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Electrochemical Capacitor with Porous Electrodes
Introduction
This model demonstrates how to set up an electrochemical supercapacitor using the Tertiary Current Distribution, Nernst–Planck (tcd) interface.
The 1D isothermal model includes the following processes:
•
Electronic current conduction in the electrodes
•
Ionic charge transport in the porous electrodes and separator
•
Double layer capacitance in the porous electrodes
The model is based on a paper by M.W. Verbrugge (Ref. 1). In the paper, the authors analyze the effect of the microstructure of the porous electrodes on the performance of a supercapacitor with a relatively high specific energy.
Model Definition
This example models the electrochemical capacitor cross section in 1D, which implies that edge effects in the length and height of the capacitor cell are neglected. The example uses the following domains:
•
Two porous electrode (right and left): 50 μm
•
Separator (flooded with electrolyte): 25 μm
Domain Conditions
The model solves for the potentials in the electrode and the binary nonaqueous electrolyte phases, in combination with a concentration dependent variable for one of the ions. The concentration for the other ion is calculated from the condition of electroneutrality.
The electric potential in the electron conducting phase, , is calculated using a charge balance based on Ohm’s law. The migrative and diffusive charge and species transport in the electrolyte are modeled using the Nernst–Planck equations, assuming electroneutrality.
The double-layer charging is defined as a source term in the porous electrodes based on the time derivative of the potential jump over the double layer according to
(1)
where av,dl (m2/m3) is the active specific surface area for double-layer charging, and Cdl is the double-layer capacitance (F/m2).
The effective electrical conductivity in porous electrodes σseff, is defined by taking porosity and tortuosity into account through the expression
where ε is the porosity parameter and τ is the tortuosity parameter. The effective ionic conductivities in the porous electrodes and the separator are also calculated similarly.
Similarly, the effective diffusion coefficient, Deff, for the electrolyte salt corrects for the porosity and the tortuosity through
The ionic charge balances are modeled according to the electroneutrality condition in the bulk of the binary 1:1 electrolyte (Ref. 1). The mobilities, ui, of the ionic species under migration are calculated via the Nernst–Einstein relation:
Boundary Conditions
For the electronic current balance, a potential of 0 V is set on the left electrode’s current collector/feeder boundary. At the right electrode current collector/feeder, either the current density or power is specified. The inner boundaries facing the separator are insulating for electric currents.
For Study 1, current density is calculated by the use of a global equation node. For Study 2, a constant power charging is specified at the right porous electrode boundary. The initial values are specified using the rest potentials. Parametric sweep is used to simulate different values of applied power and resting potential.
Events Interface - Constant Current Charge - Constant Voltage Discharge
The Events interface is used to simulate the load cycle comprising CC charge – CV discharge source. The Event Sequence feature is used to set up events for the CC step, the CV step, and the rest period.
Results and Discussion
Simulations of constant current (CC) charge-constant voltage (CV) discharge are performed in Study 1.
Current and Voltage Profile for CC-Charge CV-Discharge
The capacitor is charged to 1.8 V at a constant current of 100 A followed by constant voltage for 5 s and a rest stage of 180 s. This procedure is repeated while incrementing the maximum voltage (V_max) by 0.2 V until it reaches 2.45 V. Figure 1 shows the current-voltage response for times until the maximum voltage of 2.45 V is reached.
Voltage and Current Profiles for CC-Charge CV-Discharge Cycle
Figure 1: Voltage and current profiles for the events-based load profile.
Figure 2 shows current and voltage profiles overlapped to depict each cycle between constant current charges for incremented maximum voltage values. To achieve the overlapping, filters have been used for plotting the current and voltage profile (see Modeling Instructions).
Voltage and Current Profiles for CC Charging-CV Discharge Overlapped
Figure 2: Overlapped current and voltage profiles for the event-based load profile.
