conductPotentialTemperature

Electrically and thermally conducting solid (e.g. battery current-collector tab) solving electric potential conduction and heat conduction with Joule heating.

Overview

conductPotentialTemperature models a solid, electrically conducting region such as the copper or aluminium current-collector tab of a lithium-ion battery cell. It solves a charge-conservation (Laplace) equation for the electric potential and a transient heat-conduction equation for the temperature, with ohmic (Joule) heating from the potential gradient as the heat source. It has no electrochemistry of its own and is intended to be coupled to an NTGK battery region, which provides the current through the shared interface.

The electrode polarity of the region is set by the electrode keyword in transportProperties. It decides which potential field is used: phiPos for positive and phiNeg for negative. These field names match the potential fields of NTGK, so the same field name can be coupled across the battery/tab interface. A single region type therefore covers both tabs.

Governing Equations

Electric potential

$$ \nabla \cdot \left( \sigma \, \nabla \phi \right) = 0 $$

where $\phi$ is phiPos or phiNeg, depending on electrode. The equation has no transient term, so the potential is quasi-steady within each time step.

Energy

$$ \rho c_p \frac{\partial T}{\partial t} - \nabla \cdot \left( k \, \nabla T \right) = S_T $$

Joule heating

$$ S_T = \dot{Q}_\text{ohm} = \sigma \, \nabla\phi \cdot \nabla\phi $$

$S_T$ is evaluated explicitly with fvc::grad and stored in the field ST.

Symbol	Field / keyword	Description	Unit
$\phi$	`phiPos` / `phiNeg`	Electric potential	V
$T$	`T`	Temperature	K
$\sigma$	`sigma`	Electric conductivity	S/m
$\rho$	`rho`	Density	kg/m³
$c_p$	`cp`	Specific heat capacity	J/(kg K)
$k$	`k`	Thermal conductivity	W/(m K)
$S_T$	`ST`	Volumetric heat source (Joule heating)	W/m³

All material properties are uniform dimensionedScalars.

Numerical Algorithm

conductPotentialTemperature has no region-local solve. usesPIMPLE() returns false, and solveRegion(), prePredictor(), momentumPredictor(), pressureCorrector() and meshMotionCorrector() are all no-ops. Each time step goes through the coupled-matrix path of multiRegionSystem:

setCoupledEqns(fieldName) assembles one equation and registers it in fvScalarMatrices under the key <field><region>MeshconductPotentialTemperatureTypeEqn:
- if fieldName is the region’s potential field (phiPos or phiNeg): fvm::laplacian(sigma, phi), using the laplacian scheme name laplacian(sigma,fai);
- if fieldName is T: fvm::ddt(rho*cp, T) - fvm::laplacian(k, T) (scheme name laplacian(k,T)), with ST as an explicit source;
- any other field name, including the potential field of the opposite polarity, is ignored.
The equations are solved by the DNA (partitioned) coupling loop for each field listed in an interface’s coupledFields (iteration controls come from the DNA block in multiRegionProperties).
postSolve() calls correct(), which recomputes ST from the updated potential. The new ST is first used in the T equation of the next time step.

getMinDeltaT() returns GREAT, so this region never limits the time step.

selfCoupledFieldNames() is not overridden. It returns an empty list, so the region’s equations are assembled and solved only when their field is listed in the coupledFields of a partitioned interface in regionInterfaceProperties.

Sub-Models

None. The region does not read or apply fvOptions, and there are no turbulence, transport-model or MRF sub-models.

Required Fields

Field	Type	Description	Unit
`phiPos` or `phiNeg`	`volScalarField`	Electric potential (`phiPos` if `electrode positive;`, `phiNeg` if `electrode negative;`)	V
`T`	`volScalarField`	Temperature	K

Optional / created at runtime:

Field	Type	Description	Unit
`ST`	`volScalarField`	Joule heat source. Read if present, otherwise initialised to 0; written at write time	W/m³

Case Setup

Required dictionaries

constant/<region>/transportProperties   – electrode, sigma, rho, cp, k
system/<region>/fvSolution               – solvers for phiPos/phiNeg and T
system/<region>/fvSchemes                – ddt, grad (Joule heating), laplacian schemes

`transportProperties`

Every keyword is required; there are no defaults. Any electrode value other than positive or negative is a fatal error.

