Homogenizing a Heterogeneous Electrode Model

This example demonstrates how to calculate the effective transport properties of a heterogeneous 3D geometry of an nickel manganese cobalt (NMC) electrode. The effective flux parameter is then used to create a representative homogenized 1D model of the NMC electrode.

By homogenizing, memory requirements and computational times are reduced by orders of magnitude.

See also the Heterogeneous NMC Electrode tutorial for how to define the corresponding heterogeneous model.

Model Definition

Figure 1 shows the 3D geometry generated from tomography data (Ref. 1) using a Model Method. The geometry consists of one separator domain, one domain representing the porous conductive binder, and number of particle domains. How the geometry is created is documented in the Heterogeneous Electrode Geometry Generation tutorial.

Figure 1: Model geometry.

Model for calculating effective transport parameters

The homogenized model of the NMC electrode uses one single domain to define all transport processes occurring in the electrode and electrolyte phases. The homogenized equations make use of effective (volume-averaged) transport parameters, which are calculated by multiplying the diffusivity and conductivities of the porous conductive binder by an effective flux parameter feff. This parameter is also sometimes referred to as the McMullin number (Ref. 2).

The effective flux parameter will depend on both the volume fraction,

, and the tortuosity,

, of the porous conductive binder domain according to

(1)

For a homogeneous domain (i.e. with no particles present) with volume fraction and tortuosity both equal to unity, feff = 1.

To calculate the effective transport coefficient for the porous conductive domain, we will solve for the Laplace equation (i.e. a diffusion equation with the diffusion coefficient equal to unity) using the dependent variable u.

(2)

in the porous conductive binder domain together with the boundary conditions

(3)

at the boundary facing the separator and

(4)

at the boundary facing the NMC current collector.

By integrating the flux in the solved model at the NMC current collector boundary, our effective flux parameter felec can be calculated as

(5)

Homogenized battery model

Figure 2 shows the 1D homogenized model geometry.

Figure 2: 1D geometry used in the homogenized model.

The model is defined using the interface similarly to the Heterogeneous NMC Electrode tutorial, with the following differences:

•

The node is used to define the homogenized mixture of electrode particles, binder and electrolyte material of the electrode, considering the volume fractions of the electrode and electrolyte phases in the 3D geometry and the effective flux parameter feff .

•

A child node to the defines the equilibrium potential, and the electrode kinetics using a specific surface area parameter based on the 3D geometry.

•

The diffusion of solid lithium is solved for using an extra dimension defined by the child node to the , using an average particle radius that was calculated by the model method when creating the 3D geometry. (In the Heterogeneous NMC Electrode tutorial a separate interface was used to model the intercalated lithium diffusion.)

For more details on this homogenized 1D approach to lithium battery cell modeling, see the 1D Isothermal Lithium-Ion Battery tutorial.

Results and Discussion

Figure 3 shows the Laplacian flux through the porous conductive binder domain. Integrating the flux on the boundary and using Equation 5 results in the value feff=0.55.

Figure 3: Flux through the porous conductive binder domain when solving for the Laplace equation.

Figure 4 shows the 2C discharge curve of the battery, and compares the result to the heterogeneous model. The resulting discharge voltages of the two models match well during the discharge, apart from the final part of the discharge, where diffusion limitations of intercalated lithium in the largest particles of the heterogeneous model result in a faster drop of the discharge voltage. (Increasing the average radius parameter of the homogeneous model about 20% will make the two curves more or less overlap in this case.)

Figure 4: 2C discharge curves comparing the results from the homogenized model to the original heterogeneous model.

Figure 5 shows the Nyquist plot from the EIS simulation, compared to the heterogeneous model results. The two semi-circles, related to the charge transfer, more or less overlap, whereas the approx 45º tail, related to the intercalated lithium diffusion, show larger differences between the two models.