Study 2: Constant Power Charging
Figure 3 shows the double-layer current for the constant power discharge for the dimensionless length of the electrochemical cell. Figure 4 shows the charge density distribution over the dimensionless length of the electrochemical cell for constant power charge. Figure 3 and Figure 4 can collectively be seen as the indicator for electrode utilization. They show that for a higher cell power, there is a nonuniform charge distribution and poor electrode utilization. This, in turn, can lead to poor capacitor design and underutilization of electrodes despite including porous structure for more active area.
Figure 3: Double layer current source for constant power charging.
Figure 4: Charge density plotted over the dimensionless length of the cell.
Figure 5 shows the current and voltage profiles for different applied power parameters. For the higher charging power, the voltage change is larger.
Figure 5: Current and voltage profiles for constant voltage charging.
Notes About the COMSOL Implementation
Depositing-Dissolving Species
The charge density for the double layer kinetics can be monitored using the depositing-dissolving species and nonfaradaic reaction nodes. The reaction rate for the nonfaradaic process (capacitor charge discharge) is set proportional to the double layer current inside the porous electrode.
Porous Electrode Area and Tortuosity
The area of the porous electrodes parameter used in the model is calculated from the nominal capacitance of 3,500 F as given in Ref. 1. The result reported in Ref. 1 for the constant power can be reproduced by changing the specific area parameter to 27,470 cm2.
The value for electrode tortuosity is not reported in the reference and is assumed to be unity in the current study. For reproduction of the results for constant power, set the tortuosity 2.3.
Reference
1. M.W. Verbrugge and P. Liu, ‘‘Microstructural Analysis and Mathematical Modeling of Electric Double-Layer Supercapacitors”, J. Electrochem. Soc., vol. 152, no. 5, pp. D79–D87, 2005.
Application Library path: Battery_Design_Module/Electrochemical_Capacitors/electrochemical_capacitor_porous_electrodes
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  1D.
2
In the Select Physics tree, select Electrochemistry > Tertiary Current Distribution, Nernst–Planck > Tertiary, Electroneutrality (tcd).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies > Time Dependent.
6
Click  Done.
Global Definitions
Parameters : Electrochemical Cell
Import the parameter file for the electrochemical cell.
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 electrochemical_capacitor_porous_electrodes_cell_parameters.txt.
5
In the Label text field, type Parameters : Electrochemical Cell.
Parameters : Load Profile
1
In the Home toolbar, click  Parameters and choose Add > Parameters.
To this Parameters node, import the file containing the parameters for the different load profiles.
2
In the Settings window for Parameters, type Parameters : Load Profile in the Label text field.
3
Locate the Parameters section. Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file electrochemical_capacitor_porous_electrodes_load_parameters.txt.
Geometry 1
Interval 1 (i1)
1
In the Model Builder window, expand the Component 1 (comp1) > Definitions node.
2
Right-click Component 1 (comp1) > Geometry 1 and choose Interval.
3
In the Settings window for Interval, locate the Interval section.
4
From the Specify list, choose Interval lengths.
5
In the table, enter the following settings:
Lengths (m)
L_elec
L_sep
L_elec
6
Click  Build All Objects.
Tertiary Current Distribution, Nernst–Planck (tcd)
Species Charges 1
1
In the Model Builder window, under Component 1 (comp1) > Tertiary Current Distribution, Nernst–Planck (tcd) click Species Charges 1.
2
In the Settings window for Species Charges, locate the Charge section.
3
In the zc1 text field, type 1.
4
In the zc2 text field, type -1.
Electrolyte 1
1
In the Model Builder window, click Electrolyte 1.
2
In the Settings window for Electrolyte, locate the Diffusion section.
3
In the Dc1 text field, type D.
4
In the Dc2 text field, type D.
Initial Values 1
1
In the Model Builder window, click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Values section.
3
In the c2 text field, type c_bulk.
4
In the phis text field, type V_init.
Now set up the capacitive porous electrodes in the physics. Start by defining the transport and microstructural properties for these electrodes. Note that we will use only one porous domain node to define both electrodes. To monitor the charge flux, add a depositing-dissolving species (c1_q) to the porous electrode node.
