multiPhysicsFoam – Conduction

Docs: conductPotentialTemperature

Mon, 01 Jan 0001 00:00:00 +0000

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

Docs: conductTemperature

Mon, 01 Jan 0001 00:00:00 +0000

Overview

conductTemperature models transient heat conduction in a solid. It solves a single energy equation for the temperature T, using density rho, specific heat cv and thermal conductivity kappa. There is no velocity field, no advection and no pressure-velocity coupling. It is meant to be the solid side of a conjugate heat transfer problem, coupled to a fluid region (usually pimpleFluid transportTemperature) through a heatTransferInterface. It can also be stacked with a solid mechanics region type (e.g. solidStVK) for thermo-FSI.

Governing Equations

Energy (heat conduction)

$$ \frac{\partial \left( \rho \, c_v \, T \right)}{\partial t} = \nabla \cdot \left( \kappa \, \nabla T \right) $$

Symbol	Description	Unit
$T$	Temperature	K
$\rho$	Density	kg/m³
$c_v$	Specific heat capacity	J/(kg K) = m²/(s² K)
$\kappa$	Thermal conductivity	W/(m K) = kg m/(s³ K)

The equation is assembled as

fvm::ddt(rho_*cv_, T()) == fvm::laplacian(kappa_(), T(), "laplacian(kappa,T)")

so fvSchemes must provide a laplacian(kappa,T) entry (or a default laplacian scheme). The equation has no source term.

Numerical Algorithm

conductTemperature does not run its own solve loop. usesPIMPLE() returns false. solveRegion(), prePredictor(), momentumPredictor(), pressureCorrector(), meshMotionCorrector() and postSolve() are all no-ops. The temperature is solved only when multiRegionSystem asks for it:

The region does not override selfCoupledFieldNames(), so the base default (an empty list) applies. T is therefore solved only when it appears as a partitioned-coupled field, i.e. when the case has a DNA { T { ... } } entry in multiRegionProperties and T is listed in coupledFields of an interface in regionInterfaceProperties.
Inside each DNA coupling iteration, multiRegionSystem::assembleAndSolveEqns<fvMatrix, scalar>("T") calls setCoupledEqns("T"). This assembles the equation above and stores it under the key T<regionName>MeshconductTemperatureTypeEqn. The equation is then relaxed (relaxationFactors in fvSolution) and solved. The region Picard corrector (maxCorr / relTol / tolerance per field in fvSolution) can re-assemble and re-solve it within one DNA iteration. setCoupledEqns ignores its fieldName argument and always assembles the T equation.
correct() only calls kappa.correctBoundaryConditions().
getMinDeltaT() returns GREAT, so this region never limits the global time step.

Sub-Models

None. The region does not read fvOptions, a thermophysical model or turbulence. All material properties come from transportProperties or from optional field files.

Required Fields

Field	Type	Read mode	Description	Unit
`T`	`volScalarField`	`MUST_READ`, `AUTO_WRITE`	Temperature	K
`kappa`	`volScalarField`	`READ_IF_PRESENT`, `AUTO_WRITE`	Thermal conductivity. If there is no file, a uniform field is created from `transportProperties` with `calculated` patches.	W/(m K)
`rho`	`volScalarField`	`READ_IF_PRESENT`, `NO_WRITE`	Density. If there is no file, it is uniform from `transportProperties`.	kg/m³
`cv`	`volScalarField`	`READ_IF_PRESENT`, `NO_WRITE`	Specific heat. If there is no file, it is uniform from `transportProperties`.	m²/(s² K)

T and kappa are created through lookupOrRead. If a sibling region type on the same mesh has already registered a field with that name, that field is reused. rho and cv are always constructed by this region.

