O.

6 In the MwN2 text field, type MN2.


8 From the Effective diffusivity model list, choose Bruggeman model.


10 In the list, select 7.

11 Click Remove from Selection.



wO2 wH2Oc D_O2_H2O

wO2 wN2 D_O2_N2

wH2Oc wN2 D_N2_H2O


wO2 wH2Oc D_O2_H2O

wO2 wN2 D_O2_N2

wH2Oc wN2 D_N2_H2O


Porous Electrode Coupling 11 In the Physics toolbar, click Domains and choose Porous Electrode Coupling.

2 In the Settings window for Porous Electrode Coupling, locate the Domain Selection section.


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

Reaction Coefficients 1.

2 In the Settings window for Reaction Coefficients, locate the Model Inputs section.

3 From the iv list, choose Local current source, Porous Electrode Reaction 1 (cd/pce2/per1).


5 In the νwO2 text field, type -1.

6 In the νwH2Oc text field, type 2.



Inflow 11 In the Physics toolbar, click Boundaries and choose Inflow.

2 In the Settings window for Inflow, locate the Boundary Selection section.

3 From the Selection list, choose Cathode Inlet.

4 Locate the Inflow section. From the Mixture specification list, choose Mole fractions.

5 In the x0,wO2 text field, type x_O2_in.

6 In the x0,wH2Oc text field, type x_H2O_in.

Outflow 11 In the Physics toolbar, click Boundaries and choose Outflow.

2 In the Settings window for Outflow, locate the Boundary Selection section.

3 From the Selection list, choose Cathode Outlet.

M E S H 1

First create a mapped 2D mesh in the plane normal to the channel direction, then sweep this mesh in the channel direction.


Edge 11 In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and choose

More Operations>Edge.

2 Select Edges 3, 17, 33, 36, 48, and 51 only.

Size 11 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_ch/8.

Edge 21 In the Model Builder window, right-click Mesh 1 and choose More Operations>Edge.

2 Select Edges 13 and 65 only.

Distribution 11 Right-click Edge 2 and choose Distribution.

2 In the Settings window for Distribution, locate the Distribution section.

3 From the Distribution type list, choose Predefined.

4 In the Number of elements text field, type 8.

5 In the Element ratio text field, type 4.

Edge 31 In the Model Builder window, under Component 1 (comp1)>Mesh 1 right-click Edge 2 and

choose Duplicate.

2 In the Settings window for Edge, locate the Edge Selection section.

3 Click Clear Selection.


Distribution 11 In the Model Builder window, expand the Edge 3 node, then click Distribution 1.


3 Select the Reverse direction check box.










Larger concentration gradients are expected in the cathode catalyst layer (due to the lower diffusion coefficient of oxygen), therefore use a higher element density in the cathode.









2 Select Edges 29, 34, 45, and 49 only.



3 From the Distribution type list, choose Predefined.


5 In the Element ratio text field, type 2.


6 Select the Symmetric distribution check box.

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

2 Select Boundaries 2, 5, 8, 11, 14, 23, and 27 only.

Swept 1Right-click Mesh 1 and choose Swept.

Size 11 In the Model Builder window, 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_ch.


D E F I N I T I O N S

Add a domain probe for the integral of the electrochemical current density to get a polarization plot automatically during the parametric solver process. You can then use this probe also for creating a polarization curve during postprocessing.

Domain Probe 1 (dom1)1 In the Definitions toolbar, click Probes and choose Domain Probe.

2 In the Settings window for Domain Probe, locate the Probe Type section.

3 From the Type list, choose Integral.

4 Locate the Source Selection section. From the Selection list, choose Anode Electrode.

5 Locate the Expression section. In the Expression text field, type cd.iv_per1/((W_ch+W_rib)*L)/1e4.

S T U D Y 1

The problem is now ready for solving. Solve for the current distribution and flow first in three individually study steps. Then use an auxiliary sweep for the whole problem to solve for a range of potentials and simulate a polarization plot.

Step 1: Stationary1 In the Model Builder window, under Study 1 click Step 1: Stationary.

2 In the Settings window for Stationary, locate the Physics and Variables Selection section.


3 In the table, clear the Solve for check box for Reacting Flow in Porous Media: Reacting Flow

in Porous Media and Reacting Flow in Porous Media: Reacting Flow in Porous Media 2 .