Porous Electrode 1
1
In the Physics toolbar, click  Domains and choose Porous Electrode.
2
Select Domains 1 and 3 only.
3
In the Settings window for Porous Electrode, locate the Diffusion section.
4
In the Dc1 text field, type D.
5
In the Dc2 text field, type D.
6
Locate the Electrode Current Conduction section. From the σs list, choose User defined. In the associated text field, type sigma_s.
7
Locate the Porous Matrix Properties section. In the εs text field, type 1-eps_el.
8
In the εl text field, type eps_el.
9
Locate the Effective Transport Parameter Correction section. From the Diffusion list, choose Tortuosity.
10
Locate the Porous Matrix Properties section. In the τl text field, type tau_electrolyte.
11
Locate the Effective Transport Parameter Correction section. From the Electric conductivity list, choose Tortuosity.
12
Locate the Porous Matrix Properties section. In the τs text field, type tau_electrode.
13
Click to expand the Dissolving–Depositing Species section. Click  Add.
14
In the table, enter the following settings:
Species
Density (kg/m^3)
Molar mass (kg/mol)
c1_q
8960
0.06355
15
Clear the Add volume change to electrode volume fraction checkbox.
16
Clear the Subtract volume change from electrolyte volume fraction checkbox.
Porous Matrix Double Layer Capacitance 1
1
In the Physics toolbar, click  Attributes and choose Porous Matrix Double Layer Capacitance.
2
In the Settings window for Porous Matrix Double Layer Capacitance, locate the Porous Matrix Double Layer Capacitance section.
3
In the Cdl text field, type Cdl.
4
In the av,dl text field, type Av.
5
Locate the Stoichiometric Coefficients section. In the νc2 text field, type 0.5.
Porous Electrode Reaction 1
Since, there is no Faradaic reaction taking place at the porous electrodes, disable the porous electrode reaction.
In the Model Builder window, right-click Porous Electrode Reaction 1 and choose Disable.
Porous Electrode 1
In the Model Builder window, click Porous Electrode 1.
Nonfaradaic Reactions 1
1
In the Physics toolbar, click  Attributes and choose Nonfaradaic Reactions.
2
In the Settings window for Nonfaradaic Reactions, locate the Reaction Rate section.
3
In the Reaction rate for dissolving–depositing species table, enter the following settings:
Species
Reaction rate (mol/(m^3*s))
c1_q
tcd.ivdl/F_const
Separator 1
1
In the Physics toolbar, click  Domains and choose Separator.
2
Select Domain 2 only.
3
In the Settings window for Separator, locate the Diffusion section.
4
In the Dc1 text field, type D.
5
In the Dc2 text field, type D.
6
Locate the Porous Matrix Properties section. In the εl text field, type eps_sep.
7
Locate the Effective Transport Parameter Correction section. From the Diffusion list, choose Tortuosity.
8
Locate the Porous Matrix Properties section. In the τl text field, type tau_sep.
Electric Ground 1
1
In the Model Builder window, expand the Separator 1 node.
2
Right-click Tertiary Current Distribution, Nernst–Planck (tcd) and choose Electrode Phase > Electric Ground.
3
Select Boundary 1 only.
Electrode Current Density 1
Use the electrode current density to describe the load profile at the boundary.
1
In the Physics toolbar, click  Boundaries and choose Electrode Current Density.
2
Select Boundary 4 only.
3
In the Settings window for Electrode Current Density, locate the Electrode Current Density section.
4
In the in,s text field, type i_app_ch/A_cell.
Definitions
Define the integration operator at Boundary 4 to be used in the Events interface.
Integration 1 (intop1)
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Integration.
2
In the Settings window for Integration, locate the Source Selection section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundary 4 only.
5
In the Operator name text field, type right_el.
Component 1 (comp1)
We will use the Events interface to set up the load cycle for the electrochemical cell. The Event Sequence feature will be used to configure the three parts of the cycle.
