Page 160 - Lindens Handbook of Batteries

MATHEMATICAL MODELING OF BATTERIES 6.13

6.4.5 Distribution of Ions
Equation (6.24) relates the driving force for the generation of electricity to the concentration of the
participating chemical species at the electrode-electrolyte interface. All the concentration terms
(c and c ) are defined at the reacting interface. It is difficult to monitor the concentration of chemicals
s
at the electrode surface. A material balance for the ions relates the concentration in the bulk of the
solution to that at the electrode surface. The material balance equation states that the concentration
of the ion changes with time in accordance with the flux of the ions 11
∂c
(
k =-∇⋅ N ) + R (6.29)
∂t k k
The flux used in the material balance is consistent with the one used to determine the conductivity
of the electrolyte [Eq. (6.22)]. The term R refers to change in concentration of the ion if it were
k
consumed or generated in a reaction k. At the electrode-electrolyte interface, the change in concen-
tration of the ions is because of the electrochemical reaction, and hence Eq. (6.15) is used to relate
the amount of ions that are produced by the reaction to the amount of ions present at the electrode-
electrolyte boundary. In the case of electrolytes at higher concentrations, interactions among the
ions must be considered. For example, the diffusion of one species of ions depends on its interaction
with ions of every other kind present within the electrolyte. Complexities such as this are usually
handled by defining an effective property that considers such interactions. In this case, the following
expression is used for the flux:

ˆ
N = ˆ c ( ˆ v + v ˆ) D c+∇ (6.30)
In Eq. (6.30), properties such as the diffusion coefficient are interpreted as effective properties. Note
ˆ
that the effective flux ( )N is now a function of the electrolyte concentration (c) and not the concen-
tration of the individual ions (c ). The velocity term ˆ v now relates to an effective field within the
k
electrolyte and is usually expressed in terms of the transport number ()t + 0
0
cv ˆ = -(1 t ) i 2 (6.31)
+
F
Expressions such as Eq. (6.30) enable one to obtain values for diffusivity or conductivity of the
electrolyte experimentally measured using the actual mixture. The equations outlined in Sec. 6.4
constitute the mathematical framework for the mechanistic model of a battery. The following
sections illustrate a few examples using these equations for some commonly encountered battery
chemistries.

6.5 KINETIC MODEL OF A SILVER VANADIUM OXIDE CELL

The Silver Vanadium Oxide (SVO) cell (see Ch. 31) is commonly used as a primary cell in medical
devices, for example. The cathode reaction can be written as follows 12

+
-
+
5
+
5
0
+
+
Ag VO + 11 x + ( y Li + + x + ( y e → Li xy Ag 2- x 4-y 11 + xAg (6.32)
)
O
V
)
-
+
24
It is assumed that two electrochemical reactions take place at the cathode
+
+
+
0
V O +
Ag VO + 5 + 11 xLi + + xe → - LiAg 2- x 4 5 + 11 xAg (6.33)
24
x
+
+
+
-
O
Ag VO + 5 + 11 yLi + + ye → LiAgV 5+ y 11 (6.34)
24-
y
24