Category 13: Potential Equation#

The Potential equation is a Laplace equation for the voltage (potential) given a charge distribution in a dielectric medium or a voltage or current boundary condition in an electrically conductive medium. The following boundary conditions allow the current or voltage to be set on a boundary.

CURRENT#

BC = CURRENT SS <bc_id> <float>

Description / Usage#

(WIC/POTENTIAL)

This card specifies the electrical current density at a given boundary.

Definitions of the input parameters are as follows:

CURRENT

Name of the boundary condition (<bc_name>).

SS

Type of boundary condition (<bc_type>), where SS denotes side set in the EXODUS II database.

<bc_id>

The boundary flag identifier, an integer associated with <bc_type> that identifies the boundary location (side set in EXODUS II) in the problem domain.

<float>

Value of current density (in A/ \(m^2\) or A/ \(cm^2\), depending on units of length scale used in the problem).

Examples#

An example input card:

BC = CURRENT SS 1   -0.05

Technical Discussion#

No Discussion.

CURRENT_USER#

BC = CURRENT_USER SS <bc_id> <float_list>

Description / Usage#

(WIC/POTENTIAL)

This boundary condition card is used to define a routine for a user-defined electrical current density model. Definitions of the input parameters are as follows:

CURRENT_USER

Name of the boundary condition (<bc_name>).

SS

Type of boundary condition (<bc_type>), where SS denotes side set in the EXODUS II database.

<bc_id>

The boundary flag identifier, an integer associated with <bc_type> that identifies the boundary location (side set in EXODUS II) in the problem domain.

<float_list>

A list of float values separated by spaces which will be passed to the user-defined subroutines so the user can vary the parameters of the boundary condition. This list of float values is passed as a one-dimensional double array to the appropriate C function.

Examples#

The following is a sample input card:

BC = CURRENT_USER SS 100   10.0 3.14159

Technical Discussion#

No Discussion.

VOLT#

BC = VOLT NS <bc_id> <float1> [float2]

Description / Usage#

(DC/POTENTIAL)

This Dirichlet boundary condition card is used to set a constant voltage. Definitions of the input parameters are as follows:

VOLT

Name of the boundary condition (<bc_name>).

NS

Type of boundary condition (<bc_type>), where NS denotes node set in the EXODUS II database.

<bc_id>

The boundary flag identifier, an integer associated with <bc_type> that identifies the boundary location (node set in EXODUS II) in the problem domain.

<float1>

Value of voltage.

[float2]

An optional parameter (that serves as a flag to the code for a Dirichlet boundary condition). If a value is present, and is not -1.0, the condition is applied as a residual equation. Otherwise, it is a “hard set” condition and is eliminated from the matrix. The residual method must be used when this Dirichlet boundary condition is used as a parameter in automatic continuation sequences.

Examples#

Following is a sample card:

BC = VOLT NS 3   -0.22

Technical Discussion#

No Discussion.

CURRENT_BV#

BC = CURRENT_BV SS <bc_id> <integer> <floatlist>

Description / Usage#

(WIC/POTENTIAL)

The CURRENT_BV card enables the specification of variable electrical current density as given by Butler-Volmer kinetics and the Faraday’s law at the specified boundary (namely, an electrode surface).

The <floatlist> has seven parameters for this boundary condition; definitions of the input parameters are as follows:

CURRENT_BV

Name of the boundary condition (<bc_name>).

SS

Type of boundary condition (<bc_type>), where SS denotes side set in the EXODUS II database.

<bc_id>

The boundary flag identifier, an integer associated with <bc_type> that identifies the boundary location (side set in EXODUS II) in the problem domain.

<integer>

Species number of concentration.

<float1>

Stoichiometric coefficient.

<float2>

Kinetic rate constant.

<float3>

Reaction order.

<float4>

Anodic direction transfer coefficient.

<float5>

Cathodic direction transfer coefficient.

<float6>

Electrode potential or applied voltage.

<float7>

Theoretical open-circuit potential.