Add Physics
1
In the Home toolbar, click  Add Physics to open the Add Physics window.
2
Go to the Add Physics window.
3
In the tree, select Mathematics > ODE and DAE Interfaces > Events (ev).
4
Click the Add to Component 1 button in the window toolbar.
5
In the Home toolbar, click  Add Physics to close the Add Physics window.
Events (ev)
Discrete States 1
1
In the Physics toolbar, click  Global and choose Discrete States.
2
In the Settings window for Discrete States, locate the Discrete States section.
3
In the table, enter the following settings:
Name
Initial value (u0)
Description
V_max
1.8
Maximum voltage
Charge-discharge cycle
1
In the Physics toolbar, click  Global and choose Event Sequence.
2
In the Settings window for Event Sequence, type Charge-discharge cycle in the Label text field.
3
Locate the Sequence Control section. Select the Loop checkbox.
Constant current
1
In the Model Builder window, expand the Charge-discharge cycle node, then click Sequence Member 1.
2
In the Settings window for Sequence Member, type Constant current in the Label text field.
3
Locate the Sequence Member section. In the Discrete state name text field, type CC.
4
In the End condition expression (>0) text field, type comp1.right_el(phis)-V_max[V].
Charge-discharge cycle
In the Model Builder window, click Charge-discharge cycle.
Constant voltage
1
In the Physics toolbar, click  Attributes and choose Sequence Member.
2
In the Settings window for Sequence Member, type Constant voltage in the Label text field.
3
Locate the Sequence Member section. In the Discrete state name text field, type CV.
4
From the End condition list, choose Duration.
5
In the Duration text field, type t_cv.
Charge-discharge cycle
In the Model Builder window, click Charge-discharge cycle.
Rest
1
In the Physics toolbar, click  Attributes and choose Sequence Member.
Set the reinitialization value for V_max to V_max+0.2 to ramp up the maximum potential by 0.2 V at the end of each cycle.
2
In the Settings window for Sequence Member, type Rest in the Label text field.
3
Locate the Sequence Member section. In the Discrete state name text field, type REST.
4
From the End condition list, choose Duration.
5
In the Duration text field, type t_rest.
6
Locate the Reinitialization section. In the table, enter the following settings:
 
Expression
V_max
V_max+0.2
Tertiary Current Distribution, Nernst–Planck (tcd)
Set up the global equation for the applied current.
1
Click the  Show More Options button in the Model Builder toolbar.
2
In the Show More Options dialog, select Physics > Equation Contributions in the tree.
3
In the tree, select the checkbox for the node Physics > Equation Contributions.
4
Click OK.
Global Equations 1 (ODE1)
1
In the Physics toolbar, click  Global and choose Global Equations.
2
In the Settings window for Global Equations, locate the Global Equations section.
3
In the table, enter the following settings:
Name
f(u,ut,utt,t) (1)
Initial value (u_0) (1)
Initial value (ut_0) (1/s)
i_app_ch
(CC==1)*(i_app_ch-i_app)/i_app+(CV==1)*(right_el(phis)-V_min)/V_min+(REST==1)*(i_app_ch)/i_app
i_app
0
4
Locate the Units section. Click  Define Dependent Variable Unit.
5
In the Dependent variable quantity table, enter the following settings:
Dependent variable quantity
Unit
Custom unit
A
Electrode Current Density 1, Global Equations 1 (ODE1)
1
In the Model Builder window, under Component 1 (comp1) > Tertiary Current Distribution, Nernst–Planck (tcd), Ctrl-click to select Electrode Current Density 1 and Global Equations 1 (ODE1).
2
Right-click and choose Group.
Constant Current Charge/Constant Voltage Discharge
In the Settings window for Group, type Constant Current Charge/Constant Voltage Discharge in the Label text field.
Mesh 1
Define the user-controlled mesh.
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Sequence Type section.
3
From the list, choose User-controlled mesh.
