Lithium-ion / Doyle-Fuller-Newman model

On this page, we describe the different models available in BattMo for simulation a lithium ion battery cell. We have models available for simulating 1D and 3D geometries:

• P2D: Pseudo-two-dimensional model

• P4D Pouch: Pseudo-four-dimensional model for pouch cells

• P4D Cylindrical: Pseudo-four-dimensional model for cylindrical cells

P2D model

Charge conservation in the electrode

In the solid particles that make up the electrodes, the charge conservation equation is given by

$$-\frac{\partial }{\partial x}{\sigma }_{\text{eff}}\frac{\partial }{\partial x}{\varphi }_{\text{s}}=a_{\text{s}}Fj.$$

Here is ${\sigma }_{\text{eff}}$ the effective conductivity of the electrode, ${\varphi }_{\text{s}}$ is the potential, $a_{\text{s}}$ is the volumetric surface area (specific interfacial surface area) of the electrode, $F$ is the Faraday constant, and $j$ is the rate of lithium flux (reaction rate). This equation describes electron movement. It is a linear diffusion equation with a forcing term that models the flux of electrons. This flux is equal to the local flux of lithium from the electrode to the electrolyte.

The boundary conditions are:

$${\sigma }_{eff}^n\frac{\partial }{\partial x}{\varphi }_s^n{|}_{x=0}={\sigma }_{eff}^p\frac{\partial }{\partial x}{\varphi }_s^p{|}_{x=L^{tot}}=\frac{i_{app}}{A},$$$$\frac{\partial }{\partial x}{\varphi }_s^n{|}_{x=L^n}=\frac{\partial }{\partial x}{\varphi }_s^p{|}_{x=L^n+L^p}=0.$$

Here is $i_{app}$ the electrical current at the terminals of the cell, $L^{tot}=L^n+L^s+L^p$ where $L^n$ is the thickness of the negative electrode, $L^s$ is the thickness of the separator and $L^p$ is the thickness of the positive electrode. $A$ is the surface area of the current collector or electrode.

The initial values are:

$${\sigma }_{s,0}^n=0,{\sigma }_{s,0}^p=U_{ocp}^p({\theta }_{s,0}^p)-U_{ocp}^n({\theta }_{s,0}^n),$$

where ${\theta }_s=c_s/c_{s,max}$ is the stoichiometry of the electrode such that $0\le {\theta }_s\le 1$, and $U_{ocp}({\theta }_s)$ is the open-circuit potential (OCP) of the electrode.

Mass conservation in the electrode

The mass conversation in the solid electrode particles is defined as:

$$\frac{\partial c_s}{\partial t}=\frac{1}{r^2}\frac{\partial }{\partial r}(D_s{r}^2\frac{\partial c_s}{\partial r}),$$

where $c_s$ is the concentration of lithium in the solid electrode particles, and $D_s$ is the diffusion coefficient of the electrode. This PDE is a reformulation of Fick's second law in spherical coordinate, assuming spherical symmetry.

The boundary conditions are:

$$-4\pi r^2{D}_{\text{s}}\frac{\partial c_{\text{s}}}{\partial r}(t,x,r)=\frac{a_{\text{s}}j}{{\epsilon }_{\text{s}}}\frac{4\pi r^3}{3},D_s\frac{\partial c_s}{\partial r}{|}_{r=0}=0,$$

where ${\epsilon }_s$ is the volume fraction of the electrode.

The initial values are:

$$c_{s,0}=c_{s,max}({\theta }_0+z_0({\theta }_{100}-{\theta }_0)),$$

where $0\le z_0\le 1$ is the initial cell SOC. ${\theta }_0$ is when the SOC is $0$, and ${\theta }_{100}$ is when the SOC is $100$.

Charge conservation in the electrolyte

The charge conservation in the electrolyte is given by

$$-\frac{\partial }{\partial x}{\mathbf{\text{j}}}_{\text{e}}=a_{\text{s}}Fj,$$

where

$${\mathbf{\text{j}}}_{\text{e}}=-{\kappa }_{\text{e},\text{eff}}\frac{\partial {\varphi }_{\text{e}}}{\partial x}-{\kappa }_{\text{e},\text{eff}}\frac{1-t_{+}}{z_{+}F}\left(\frac{\partial \mu }{\partial c}\right)\frac{\partial c_{\text{e}}}{\partial x}.$$

Here $\mu =2RT\log (c_{\text{e}})$ is the chemical potential, with $R$, the universal gas constant, $T$, the temperature, and $c_{\text{e}}$ the concentration of lithium in the electrolyte. ${\kappa }_{e,eff}$ is the effective conductivity of the electrolyte, $z_{+}$ is the charge number, and $t_{+}$ is the transference number of the positive ion in the electrolyte with respect to the solvent. The effective quantities are computed from the intrinsic properties and the volume fraction using a Bruggemann coefficient, denoted $b$, which yields ${\kappa }_{\text{e},\text{eff}}={\epsilon }_{\text{e}}^b{\kappa }_{\text{e}}$. For the electrolyte, we have a spatially dependent Bruggeman coefficient.

