Docs: NTGK

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

Overview

NTGK implements a semi-empirical electrochemical and thermal model for lithium-ion (Li-ion) battery cells. The acronym stands for the names of the fitting functions used for the electrochemical conductance (NTGK) and open circuit potential, which are polynomial functions of the depth of discharge (DOD) with temperature corrections. The model is based on the formulations of Kwon et al. (2006) and Kim et al. (2007).

The region type solves:

Electric potential transport in the positive and negative electrodes
Volumetric heat generation inside the cell (electrochemical and ohmic sources)
Cell energy equation (temperature)
Optional thermal abuse reactions (SEI decomposition, negative/positive solvent reactions, electrolyte decomposition) for over-temperature safety analysis

It is intended for use as the jelly-roll region of a battery cell, and can be coupled to current-collector tab regions (conductPosPotentialTemperature, conductNegPotentialTemperature) via defaultInterface for realistic battery geometries.

Governing Equations

Electrode potential (positive electrode)

$$ \nabla \cdot \left( \sigma^+ \nabla \phi^+ \right) = \frac{1}{D^+} j $$

Electrode potential (negative electrode)

$$ \nabla \cdot \left( \sigma^- \nabla \phi^- \right) = -\frac{1}{D^-} j $$

Both equations are assembled with a linearised source term (semi-implicit) to improve convergence:

$$ \frac{j}{D^\pm} = \frac{Y}{D^\pm}\left(\phi^\pm \mp \phi^\mp \mp U_\text{OCV}\right) $$

Volumetric exchange current density (NTGK model)

$$ j = Y \left( \phi^+ - \phi^- - U_\text{OCV} \right) $$

The temperature- and DOD-dependent electrochemical conductance $Y$ and open circuit voltage $U_\text{OCV}$ are computed as 5th-order polynomial fits in DOD with Arrhenius/linear temperature corrections:

$$ Y_0(\text{DOD}) = \sum_{i=0}^{5} a_i \,\text{DOD}^i \qquad Y = Y_0 \cdot \exp\!\left(-C_1\left(\frac{1}{T} - \frac{1}{T_\text{ref}}\right)\right) $$ $$ U_0(\text{DOD}) = \sum_{i=0}^{5} b_i \,\text{DOD}^i \qquad U_\text{OCV} = U_0 + C_2 \left(T - T_\text{ref}\right) $$

Depth of discharge

DOD is updated globally (cell-average) each time step:

$$ \text{DOD}^{n+1} = \text{DOD}^n - \frac{\Delta t}{Q_\text{bat}} \sum_\text{cells} j \, V_\text{cell} / D $$

Energy equation

$$ \rho c_p \frac{\partial T}{\partial t} - \nabla \cdot \left( k \nabla T \right) = \dot{Q}_\text{ech} + \dot{Q}_\text{ohm} + \dot{Q}_\text{abuse} $$

The electrochemical heat source and ohmic dissipation are:

$$ \dot{Q}_\text{ech} = \frac{j}{D}\left(\phi^+ - \phi^- - U_\text{OCV} + C_2 T\right) $$ $$ \dot{Q}_\text{ohm} = \sigma^+ \left|\nabla\phi^+\right|^2 + \sigma^- \left|\nabla\phi^-\right|^2 $$

Thermal abuse reactions

Four exothermic decomposition reactions are activated progressively as the cell temperature exceeds threshold values:

Reaction	Onset temperature	Rate law
SEI decomposition	$T > 363.15\,\text{K}$	$R_\text{SEI} = A_\text{SEI} \exp(-E_{A,\text{SEI}}/RT)\, c_\text{SEI}^{m_\text{SEI}}$
Negative-electrode solvent	$T > 393.15\,\text{K}$	$R_\text{NE} = A_\text{NE} \exp(-t_\text{SEI}/t_{\text{SEI},\text{ref}})\, c_\text{NE}^{m_\text{NE}} \exp(-E_{A,\text{NE}}/RT)$
Positive-electrode solvent	$T > 393.15\,\text{K}$	$R_\text{PE} = A_\text{PE}\, \alpha^{m_{PE,1}} (1-\alpha)^{m_{PE,2}} \exp(-E_{A,\text{PE}}/RT)$
Electrolyte decomposition	$T > 473.15\,\text{K}$	$R_\text{ELE} = A_\text{ELE} \exp(-E_{A,\text{ELE}}/RT)\, c_\text{ELE}^{m_\text{ELE}}$

Each reaction contributes a heat release $\dot{Q}_k = H_k W_k R_k$ to $\dot{Q}_\text{abuse}$.

The concentrations governing these reactions are tracked by separate transport equations (no spatial diffusion):

$$ \frac{\partial c_\text{SEI}}{\partial t} = -R_\text{SEI}, \quad \frac{\partial c_\text{NE}}{\partial t} = -R_\text{NE}, \quad \frac{\partial t_\text{SEI}}{\partial t} = R_\text{NE}, \quad \frac{\partial \alpha}{\partial t} = R_\text{PE}, \quad \frac{\partial c_\text{ELE}}{\partial t} = -R_\text{ELE} $$

