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Transport in Porous Media

, Volume 88, Issue 2, pp 249–270 | Cite as

Derivation of Micro/Macro Lithium Battery Models from Homogenization

  • Francesco Ciucci
  • Wei Lai
Article

Abstract

In this article, we develop a micro–macroscopic coupled model aimed at studying the interplay between electrokinetics and transport in lithium ion batteries. The system studied consists of a solid (electrode material) and a liquid phase (electrolyte) with periodic microscopic features. In this work, homogenization of generalized Poisson–Nernst–Planck (PNP) equation set leads to a micro/macro formulation similar in nature to the one developed in Newman’s model for lithium batteries. Underlying conservation equations are derived for each phase using asymptotic expansions and mathematical tools from homogenization theory, starting from a PNP micromodel, and in particular Newman’s model is obtained as a corollary of the micro/macro approach developed here. The advantage of homogenization lies in the fact that effective parameters can be derived directly from the analysis of the periodic microstructure and from the application of the theory developed in this article. In addition, the advantages of using homogenization in Lithium ion battery modeling are outlined. Lastly, this work is a necessary step toward more general homogenized models and toward mathematical proofs, and it is also needed preliminary analysis for multiscale computational schemes.

Keywords

Newman’s model Porous electrode Lithium batteries Poisson–Nernst–Planck Homogenization 

List of symbols

+

Li+

e in the solid and X in the liquid

ν

Normal of Γ oriented from the liquid towards the solid

δ

Scale separation = l micro/l MACRO

δk(f)k

Approximation of f at order k in δ

Γ

Interface between liquid and solid phase oriented from the liquid towards the solid

γ

Thermodynamic factor \({=\gamma=1+\frac{\partial \log f(c)}{\partial \log c}}\)

\({\hat{\mu}}\)

Dimensionless electrochemical potential \({=\frac{e}{U_{\rm T}}\mu + z \phi}\)

Λ

Effective dimensionless diffusivity matrix

(f)0

Approximation of f at order 0 in δ

j

Current density

x

Macroscopic coordinate

y

Microscopic coordinate = x/δ

\({\mathcal L}\)

Liquid phase

\({\mathcal S}\)

Solid phase

μ

Chemical potential = \({\mu = k_{\rm B} T \log\left(f\pm(c)\frac{c}{\left(c\right)^0}\right)}\)

\({\nabla}\)

Gradient operator with respect to the dimensional coordinates

\({\nabla_x}\)

Gradient operator with respect to the macro coordinates

\({\nabla_y}\)

Gradient operator with respect to the micro coordinates

\({\phi}\)

Electrical potential

Φ1

Order 0 approximation of the reduced electrochemical potential of e in \({\mathcal S = \Phi_1 =-\frac{\left(\tilde \mu_-^{\rm s}\right)_0}{e}}\)

Φ2

Order 0 approximation of the reduced electrochemical potential of Li+ in \({\mathcal L = \Phi_ 2=\frac{\left(\tilde \mu_+^{\rm l}\right)_0}{e}}\)

Σ

Effective dimensionless conductivity tensor

σ

Conductivity \({=Dc \frac{e^2}{k_{\rm B}T}}\)

τ2

Tortuosity factor

\({\tilde \mu}\)

Electrochemical potential \({= \mu + ez\phi}\)

\({\tilde \mu^\star}\)

Reduced electrochemical potential \({=\frac{\hat{\mu}}{z e}}\)

\({\tilde \nabla}\)

Gradient operator with respect to the dimensionless coordinates

\({\tilde{{\bf x}}}\)

Dimensionless macroscopic coordinate

\({\tilde{{\bf y}}}\)

Dimensionless microscopic coordinate \({=\tilde{\bf x}/\delta}\)

ε

Porosity

ε0

Permittivity of vacuum

εr

Relative dielectric constant

c

Concentration

D

Diffusivity

kB

Boltzmann’s constant

l

Liquid

lMACRO

Macroscopic lengthscale

lmicro

Microscopic lengthscale

s

Solid

T

Temperature

UT

Thermal voltage \({=\frac{k_{\rm B} T}{e}}\)

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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Heidelberg Graduate School of Mathematical and Computational Methods for the Sciences, Interdisciplinary Center for Scientific ComputingUniversity of HeidelbergHeidelbergGermany
  2. 2.Department of Chemical Engineering and Materials ScienceMichigan State UniversityEast LansingUSA

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