Page 165 - Lindens Handbook of Batteries

6.18 PRINCIPLES OF OPERATION

Aluminum Copper
current Active Binder + filler Active Binder + filler current
collector material material Electrolyte material material collector
Model in the
r-coordinate shows
changes within the
particles

Cathode Separator Anode
Model in the x-
coordinate x = 0 x = l p + l s + l n
shows changes 1-Dimensional model
across the
electrode l p l s l n
thickness
FiguRE 6.11 Schematic of a lithium-ion cell used to develop a one-dimensional model along the thickness of the electrodes.

6.8 INTERCALATION IN POROUS ELECTRODES

The intercalation mechanism in porous electrodes has been represented in mathematical models in
several ways. The simplest treatment considers the phenomenon as diffusion of ions into a solid
solution. Fick’s law is used to represent this process. The electrode particles are usually assumed to
be of a regular geometry (see Fig. 6.11). For example, the diffusion of ions within spherical particles
is governed by the following equation:
2
∂c  ∂ c 2 ∂ 
c
s = D s  s + s  (6.46)
∂ 
∂t  ∂r 2 r r
The subscript s in Eq. (6.46) is used to refer to the solid particles. The concentration of the ions at the
surface of the particles is mathematically connected to the electrolyte concentration at the interface
through the Butler-Volmer equation [(see Eq. (6.40)].
Figure 6.12 summarizes the utility of a mechanistic model in cell design. Several thought experi-
ments can be simulated by altering the different design parameters, such as the particle size, as well
as material properties, such as the conductivity. The model is used to identify the critical factor that
limits performance of the cell at high rates.

6.9 ENERGY BALANCE

Temperature control has been a concern with a lot of battery chemistries. In some cases, the effect
of an abnormal temperature is a reduction in performance, whereas in others it may lead to concerns
over safe operation of the battery. Heat generation within a battery is usually modeled using an
energy balance equation that relates the heat generated due to Joule heating, chemical/electrochemi-
cal reactions etc., to the heat exchange with the environment in which the battery operates. A general
form of the material balance equation is shown below 11
∂(ρcT )
l
P =∇⋅∇ ) T + q (6.47)
(
∂t