The boundary conditions must be such that all current at the current collector boundaries are electronic, and all current at the separator boundaries are ionic:

$${\mathbf{\text{j}}}_{\text{e}}{|}_{x=0,x=L^{tot}}=0,$$$${\mathbf{\text{j}}}_{\text{e}}{|}_{x=L^n,x=L^n+L^s}=\frac{i_{app}}{A}.$$

The initial condition is:

$${\varphi }_{e,0}=-U_{ocp({\theta }_{s,0}^n)}^n$$

Mass conservation in the electrolyte

The mass conservation in the electrolyte is modeled as:

$$\frac{\partial }{\partial t}({\epsilon }_{\text{e}}{c}_{\text{e}})+\frac{\partial }{\partial x}{\mathbf{\text{N}}}_{\text{e}}=a_{\text{s}}j,$$

where ${\epsilon }_e$ is thevolume fraction of the electrolyte, and the flux ${\mathbf{\text{N}}}_{\text{e}}$ is equal to:

$${\mathbf{\text{N}}}_{\text{e}}=-D_{\text{e},\text{eff}}\frac{\partial c_{\text{e}}}{\partial x}+\frac{t_{+}}{z_{+}F}{\mathbf{\text{j}}}_{\text{e}}.$$

The effective diffusion coefficient is calculated by $D_{\text{e},\text{eff}}={\epsilon }_{\text{e}}^b{D}_{\text{e}}$. The boundary conditions enforce continuity of electrolyte concentration and flux of lithium across the cell, and enforces that there is no movement of lithium from the inside of the cell to the exterior of the cell:

$$\frac{\partial c_e^n}{\partial x}{|}_{x=0}=\frac{\partial c_e^p}{\partial x}{|}_{x=L^{tot}}=0$$$$D_{e,eff}^n\frac{\partial c_e^n}{\partial x}{|}_{x=L^n}=D_{e,eff}^s\frac{\partial c_e^s}{\partial x}{|}_{x=L^n}$$$$D_{e,eff}^s\frac{\partial c_e^s}{\partial x}{|}_{x=L^n+L^s}=D_{e,eff}^p\frac{\partial c_e^p}{\partial x}{|}_{x=L^n+L^s}$$$$c_e^n{|}_{x=L^n}=c_e^s{|}_{x=L^n}$$$$c_e^s{|}_{x=L^n+L^s}=c_e^p{|}_{x=L^n+L^s}$$

The initial values are>

$$c_e=c_{e,0}$$

Reaction kinetics

The reaction rate is equal to the rate of lithium flux from the electrode particles into the electrolyte:

$$j=j_0(c_{\text{s}},c_{\text{e}},T)(e^{\alpha F\frac{{\eta }_{\text{s}}}{RT}}-e^{-(1-\alpha )F\frac{{\eta }_{\text{s}}}{RT}}).$$

where ${\eta }_{\text{s}}$ and $j_0$ denote the overpotential and the reaction exchange current density. The overpotential ${\eta }_{\text{s}}$ is given by

$${\eta }_{\text{s}}={\varphi }_{\text{s}}-{\varphi }_{\text{e}}-U_{\text{ocp}}(c_{\text{s}},T).$$

where $U_{\text{ocp}}$ denotes the open circuit potential, given as a function of the Lithium concentration in the electrode and the temperature. The exchange current density is given by

$$j_0=k_{\text{s},0}{(c_{\text{e}}(c_{\text{s},max}-c_{\text{s}})c_{\text{s}})}^{\frac{1}{2}}nF.$$

DFN Model Parameters (BattMo)

This table lists all required parameters from the DFN model used in BattMo.

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Negative Electrode	Separator	Positive Electrode	Description	BattMo Name
${\sigma }_{\text{eff}}^n$		${\sigma }_{\text{eff}}^p$	Effective electrode conductivity	ElectronicConductivity
$a_{\text{s}}^n$		$a_{\text{s}}^p$	Specific interfacial surface area	VolumetricSurfaceArea
$D_s^n$		$D_s^p$	Lithium diffusivity in solid phase	DiffusionCoefficient
${\epsilon }^n$	${\epsilon }^s$	${\epsilon }^p$	Porosity	Porosity
$c_{s,\text{max}}^n$		$c_{s,\text{max}}^p$	Max lithium concentration in solid phase	MaximumConcentration
${\kappa }_{\text{e}}$	${\kappa }_{\text{e}}$	${\kappa }_{\text{e}}$	Electrolyte conductivity	IonicConductivity
$D_{\text{e}}$	$D_{\text{e}}$	$D_{\text{e}}$	Electrolyte diffusivity	DiffusionCoefficient
$U_{\text{ocp}}^n(\theta )$		$U_{\text{ocp}}^p(\theta )$	Open circuit potential as function of stoichiometry	OpenCircuitPotential
${\theta }_0^n$		${\theta }_0^p$	Stoichiometry at 0% SOC	StoichiometricCoefficientAtSOC0
${\theta }_{100}^n$		${\theta }_{100}^p$	Stoichiometry at 100% SOC	StoichiometricCoefficientAtSOC100
$z_0$			Initial state of charge	InitialStateOfCharge
$k_s^{n}0$		$k_s^{p}0$	Reaction rate constant	ReactionRateConstant
$E_a^n$		$E_a^p$	Activation energy of the reaction	ActivationEnergyOfReaction
${\alpha }^n$		${\alpha }^p$	Charge transfer coefficient	ChargeTransferCoefficient
$A$		$A$	Electrode surface area	ElectrodeGeometricSurfaceArea
$T$	$T$	$T$	Temperature	InitialTemperature
$t_{+}$	$t_{+}$	$t_{+}$	Transference number	TransferenceNumber
$z_{+}$	$z_{+}$	$z_{+}$	Charge number of positive ion	ChargeNumber
$\text{c\_e\_0}$	$\text{c\_e\_0}$	$\text{c\_e\_0}$	Initial electrolyte concentration	Concentration
$b^n$	$b^s$	$b^p$	Bruggeman coefficient (porosity scaling)	BruggemanCoefficient
$L^n$	$L^s$	$L^p$	Thickness	Thickness
$r^n$		$r^p$	Radius of particles in electrode	ParticleRadius

P4D Pouch

P4D Cylindrical