Variable table

Symbol	Field name	Description	Unit
$\phi^+$	`phiPos`	Positive electrode electric potential	V
$\phi^-$	`phiNeg`	Negative electrode electric potential	V
$T$	`T`	Temperature	K
$j$	`j`	Volumetric exchange current density	A/m²
$Y$	`Y_ech`	Electrochemical conductance	S/m²
$U_\text{OCV}$	`U`	Open circuit voltage	V
$\text{DOD}$	`DOD`	Depth of discharge	—
$\dot{Q}$	`ST`	Total volumetric heat source	W/m³
$c_\text{SEI}$	`cSEI`	Dimensionless SEI reactant concentration	—
$c_\text{NE}$	`cNE`	Dimensionless negative-electrode reactant	—
$t_\text{SEI}$	`tSEI`	Dimensionless SEI layer thickness	—
$\alpha$	`alpha`	Degree of conversion (positive electrode)	—
$c_\text{ELE}$	`cELE`	Dimensionless electrolyte concentration	—

Numerical Algorithm

The solution proceeds as follows each time step:

correct() (called at initialisation and after each solve):
- Update DOD from the current exchange current density $j$.
- Evaluate $Y(\text{DOD}, T)$ and $U_\text{OCV}(\text{DOD}, T)$.
- Update $j = Y(\phi^+ - \phi^- - U_\text{OCV})$.
- Compute heat sources $\dot{Q}_\text{ech}$ and $\dot{Q}_\text{ohm}$.
- Compute thermal abuse reaction rates and heat sources.
solveRegion(): Integrate the five abuse-species ODEs explicitly ($c_\text{SEI}$, $c_\text{NE}$, $t_\text{SEI}$, $\alpha$, $c_\text{ELE}$).
setCoupledEqns(): Assemble the FVM matrices for $\phi^+$, $\phi^-$, and $T$ into the global block system for monolithic coupling with adjacent tab regions.
postSolve(): Calls correct() again to update all derived fields after the potentials and temperature have been updated by the coupled solve.

Required Fields

Field	Type	Description	Unit
`phiPos`	`volScalarField`	Positive electrode potential	V
`phiNeg`	`volScalarField`	Negative electrode potential	V
`T`	`volScalarField`	Temperature	K

Required Dictionaries

constant/<region>/transportProperties – rho, cp, k, sigmaPos, sigmaNeg
constant/<region>/electrochemicalProperties – geometry (D, Dp, Dn, Dech),
 fitting coefficients (a0–a5, b0–b5,
 C1, C2, TRef), QBat, DOD
constant/<region>/thermalAbuseProperties – Arrhenius parameters (A*, EA*,
 m*, H*, W*) and initial species
 concentrations (cSEI, cNE, tSEI,
 alpha, cELE)

Case Setup

`multiRegionProperties` entry (standalone battery cell)

regions
(
 ( batteryRegion (NTGK) )
);

`multiRegionProperties` entry (battery with current-collector tabs)

regions
(
 ( batteryRegion (NTGK) )
 ( negTabRegion (conductNegPotentialTemperature) )
 ( posTabRegion (conductPosPotentialTemperature) )
);

DNA
{
 phiPos { maxCoupleIter 50; residualControl { maxJumpRes 1e-5; maxFluxRes 1e-5; } }
 phiNeg { maxCoupleIter 25; residualControl { maxJumpRes 1e-5; maxFluxRes 1e-5; } }
 T { maxCoupleIter 10; residualControl { maxJumpRes 1e-5; maxFluxRes 1e-5; } }
}

`controlDict` (singleRegionMode)

singleRegionMode yes;
regionType NTGK;

endTime 3700; // typical: one-hour discharge (3600 s) + margin
deltaT 1; // 1 s time step is typical

Coupling

Interface type	Coupled region types	Exchanged fields	Coupling mode
`defaultInterface`	`conductPosPotentialTemperature`	`phiPos`, `T`	partitioned (DNA)
`defaultInterface`	`conductNegPotentialTemperature`	`phiNeg`, `T`	partitioned (DNA)

The electrode potential and temperature boundary conditions at the tab interfaces are enforced via genericRegionCoupledJump / genericRegionCoupledFlux patch fields on the battery side, matching the normal current flux and temperature continuity across the jelly-roll / tab interface.

Known Limitations

No spatial charge transport in electrolyte — the model uses a volume-averaged (lumped) DOD rather than a spatially resolved lithium concentration in the electrolyte. Internal concentration gradients within the cell thickness are not captured.
Homogeneous jelly-roll — the electrode stack is treated as a single homogeneous material with effective properties ($\rho$, $c_p$, $k$). Layered electrode/separator geometry is not resolved.
Polynomial fit validity — the $Y$ and $U_\text{OCV}$ fits are only valid over the DOD and temperature range used for calibration. Extrapolation outside the calibration window will produce unphysical results.
Thermal abuse thresholds hard-coded — the onset temperatures for abuse reactions ($363.15\,\text{K}$, $393.15\,\text{K}$, $473.15\,\text{K}$) are fixed constants in the source code and cannot be changed via dictionary input.
Global (cell-average) DOD — the depth of discharge is tracked as a single scalar value integrated over the whole region volume. Local DOD variations within the cell are not resolved.
No electrochemical–mechanical coupling — swelling, lithiation-induced stress, and mechanical degradation are not modelled.

Tutorials

Tutorial path	Description
`tutorials/singleRegion/NTGK/cylindricalCell18650`	Standalone 18650 cylindrical cell discharge (singleRegionMode)
`tutorials/electrochemistry/NTGKBatteryWithTabs`	Prismatic battery cell with positive and negative current-collector tabs coupled via `defaultInterface`

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

multiPhysicsFoam – Thermal