Case Setup

Required dictionaries

constant/<region>/transportProperties – rho, cv, kappa (all required, dimensioned)
system/<region>/fvSolution – T solver, relaxationFactors
system/<region>/fvSchemes – ddt and laplacian(kappa,T) schemes
0/<region>/T – temperature with coupled BC on the interface patch

`multiRegionProperties` entry

regions
(
 ( fluid (pimpleFluid transportTemperature) )
 ( solid (conductTemperature) )
);

DNA
{
 T
 {
 maxCoupleIter 20;

 residualControl
 {
 maxJumpRes 1e-07;
 outputJumpResField no;
 maxFluxRes 1e-07;
 outputFluxResField no;
 }
 }
}

`transportProperties`

All three entries are looked up as dimensionedScalar and are required, even when a field file overrides them:

rho rho [1 -3 0 0 0 0 0] 1.0;
cv cv [0 2 -2 -1 0 0 0] 100.0;
kappa kappa [1 1 -3 -1 0 0 0] 100.0;

Key `fvSolution` settings

solvers
{
 T
 {
 solver PCG;
 preconditioner DIC;
 tolerance 1e-15;
 relTol 0;
 minIter 2;
 maxIter 250;
 }
}

relaxationFactors
{
 T 1.0;
}

Interface boundary condition on `T`

On the solid side of a partitioned heatTransferInterface (flux side; the fluid side then uses regionCoupledTemperatureJump):

top
{
 type regionCoupledHeatFlux;
 interfaceType (heatTransferInterface);
 secondOrder false;
 nonOrthCorr false;
 value $internalField;
}

The coupled patch fields look up the diffusivity field kappa by default (set the patch keyword diffusivity to use another name). This matches the field that conductTemperature registers.

Coupling

Interface type	Exchanged fields	Coupling mode
`heatTransferInterface`	`T` (temperature jump / heat flux via `regionCoupledTemperatureJump` / `regionCoupledHeatFlux`, using `kappa`)	partitioned (DNA)

heatTransferInterface only provides the jumpCoefficient (from heatTransferInterfaceCoeffs). The patch fields do the actual transfer. Monolithic coupling is commented out in multiRegionSystem::solve(), so only partitioned coupling is available at the moment.

Known Limitations

Coupled use only. selfCoupledFieldNames() is not overridden, so T is never solved unless it is DNA-coupled. A standalone (single-region) conductTemperature case does not advance T. Use laplacianFluid for standalone conduction.
No source terms. fvOptions are not read or applied.
Static mesh only. meshMotionCorrector() is a no-op and the equation contains no mesh-flux terms.
No time-step control. getMinDeltaT() returns GREAT; deltaT must be chosen by the user or limited by other regions.
Constant rho and cv in time. They are read once and never updated (NO_WRITE). Only kappa gets a boundary refresh in correct(), and its values are never recomputed (for example as a function of T).
No non-orthogonal corrector loop. The coupled solve path has no correctNonOrthogonal() loop, so nNonOrthogonalCorrectors in fvSolution is not used by this region.
No output/post-processing. postSolve() is a no-op. No derived fields (e.g. heat flux) are written.

Tutorials

Tutorial path	Description
`tutorials/conjugateHeatTransfer/flowOverHeatedPlate`	Flow over a heated plate (`pimpleFluid transportTemperature` + `conductTemperature`)
`tutorials/conjugateHeatTransfer/ShellAndTubeHeatExchanger`	Shell-and-tube heat exchanger, two fluid regions + `conductTemperature` solid
`tutorials/fluidStructureInteraction/heatedBeamInCrossFlow`	Heated beam in cross-flow, thermo-FSI (`solidStVK conductTemperature` solid)

References

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

multiPhysicsFoam – Conduction

Docs: conductPotentialTemperature

Overview

Governing Equations

Electric potential

Energy

Joule heating

Numerical Algorithm

Sub-Models

Required Fields

Case Setup

Required dictionaries

transportProperties

multiRegionProperties entry

Key fvSolution settings

Interface boundary conditions

Coupling

Known Limitations

Tutorials

References

Docs: conductTemperature

Overview

Governing Equations

Energy (heat conduction)

Numerical Algorithm

Sub-Models

Required Fields

Case Setup

Required dictionaries

multiRegionProperties entry

transportProperties

Key fvSolution settings

Interface boundary condition on T

Coupling

Known Limitations

Tutorials

References

`transportProperties`

`multiRegionProperties` entry

Key `fvSolution` settings

`multiRegionProperties` entry

`transportProperties`

Key `fvSolution` settings

Interface boundary condition on `T`