Examples#

An example input card:

BC = CURRENT_BV SS 1 0   -1.0 0.000002 1.0 0.21 0.21 -0.65   -0.22

Technical Discussion#

Users are referred to Chen (2000) for details of the Butler-Volmer model and also Newman (1991), particularly Equations 8.6 and 8.10 and Chapter 8, pp. 188-189 in the latter.

References#

GTM-025.0: Modeling diffusion and migration transport of charged species in dilute electrolyte solutions: GOMA implementation and sample computed predictions from a case study of electroplating, K. S. Chen, September 21, 2000

    1. Newman, “Electrochemical Systems”, Second Edition, Prentice-Hall, Inc. (1991).

CURRENT_HOR#

BC = CURRENT_HOR SS <bc_id> <integer> <floatlist>

Description / Usage#

(WIC/POTENTIAL)

The CURRENT_HOR card enables the specification of the variable current density as given by linearized Butler-Volmer kinetics (such as that for the hydrogen-oxidation reaction in polymer-electrolyte-membrane fuel cells) at the specified boundary (i.e., at the electrode surface).

The <floatlist> consists of 9 values; definitions of the input parameters are as follows:

CURRENT_HOR

Name of the boundary condition (<bc_name>).

SS

Type of boundary condition (<bc_type>), where SS denotes side set in the EXODUS II database.

<bc_id>

The boundary flag identifier, an integer associated with <bc_type> that identifies the boundary location (side set in EXODUS II) in the problem domain.

<integer>

Species number of concentration.

<float1>

Product of interfacial area per unit volume by exchange current density, \(ai_0\), in units of A/ \(cm^3\).

<float2>

Catalyst layer or catalyzed electrode thickness, H, in unit of cm.

<float3>

Reference concentration, \(c_{ref}\), in units of moles/ \(cm^3\).

<float4>

Anodic direction transfer coefficient, \(\alpha_a\).

<float5>

Cathodic direction transfer coefficient, \(\alpha_c\).

<float6>

Temperature, T, in unit of K.

<float7>

Theoretical open-circuit potential, \(U_0\), in unit of V.

<float8>

Reaction order, \(\beta\).

<float9>

Electrode potential, V, in unit of V.

Examples#

The following is a sample input card:

BC = CURRENT_HOR SS 14 0 1000. 0.001 4.e-5 1. 1. 353. 0. 0.5 0.

Technical Discussion#

For electrochemical reactions such as the hydrogen-oxidation reaction (HOR), surface overpotential is relatively small such that the Butler-Volmer kinetic model can be linearized to yield a simplified equation for computing current density:

../../_images/186_goma_physics.png

where j is current density in units of A/ \(cm^2\); \(ai_0\) denotes the product of interfacial area per unit volume by exchange current density, which has units of A/ \(cm^3\); H is the catalyst layer or catalyzed electrode thickness in unit of cm; c and \(c_{ref}\) are, respectively, species and reference molar concentrations in units of moles/ \(cm^3\); \(\beta\) is reaction order; \(\alpha_a\) and \(\alpha_c\) are, respetively, the anodic and cathodic transfer coefficients; F is the Faraday’s constant ( 96487 C/mole); R is the universal gasl constant ( 8.314 J/mole-K); T is temperature in unit of K; V and \(\phi\) are, respectively, the electrode and electrolyte potentials in unit of V; \(U_0\) and is the open-circuit potential in unit of V.

References#

  1. Newman, Electrochemical Systems, 2nd Edition, Prentice-Hall, NJ (1991).

K. S. Chen and M. A. Hickner, “Modeling PEM fuel cell performance using the finiteelement method and a fully-coupled implicit solution scheme via Newton’s technique”, in ASME Proceedings of FUELCELL2006-97032 (2006).

CURRENT_ORR#

BC = CURRENT_ORR SS <bc_id> <integer> <floatlist>

Description / Usage#

(WIC/POTENTIAL)

The CURRENT_ORR card enables the specification of the variable current density as given by the Tafel kinetics (such as that for the oxygen-reduction reaction in polymerelectrolyte- membrane fuel cells) at the specified boundary (i.e., at the electrode surface).