Size 1
1
In the Model Builder window, right-click Edge 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
From the Selection list, choose All boundaries.
5
Locate the Element Size section. From the Predefined list, choose Extremely fine.
6
Click  Build All.
Study 1: CC Charge CV Discharge
Set up the time-dependent study for the electrochemical cell for the load cycle defined as events.
1
In the Model Builder window, click Study 1.
2
In the Settings window for Study, type Study 1: CC Charge CV Discharge in the Label text field.
3
Locate the Study Settings section. Clear the Generate default plots checkbox.
Step 1: Time Dependent
1
In the Model Builder window, under Study 1: CC Charge CV Discharge click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, locate the Study Settings section.
3
From the Time unit list, choose min.
4
In the Output times text field, type 0 40.
Solution 1 (sol1)
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: CC Charge CV Discharge > Solver Configurations > Solution 1 (sol1) > Dependent Variables 1 node, then click Global Equations 1 (comp1.ODE1).
4
In the Settings window for State, locate the Scaling section.
5
From the Method list, choose Manual.
6
In the Scale text field, type i_app.
7
In the Model Builder window, under Study 1: CC Charge CV Discharge > Solver Configurations > Solution 1 (sol1) click Time-Dependent Solver 1.
8
In the Settings window for Time-Dependent Solver, locate the General section.
9
From the Times to store list, choose Steps taken by solver.
10
Click to expand the Time Stepping section. From the Steps taken by solver list, choose Strict.
11
Select the Initial step checkbox.
12
In the Event tolerance text field, type 0.001.
13
Click to expand the Results While Solving section. Right-click Study 1: CC Charge CV Discharge > Solver Configurations > Solution 1 (sol1) > Time-Dependent Solver 1 and choose Stop Condition.
Add the stop condition for the time-dependent solver to stop at 2.45 V.
14
In the Settings window for Stop Condition, locate the Stop Expressions section.
15
Click  Add.
16
In the table, enter the following settings:
Stop expression
Stop if
Active
Description
comp1.V_max>2.45
True (>=1)
√
Stop expression 1
17
Locate the Output at Stop section. From the Add solution list, choose Steps before and after stop.
18
Clear the Add information checkbox.
19
In the Study toolbar, click  Compute.
Results
Voltage and Current Profile for CC Charge - CV Discharge
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Voltage and Current Profile for CC Charge - CV Discharge in the Label text field.
Current
1
Right-click Voltage and Current Profile for CC Charge - CV Discharge and choose Global.
2
In the Settings window for Global, type Current in the Label text field.
3
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
comp1.i_app_ch
A
Cell Current
4
Locate the x-Axis Data section. From the Unit list, choose s.
5
In the Voltage and Current Profile for CC Charge - CV Discharge toolbar, click  Plot.
Potential
1
In the Model Builder window, right-click Voltage and Current Profile for CC Charge - CV Discharge and choose Point Graph.
2
In the Settings window for Point Graph, type Potential in the Label text field.
3
Select Boundary 4 only.
4
Locate the y-Axis Data section. In the Expression text field, type phis.
5
Select the Description checkbox. In the associated text field, type Cell Potential.
6
Locate the x-Axis Data section. From the Unit list, choose s.
7
Click to expand the Coloring and Style section. Find the Line style subsection. From the Line list, choose Dash-dot.
8
Click to expand the Legends section. Select the Show legends checkbox.
9
Find the Include subsection. Clear the Point checkbox.
10
Select the Description checkbox.
Voltage and Current Profile for CC Charge - CV Discharge
1
In the Model Builder window, click Voltage and Current Profile for CC Charge - CV Discharge.
2
In the Settings window for 1D Plot Group, locate the Plot Settings section.
3
Select the x-axis label checkbox. In the associated text field, type t-t <sub>CCstart</sub>.
4
Select the Two y-axes checkbox.
5
In the table, select the Plot on secondary y-axis checkbox for Current.
6
Click to expand the Title section. From the Title type list, choose Label.
