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Description

The electrostaticDeposition is a boundary condition to calculate electric potential (V) on a given boundary based on film thickness (h) and film resistance (R) fields which are updated based on a given patch-normal current density field (jn), Coulombic efficiency and film resistivity.

The governing equations are as follows:

\[j_n = - \sigma \nabla^\perp_p V = - \sigma (\vec{n}\cdot(\nabla V)_p)\] \[\frac{dh}{dt} = C_{eff} (j_n - j_{min})\] \[\frac{dR}{dt} = \rho \frac{dh}{dt} = \rho C_{eff} (j_n - j_{min})\] \[V_{film}^n = V_{film}^o + j_n R_\Delta\] \[V_{body} = j_n R_{body}\] \[V_p^n = V_i + V_{body} + V_{film}^n\]

where:

Property Description
\(j_n\) Patch-normal current density [A/m^2]
\(V_p^n\) Electric potential on film-fluid interface [volt = kg m^2/(A s^3)]
\(V_p^o\) Previous time-step electric potential on the interface [volt]
\(V_{film}\) Electric potential due to film resistance [volt]
\(V_{body}\) Electric potential due to body resistance [volt]
\(V_i\) Initial electric potential [volt]
\(R_\Delta\) Film resistance (finite increment) [ohm m^2 = kg m^4/(A^2 s^3)]
\(R_{body}\) Body resistance [ohm m^2 = kg m^4/(A^2 s^3)]
\(\rho\) Isotropic film resistivity [ohm m = kg m^3/(A^2 s^3)]
\(h\) Film thickness [m]
\(C_{eff}\) Volumetric Coulombic efficiency [m^3/(A s)]
\(j_{min}\) Minimum current density for deposition onset [A/m^2]
\(\sigma\) Isotropic conductivity of mixture [S/m = A^2 s^3/(kg m^3)]
\(\vec{n}\) Patch-normal unit vector [-]

Usage

The condition requires entries in both the boundary and field files.

Boundary file 

<patchName>
{
    type            patch;
    ...
}

Field file 

<patchName>
{
    // Mandatory entries
    type                    electrostaticDeposition;
    h                       <scalarField>;
    CoulombicEfficiency     <PatchFunction1>;
    resistivity             <PatchFunction1>;

    // Conditional mandatory entries

        // Option-1: single-phase
        sigma       <scalar>;

        // Option-2: multiphase
        phases
        {
            alpha.air
            {
                sigma       <scalar>;
            }
            alpha.water
            {
                sigma       <scalar>;
            }
            alpha.mercury
            {
                sigma       <scalar>;
            }
            ...
        }

    // Optional entries
    jMin                    <scalar>;
    qMin                    <scalar>;
    Rbody                   <scalar>;
    Vi                      <scalar>;
    Vanode                  <scalar>;
    qCumulative             <scalarField>;

    // Inherited entries
    ...
}

where:

Property Description Type Required Default
type Type name: electrostaticDeposition word yes -
h Film thickness scalarField yes -
CoulombicEfficiency Coulombic efficiency PatchFunction1<scalar> yes -
resistivity Isotropic film resistivity PatchFunction1<scalar> yes -
sigma Isotropic electrical conductivity of phase scalar yes -
jMin Minimum current density for deposition onset scalar no 0
qMin Minimum accumulative specific charge for deposition onset scalar no 0
Rbody Resistance due to main body and/or pretreatment layers scalar no 0
Vi Initial electric potential scalar no 0
Vanode Anode electric potential scalar no GREAT
qCumulative Accumulative specific charge [A s/m^2] scalarField no 0

The inherited entries are elaborated in:

  • Foam::fixedValueFvPatchField

  • Foam::PatchFunction1

  • Depletion or abrasion of material due to negative current is not allowed.
  • When accumulative specific charge (qCumulative) is less than minimum accumulative specific charge (qMin), no deposition occurs.
  • Boundary-condition updates are not allowed during outer corrections to prevent spurious accumulation of film thickness.
  • resistivity, jMin, qMin and Rbody are always non-negative.

Further information

Tutorial:

Source code:

API:

History:

  • Introduced in version v2112