4 In the table, enter the following settings:

Stationary 21 In the Study toolbar, click Study Steps and choose Stationary>Stationary.


3 In the table, clear the Solve for check box for Secondary Current Distribution: Secondary

Current Distribution (cd) and Reacting Flow in Porous Media: Reacting Flow in Porous Media

2 .




3 In the table, clear the Solve for check box for Secondary Current Distribution: Secondary

Current Distribution (cd) and Reacting Flow in Porous Media: Reacting Flow in Porous Media

.



2 In the Settings window for Stationary, click to expand the Study Extensions section.

3 Select the Auxiliary sweep check box.

Multiphysics couplings Solve for

Reacting Flow 1 (rf1)









4 Click Add.

Use the range operator to create a list of potentials from 0.95 to 0.4 V, using steps -50 mV for the first steps, and then -100 mV towards the end.


Solution 1 (sol1)Modify the order of the segragated steps of the last study step of the default solver sequence. By solving for the potential variables first the computation time gets shorter.

1 In the Study toolbar, click Show Default Solver.

2 In the Model Builder window, expand the Solution 1 (sol1) node.

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

Solution 1 (sol1)>Stationary Solver 4>Segregated 1 node.

4 Right-click Mass fractions (2) and choose Move Up.




8 In the Model Builder window, click Study 1.

9 In the Settings window for Study, locate the Study Settings section.

10 Clear the Generate default plots check box.

11 In the Study toolbar, click Compute.

R E S U L T S

Probe Plot Group 1Modify the probe plot as follows to reproduce the polarization plot in Figure 2.

1 In the Model Builder window, expand the Results>Probe Plot Group 1 node, then click Probe Plot Group 1.

2 In the Settings window for 1D Plot Group, click to expand the Title section.

3 From the Title type list, choose Manual.

4 In the Title text area, type Polarization plot.

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

Parameter name Parameter value list Parameter unit

V_cell (Cell voltage) range(0.95,-0.05,0.85) range(0.8,-0.1,0.4)


6 In the associated text field, type Cell average current density (A/cm<sup>2</sup>).

7 Select the y-axis label check box.

8 In the associated text field, type Cell voltage (V).

9 Select the Flip the x- and y-axes check box.

10 Locate the Legend section. Clear the Show legends check box.

11 In the Probe Plot Group 1 toolbar, click Plot.

DatasetsNext, reproduce the ionic current plot of Figure 3. Begin by creating a Cut Plane data set.

Cut Plane 11 In the Results toolbar, click Cut Plane.

2 In the Settings window for Cut Plane, locate the Plane Data section.

3 From the Plane list, choose xy-planes.

4 In the z-coordinate text field, type H_ch+H_gdl+H_electrode+H_membrane/2.

2D Plot Group 2In the Results toolbar, click 2D Plot Group.

Contour 11 Right-click 2D Plot Group 2 and choose Contour.

2 In the Settings window for Contour, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Model>Component 1>

Secondary Current Distribution>Electrolyte current density vector - A/m²>cd.Ilz -

Electrolyte current density vector, z component.

3 Locate the Levels section. In the Total levels text field, type 10.

4 In the 2D Plot Group 2 toolbar, click Plot.

3D Plot Group 3The remaining instructions reproduce the hydrogen and water concentration plots shown in Figure 4 and Figure 5.

In the Home toolbar, click Add Plot Group and choose 3D Plot Group.

Surface 11 Right-click 3D Plot Group 3 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 Model>Component 1>


Transport of Concentrated Species - Anode>Species wH2>tcs.c_wH2 -

Molar concentration - mol/m³.


Compare the result with the upper plot in Figure 4.

3D Plot Group 4In the Home toolbar, click Add Plot Group and choose 3D Plot Group.



Transport of Concentrated Species 2 - Cathode>Species wO2>tcs2.c_wO2 -



Compare with the lower plot in Figure 4.




Transport of Concentrated Species - Anode>Species wH2Oa>tcs.c_wH2Oa -



The plot should look like the upper one in Figure 5.




Transport of Concentrated Species 2 - Cathode>Species wH2Oc>tcs2.c_wH2Oc -




Compare with the lower plot in Figure 5.


mass transport analysis of a high temperature pem fuel cell

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