7
Locate the Legend section. From the Position list, choose Lower right.
8
In the Voltage and Current Profile for CC Charge - CV Discharge toolbar, click  Plot.
9
Click the  Zoom Extents button in the Graphics toolbar.
Voltage and Current Profile for CC Charge - CV Discharge (Overlapped)
1
Right-click Voltage and Current Profile for CC Charge - CV Discharge and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Voltage and Current Profile for CC Charge - CV Discharge (Overlapped) in the Label text field.
Current
1
In the Model Builder window, expand the Voltage and Current Profile for CC Charge - CV Discharge (Overlapped) node, then click Current.
2
In the Settings window for Global, locate the x-Axis Data section.
3
From the Parameter list, choose Expression.
4
In the Expression text field, type t-t_CC_start.
Filter 1
1
Right-click Current and choose Filter.
2
In the Settings window for Filter, locate the Point Selection section.
3
In the Logical expression for inclusion text field, type (REST!=1)&&(V_max>1.8).
4
Locate the Line Segment Selection section. Clear the Decreasing x checkbox.
Color Expression 1
1
Right-click Current and choose Color Expression.
2
In the Settings window for Color Expression, locate the Expression section.
3
In the Expression text field, type V_max.
4
Locate the Coloring and Style section. Clear the Color legend checkbox.
Potential
1
In the Model Builder window, under Results > Voltage and Current Profile for CC Charge - CV Discharge (Overlapped) click Potential.
2
In the Settings window for Point Graph, locate the x-Axis Data section.
3
From the Parameter list, choose Expression.
4
In the Expression text field, type t-t_CC_start.
Filter 1
1
Right-click Potential and choose Filter.
2
In the Settings window for Filter, locate the Point Selection section.
3
In the Logical expression for inclusion text field, type (REST!=1)&&(V_max>1.8).
4
Locate the Line Segment Selection section. Clear the Decreasing x checkbox.
Color Expression 1
1
Right-click Potential and choose Color Expression.
2
In the Settings window for Color Expression, locate the Expression section.
3
In the Expression text field, type V_max.
4
Locate the Coloring and Style section. Clear the Color legend checkbox.
Voltage and Current Profile for CC Charge - CV Discharge (Overlapped)
1
In the Model Builder window, under Results click Voltage and Current Profile for CC Charge - CV Discharge (Overlapped).
2
In the Settings window for 1D Plot Group, locate the Axis section.
3
Select the Manual axis limits checkbox.
4
In the x minimum text field, type 0.
5
In the x maximum text field, type 40.
6
In the y minimum text field, type 0.
7
In the y maximum text field, type 2.6.
8
In the Secondary y minimum text field, type -2000.
9
In the Secondary y maximum text field, type 300.
10
Locate the Legend section. From the Position list, choose Lower left.
11
In the Voltage and Current Profile for CC Charge - CV Discharge (Overlapped) toolbar, click  Plot.
Tertiary Current Distribution, Nernst–Planck (tcd)
Define the constant power charge condition for the electrochemical capacitor.
Electrode Power 1
1
In the Physics toolbar, click  Boundaries and choose Electrode Power.
2
Select Boundary 4 only.
3
In the Settings window for Electrode Power, locate the Electrode Power section.
4
From the list, choose Average power density.
5
In the paverage text field, type -P_app/A_cell.
6
In the ϕs,bnd,init text field, type V_rest.
Initial Values - Constant Power
1
In the Physics toolbar, click  Domains and choose Initial Values.
2
Select Domain 3 only.
3
In the Settings window for Initial Values, locate the Initial Values section.
4
In the c2 text field, type c_bulk.
5
In the phis text field, type V_rest.
6
In the Label text field, type Initial Values - Constant Power.
Electrode Power 1, Initial Values - Constant Power
1
In the Model Builder window, under Component 1 (comp1) > Tertiary Current Distribution, Nernst–Planck (tcd), Ctrl-click to select Electrode Power 1 and Initial Values - Constant Power.