The <floatlist> consists of 8 values; definitions of the input parameters are as follows:

CURRENT_ORR

Name of the boundary condition (<bc_name>).

SS

Type of boundary condition (<bc_type>), where SS denotes side set in the EXODUS II database.

<bc_id>

The boundary flag identifier, an integer associated with <bc_type> that identifies the boundary location (side set in EXODUS II) in the problem domain.

<integer>

Species number of concentration.

<float1>

Product of interfacial area per unit volume by exchange current density, \(ai_0\), in units of A/ \(cm^3\).

<float2>

Catalyst layer or catalyzed electrode thickness, H, in unit of cm.

<float3>

Reference concentration, \(c_{ref}\), in units of moles/ \(cm^3\).

<float4>

Cathodic direction transfer coefficient, \(\alpha_c\).

<float5>

Temperature, T, in unit of K.

<float6>

Electrode potential, V, in unit of V.

<float7>

Theoretical open-circuit potential, \(U_0\), in unit of V.

<float8>

Reaction order, \(\beta\).

Examples#

The following is a sample input card:

BC = CURRENT_ORR SS 15 1 0.01 0.001 4.e-5 1. 353. 0.7 1.18 1.

Technical Discussion#

For electrochemical reactions such as the oxygen-reduction reaction (ORR), surface overpotential is large and negative such that the first exponential term in the Butler- Volmer kinetic model is much smaller than the second term and thus can be dropped to yield the Tafel kinetic model for computing current density:

../../_images/187_goma_physics.png

here j is current density in units of A/ \(cm^2\); \(ai_0\) denotes the product of interfacial area per unit volume by exchange current density, which has units of A/ \(cm^3\); H is the catalyst layer or catalyzed electrode thickness in unit of cm; c and \(c_{ref}\) are, respectively, species and reference molar concentrations in units of moles/ \(cm^3\); \(\beta\) is reaction order; \(\alpha_c\) is the anodic and cathodic transfer coefficient; F is the Faraday’s constant ( 96487 C/mole); R is the universal gasl constant ( 8.314 J/mole-K); T is temperature in unit of K; V and \(\phi\) are, respectively, the electrode and electrolyte potentials in unit of V; and \(U_0\) is the open-circuit potential in unit of V.

References#

  1. Newman, Electrochemical Systems, 2nd Edition, Prentice-Hall, NJ (1991).

K. S. Chen and M. A. Hickner, “Modeling PEM fuel cell performance using the finiteelement method and a fully-coupled implicit solution scheme via Newton’s technique”, in ASME Proceedings of FUELCELL2006-97032 (2006).

VOLT_USER#

BC = VOLT_USER SS <bc_id> <float_list>

Description / Usage#

(WIC/POTENTIAL)

This boundary condition card is used to specify a voltage or potential computed via a user-defined function. Definitions of the input parameters are as follows:

VOLT_USER

Name of the boundary condition (<bc_name>).

SS

Type of boundary condition (<bc_type>), where SS denotes side set in the EXODUS II database.

<bc_id>

The boundary flag identifier, an integer associated with <bc_type> that identifies the boundary location (side set in EXODUS II) in the problem domain.

<float_list>

A list of float values separated by spaces which will be passed to the user-defined subroutines so the user can vary the parameters of the boundary condition. This list of float values is passed as a one-dimensional double array to the appropriate C function.

Examples#

The following is a sample input card:

BC = VOLT_USER SS 14 0.33 1000. 0.001 4e-5 1. 1. 353. 0.

Technical Discussion#

In the VOLT_USER model currently implemented in GOMA, the electrolyte potential is given by the linearized Butler-Volmer kinetic model as in the hydrogen-oxidation reaction of a hydrogen-fueled polymer-electrolyte-membrane fuel cell. See the user_bc.c routine for details.

References#

No References.