electrode       negative;                               // positive -> phiPos, negative -> phiNeg

rho             rho   [1 -3 0 0 0 0 0]   8933;          // kg/m3   (copper)
cp              cp    [0 2 -2 -1 0 0 0]  385;           // J/kg/K
k               k     [1 1 -3 -1 0 0 0]  398;           // W/m/K
sigma           sigma [-1 -3 3 0 0 2 0]  59.6e6;        // S/m

`multiRegionProperties` entry

regions
(
    ( batteryRegion (NTGK) )
    ( negTabRegion  (conductPotentialTemperature) )
    ( posTabRegion  (conductPotentialTemperature) )
);

DNA
{
    phiPos { maxCoupleIter 50; residualControl { maxJumpRes 1e-05; maxFluxRes 1e-05; } }
    phiNeg { maxCoupleIter 25; residualControl { maxJumpRes 1e-05; maxFluxRes 1e-05; } }
    T      { maxCoupleIter 10; residualControl { maxJumpRes 1e-05; maxFluxRes 1e-05; } }
}

Key `fvSolution` settings

solvers
{
    phiNeg      // or phiPos, depending on electrode
    {
        solver          PCG;
        preconditioner  DIC;
        tolerance       1e-8;
        relTol          0;
        minIter         2;
        maxIter         250;
    }

    T
    {
        solver          PCG;
        preconditioner  DIC;
        tolerance       1e-8;
        relTol          0;
        minIter         2;
        maxIter         250;
    }
}

Interface boundary conditions

On the tab side of the battery/tab interface, the tutorial uses regionCoupledScalarFlux (or regionCoupledScalarJump) with diffusivity set to the name of the matching transportProperties entry: sigma for the potential and k for T.

toCellFromNeg
{
    type            regionCoupledScalarFlux;
    interfaceType   (defaultInterface);
    diffusivity     "sigma";        // "k" for T
    secondOrder     false;
    nonOrthCorr     false;
    value           $internalField;
}

The external terminal is set by an ordinary boundary condition on the potential: fixedValue 0 on the negative tab and fixedGradient (applied current) on the positive tab.

Coupling

Interface type	Partner region type	Exchanged fields	Coupling mode
`defaultInterface`	`NTGK`	`phiPos` (positive tab) or `phiNeg` (negative tab), `T`	partitioned (DNA)

No interface type contains code specific to this region type. Coupling goes through the generic regionCoupledScalarJump / regionCoupledScalarFlux patch fields, which read sigma / k from the region’s transportProperties. An interface’s coupledFields may list both phiPos and phiNeg, because the region ignores the potential field of the opposite polarity.

Known Limitations

Static mesh only: meshMotionCorrector() is a no-op.
No fvOptions: additional sources cannot be added to the potential or temperature equation.
Uniform, constant properties: sigma, rho, cp and k are dimensionedScalars; temperature-dependent or spatially varying properties are not supported.
Quasi-steady potential: the potential equation has no transient term. The member C_ (a specific capacitance meant for a transient term) is declared and initialised to 1 but never used.
Explicit, lagged Joule heating: ST is updated only in postSolve(), so the temperature equation uses the source from the previous time step. ST is not computed at construction, so the first time step uses the initial ST (0 unless an ST field is supplied).
Not self-coupled: with an empty selfCoupledFieldNames(), the region cannot be run standalone (e.g. in singleRegionMode). Its fields are only solved through DNA coupling.
Monolithic coupling not supported: the only tutorial runs partitioned coupling only (its Allrun -m reports monolithic coupling as unsupported).

Tutorials

Tutorial path	Description
`tutorials/electrochemistry/NTGKBatteryWithTabs`	Prismatic NTGK battery cell with copper negative tab and aluminium positive tab (`conductPotentialTemperature`), coupled via `defaultInterface` (`./Allrun -p -S`)

References

Kwon, K. H. et al., A two-dimensional modeling of a lithium-polymer battery, Journal of Power Sources 163(1), 151–157 (2006). DOI: 10.1016/j.jpowsour.2005.12.051
Kim, G.-H., Pesaran, A., Spotnitz, R., A three-dimensional thermal abuse model for lithium-ion cells, Journal of Power Sources 170(2), 476–489 (2007). DOI: 10.1016/j.jpowsour.2007.04.018
Alkfri, A., Habes, C., Marschall, H., multiRegionFoam: A Unified Multiphysics Framework for Multi-Region Coupled Continuum-Physical Problems, Engineering with Computers, 2023. DOI: 10.1007/s00366-024-01974-4