2
Right-click and choose Group.
Constant Power Charge
1
In the Settings window for Group, type Constant Power Charge in the Label text field.
Set up the study for the constant power charge.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies > Time Dependent.
4
Click the Add Study button in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 2
Step 1: Time Dependent
1
In the Settings window for Time Dependent, locate the Physics and Variables Selection section.
2
Select the Modify model configuration for study step checkbox.
3
In the tree, select Component 1 (comp1) > Tertiary Current Distribution, Nernst–Planck (tcd) > Constant Current Charge/Constant Voltage Discharge.
4
Right-click and choose Disable.
5
In the tree, select Component 1 (comp1) > Events (ev).
6
Click  Disable in Model.
7
Locate the Study Settings section. In the Output times text field, type 0 2.
Solution 2 (sol2)
In the Study toolbar, click  Show Default Solver.
Step 1: Time Dependent
1
In the Model Builder window, under Study 2 click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, click to expand the Study Extensions section.
3
Click in the Graphics window and then press Ctrl+A to select all domains.
4
Select the Auxiliary sweep checkbox.
5
Click  Add twice.
6
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
P_app (Applied Power)
0.5[kW] 0.85[kW]
W
V_rest (Rest potential for applied power)
-2.1[V] -2.5[V]
V
7
In the Model Builder window, click Study 2.
8
In the Settings window for Study, type Study 2: Constant Power Charge in the Label text field.
9
Locate the Study Settings section. Clear the Generate default plots checkbox.
Solution 2 (sol2)
1
In the Model Builder window, expand the Study 2: Constant Power Charge > Solver Configurations > Solution 2 (sol2) node, then click Time-Dependent Solver 1.
2
In the Settings window for Time-Dependent Solver, locate the General section.
3
From the Times to store list, choose Steps taken by solver.
Step 1: Time Dependent
In the Study toolbar, click  Compute.
Results
Double Layer Current
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Double Layer Current in the Label text field.
3
Locate the Title section. From the Title type list, choose Label.
Line Graph 1
1
Right-click Double Layer Current and choose Line Graph.
2
Select Domains 1 and 3 only.
3
In the Settings window for Line Graph, locate the Data section.
4
From the Dataset list, choose Study 2: Constant Power Charge/Solution 2 (sol2).
5
From the Time selection list, choose Last.
6
Locate the y-Axis Data section. In the Expression text field, type tcd.ivdl.
7
Locate the x-Axis Data section. From the Parameter list, choose Expression.
8
In the Expression text field, type x/(L_elec+L_sep+L_elec).
9
In the Double Layer Current toolbar, click  Plot.
10
Select the Description checkbox. In the associated text field, type Dimensionless length.
11
In the Double Layer Current toolbar, click  Plot.
12
Click to expand the Legends section. Select the Show legends checkbox.
13
From the Legends list, choose Manual.
14
In the table, enter the following settings:
Legends
60: t=2 s, P<sub>app</sub>=500 W, V<sub>rest</sub>=-2.1 V
125: t=2 s, P<sub>app</sub>=850 W, V<sub>rest</sub>=-2.5 V
Double Layer Current
1
In the Model Builder window, click Double Layer Current.
2
In the Settings window for 1D Plot Group, locate the Legend section.
3
From the Position list, choose Middle left.
4
In the Double Layer Current toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
Charge Density for Charging Power
1
In the Results toolbar, click  1D Plot Group.
Plot the charge density using the concentration from depositing-dissolving species node.
2
In the Settings window for 1D Plot Group, type Charge Density for Charging Power in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 2: Constant Power Charge/Solution 2 (sol2).
4
From the Time selection list, choose Last.
5
Locate the Title section. From the Title type list, choose Label.
Line Graph 1
1
Right-click Charge Density for Charging Power and choose Line Graph.
2
In the Settings window for Line Graph, locate the Selection section.
3
From the Selection list, choose All domains.
