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 (). SS Type of boundary condition (), where SS denotes side set in the EXODUS II database. The boundary flag identifier, an integer associated with that identifies the boundary location (side set in EXODUS II) in the problem domain. 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

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 (). SS Type of boundary condition (), where SS denotes side set in the EXODUS II database. The boundary flag identifier, an integer associated with that identifies the boundary location (side set in EXODUS II) in the problem domain. 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

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 (). NS Type of boundary condition (), where NS denotes node set in the EXODUS II database. The boundary flag identifier, an integer associated with that identifies the boundary location (node set in EXODUS II) in the problem domain. 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

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 (). SS Type of boundary condition (), where SS denotes side set in the EXODUS II database. The boundary flag identifier, an integer associated with that identifies the boundary location (side set in EXODUS II) in the problem domain. Species number of concentration. Stoichiometric coefficient. Kinetic rate constant. Reaction order. Anodic direction transfer coefficient. Cathodic direction transfer coefficient. Electrode potential or applied voltage. 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 (). SS Type of boundary condition (), where SS denotes side set in the EXODUS II database. The boundary flag identifier, an integer associated with that identifies the boundary location (side set in EXODUS II) in the problem domain. Species number of concentration. Product of interfacial area per unit volume by exchange current density, $$ai_0$$, in units of A/ $$cm^3$$. Catalyst layer or catalyzed electrode thickness, H, in unit of cm. Reference concentration, $$c_{ref}$$, in units of moles/ $$cm^3$$. Anodic direction transfer coefficient, $$\alpha_a$$. Cathodic direction transfer coefficient, $$\alpha_c$$. Temperature, T, in unit of K. Theoretical open-circuit potential, $$U_0$$, in unit of V. Reaction order, $$\beta$$. 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:

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 (). SS Type of boundary condition (), where SS denotes side set in the EXODUS II database. The boundary flag identifier, an integer associated with that identifies the boundary location (side set in EXODUS II) in the problem domain. Species number of concentration. Product of interfacial area per unit volume by exchange current density, $$ai_0$$, in units of A/ $$cm^3$$. Catalyst layer or catalyzed electrode thickness, H, in unit of cm. Reference concentration, $$c_{ref}$$, in units of moles/ $$cm^3$$. Cathodic direction transfer coefficient, $$\alpha_c$$. Temperature, T, in unit of K. Electrode potential, V, in unit of V. Theoretical open-circuit potential, $$U_0$$, in unit of V. 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:

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 (). SS Type of boundary condition (), where SS denotes side set in the EXODUS II database. The boundary flag identifier, an integer associated with that identifies the boundary location (side set in EXODUS II) in the problem domain. 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.

No References.