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

Here is the effective conductivity of the electrode, is the potential, is the volumetric surface area (specific interfacial surface area) of the electrode, is the Faraday constant, and 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:

Here is the electrical current at the terminals of the cell,     where is the thickness of the negative electrode, is the thickness of the separator and is the thickness of the positive electrode. is the surface area of the current collector or electrode.

The initial values are:

where   is the stoichiometry of the electrode such that   , and 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:

where is the concentration of lithium in the solid electrode particles, and 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:

where is the volume fraction of the electrode.

The initial values are:

where    is the initial cell SOC. is when the SOC is , and is when the SOC is .

Charge conservation in the electrolyte

The charge conservation in the electrolyte is given by

where

Here   is the chemical potential, with , the universal gas constant, , the temperature, and the concentration of lithium in the electrolyte. is the effective conductivity of the electrolyte, is the charge number, and 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 , which yields  . 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:

The initial condition is:

Mass conservation in the electrolyte

The mass conservation in the electrolyte is modeled as:

where is thevolume fraction of the electrolyte, and the flux is equal to:

The effective diffusion coefficient is calculated by  . 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:

The initial values are>

Reaction kinetics

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

where and denote the overpotential and the reaction exchange current density. The overpotential is given by

where 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

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
			Effective electrode conductivity	ElectronicConductivity
			Specific interfacial surface area	VolumetricSurfaceArea
			Lithium diffusivity in solid phase	DiffusionCoefficient
			Porosity	Porosity
			Max lithium concentration in solid phase	MaximumConcentration
			Electrolyte conductivity	IonicConductivity
			Electrolyte diffusivity	DiffusionCoefficient
			Open circuit potential as function of stoichiometry	OpenCircuitPotential
			Stoichiometry at 0% SOC	StoichiometricCoefficientAtSOC0
			Stoichiometry at 100% SOC	StoichiometricCoefficientAtSOC100
			Initial state of charge	InitialStateOfCharge
			Reaction rate constant	ReactionRateConstant
			Activation energy of the reaction	ActivationEnergyOfReaction
			Charge transfer coefficient	ChargeTransferCoefficient
			Electrode surface area	ElectrodeGeometricSurfaceArea
			Temperature	InitialTemperature
			Transference number	TransferenceNumber
			Charge number of positive ion	ChargeNumber
			Initial electrolyte concentration	Concentration
			Bruggeman coefficient (porosity scaling)	BruggemanCoefficient
			Thickness	Thickness
			Radius of particles in electrode	ParticleRadius

P4D Pouch

P4D Cylindrical