4
Locate the y-Axis Data section. In the Expression text field, type -tcd.c_pce1_c1_q*F_const.
5
Select the Description checkbox. In the associated text field, type Charge density.
6
Locate the x-Axis Data section. From the Parameter list, choose Expression.
7
In the Expression text field, type x/(L_sep+2*L_elec).
8
Locate the Legends section. Select the Show legends checkbox.
9
From the Legends list, choose Manual.
10
In the table, enter the following settings:
Legends
60: t=2 s, P<sub>app</sub>=500 W, V<sub>rest</sub>=-2.1 V
125: t=2 s, P<sub>app</sub>=850 W, V<sub>rest</sub>=-2.5 V
Charge Density for Charging Power
1
In the Model Builder window, click Charge Density for Charging Power.
2
In the Settings window for 1D Plot Group, locate the Plot Settings section.
3
Select the x-axis label checkbox. In the associated text field, type Dimensionless length (1).
4
Locate the Legend section. From the Position list, choose Upper left.
5
In the Charge Density for Charging Power toolbar, click  Plot.
6
Click the  Zoom Extents button in the Graphics toolbar.
Voltage and Current Profiles for Constant Power Charge
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Voltage and Current Profiles for Constant Power Charge in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 2: Constant Power Charge/Solution 2 (sol2).
4
Locate the Title section. From the Title type list, choose Label.
Current
1
Right-click Voltage and Current Profiles for Constant Power Charge and choose Point Graph.
2
In the Settings window for Point Graph, type Current in the Label text field.
3
Select Boundary 4 only.
4
Locate the y-Axis Data section. In the Expression text field, type right_el(tcd.nIs)*A_cell.
5
Locate the Legends section. Select the Show legends checkbox.
6
Locate the Coloring and Style section. From the Color list, choose Cycle (reset).
7
Locate the Legends section. From the Legends list, choose Evaluated.
8
In the Legend text field, type Current (P = eval(P_app) kW).
9
In the Voltage and Current Profiles for Constant Power Charge toolbar, click  Plot.
Potential
1
In the Model Builder window, right-click Voltage and Current Profiles for Constant Power Charge and choose Global.
2
In the Settings window for Global, type Potential in the Label text field.
3
Click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1) > Tertiary Current Distribution, Nernst–Planck > tcd.phis0_epow1 - Electric potential on boundary - V.
4
Click to expand the Coloring and Style section. Find the Line style subsection. From the Line list, choose Dash-dot.
5
From the Color list, choose Cycle (reset).
6
Click to expand the Legends section. From the Legends list, choose Evaluated.
7
In the Legend text field, type Potential (P = eval(P_app) kW).
8
In the Voltage and Current Profiles for Constant Power Charge toolbar, click  Plot.
Voltage and Current Profiles for Constant Power Charge
1
In the Model Builder window, click Voltage and Current Profiles for Constant Power Charge.
2
In the Settings window for 1D Plot Group, locate the Plot Settings section.
3
Select the y-axis label checkbox. In the associated text field, type Current (A).
4
Select the Two y-axes checkbox.
5
Select the Secondary y-axis label checkbox. In the associated text field, type Electric Potential (V).
6
In the table, select the Plot on secondary y-axis checkbox for Potential.
7
In the Voltage and Current Profiles for Constant Power Charge toolbar, click  Plot.
8
Locate the Axis section. Select the Manual axis limits checkbox.
9
In the y minimum text field, type 200.79068524910107.
10
In the y maximum text field, type 340.58128603840646.
11
In the Voltage and Current Profiles for Constant Power Charge toolbar, click  Plot.
12
In the Secondary y maximum text field, type -0.45314923375307226.
13
In the Secondary y minimum text field, type -3.
14
In the Voltage and Current Profiles for Constant Power Charge toolbar, click  Plot.
15
Locate the Legend section. From the Position list, choose Center.
16
In the Voltage and Current Profiles for Constant Power Charge toolbar, click  Plot.