Mathematical Modeling of Lithium Batteries

  • Karen E. Thomas
  • John Newman
  • Robert M. Darling

Abbreviations

List of Symbols

a

surface area of active material per volume of electrode m−1

c

salt concentration in the electrolyte (mol/m3 of solution)

ci

concentration of species i (mol/m3)

cs

concentration of lithium in the solid insertion electrode (mol/m3)

cT

total concentration of salt and solvent (mol/m3)

C

double-layer capacitance (F/m2)

Cp

heat capacity (J/m2·K)

Di

diffusion coefficient of species i in dilute solution theory (m2/s)

D

salt diffusion coefficient (m2/s)

\( \mathcal{D} \)

diffusion coefficient based on thermodynamic driving force (m2/s)

Ds

diffusion coefficient of lithium in an insertion electrode (m2/s)

\( \mathcal{D}_{i,j} \)

diffusion coefficient for interaction of species i and j (m2/s)

f±

mean molar activity coefficient of an electrolyte

F

Faraday’s constant, 96487 C/equiv

in

transfer current normal to the surface of the active material (A/m2)

i0

exchange current density (A/m2)

i2

current density (A/m2 superficial area) in the electrolyte

I

total current density in the cell (A/m2)

j

total fiux due to reaction (mol/s·m2 of active material)

k

thermal conductivity (W/m2·K)

ka, kc

rate constants for the anodic and cathodic directions of a reaction

km

mass transfer coefficient (m/s)

L

thickness of an electrode (m)

L+, Ls, L

thickness of positive electrode, separator, or negative electrode (m)

m

molality (mol/kg)

Mi

symbol for the chemical formula of species i or molecular weight (g/mol)

n

number of electrons involved in a half reaction

Ni

flux of species/(mol/s·m2 of apparent area)

q

charge on the electrode side of the double layer (C/m2)

qi

surface charge density of species i on the solution side of the double layer (C/m2)

Q

coulombic capacity of an electrode (C, C/m2, or C/m3)

\( \dot Q \)

heat-generation rate (W/m2)

r

radial position across a spherical particle (m)

rk

rate of chemical reaction k(mol/s·m3)

R

universal gas constant, 8.3143 J/mol K, or radius of a particle (m)

Rfilm

effective resistance of a solid-electrolyte interphase (Ωm2)

s

stoichiometric coefficient, positive for anodic reactants

t

time (s)

ti0

transference number of species i with respect to the solvent velocity

T

temperature, K

u

mobility (m2·mol/J·s)

U

thermodynamic potential measured with respect to a lithium reference electrode (V)

v

velocity (m/s)

V

cell potential (V)

\( \bar V \)

molar volume (m3/mol)

x

position across cell (m)

y

stoichiometry of lithium in an insertion electrode

zi

charge of ion i

Greek

α

transfer coefficient

β

symmetry factor for an elementary reaction

ε

volume fraction (of electrolyte unless otherwise specified)

Γi

excess concentration of species i in the double layer (mol/m2)

κ

effective ionic conductivity (S/m)

μ

chemical potential (J/mol)

νi

moles of ion i produced when a mole of its salt dissociates

ν

number of moles of ions into which a mole of electrolyte dissociates

ρ

density (kg/m3)

σ

effective electronic conductivity of a porous electrode (S/m)

Θ

fraction of total lithium insertion sites which are occupied by lithium

Θs

site on the lattice of the insertion material which can be occupied by lithium

Θp

site on the lattice of a crystalline polymer which can be occupied by lithium salt

Φ

potential

γ±

mean molal activity coefficient

γl

exponent for the dependence of i0 on the concentration of species i

Superscripts

o

property is with respect to solvent velocity or initial condition

Θ

secondary reference state of the chemical potential

Subscripts

a

anodic

c

cathodic

dl

double-layer

e

electrolyte

f

faradaic

i

species i

lim

limiting current

m

main reaction

n

electrochemical flux normal to surface of active material

o

solvent in an electrolytic solution

s

side reaction

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Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Karen E. Thomas
    • 1
  • John Newman
    • 1
  • Robert M. Darling
    • 2
  1. 1.Department of Chemical EngineeringUniversity of California at Berkeley & Lawrence Berkeley National LabsBerkeleyUSA
  2. 2.International Fuel CellsSouth WindsorUSA

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