Page 151 - Lindens Handbook of Batteries

6.4 PRINCIPLES OF OPERATION

(e.g., charge, mass, momentum, and energy balances) to characterize the behavior of the individual
components within a cell as functions of their material properties. These component models can then
be integrated to form rigorous mechanistic models to describe the behavior of the cell. Approximation
of properties of composite electrodes over macroscopic volume elements has been used to develop
mathematical models that use practically measurable physical properties such as effective conduc-
tivities and diffusivities. Today, people in industry are using sophisticated physics-based models to
determine the precise design measurements, such as actual geometry of the electrodes and the elec-
tronic circuitry needed to integrate batteries with other components in a device. 5
The difficulties presented by the mathematical tedium of the models have by and large been
™
removed by the development of excellent user-friendly interfaces such as Matlab and COMSOL
™
Multiphysics . As a result, the demand for development of universal standards for batteries and
the use of models to determine such standards have reached an all-time high. Thus battery models
have evolved significantly over the empirical relationships that served as rules of thumb in the bat-
tery industry. However, the principal objective behind the models has remained the same: to pre-
dict whether the battery can deliver the required output (in terms of energy or power) through the
required amount of time. The following sections describe the process of developing a mathematical
model for a cell, the choice of model equations and parameters, and some examples of implementing
such models for optimal design of batteries.

6.2 DEVELOPMENT OF A MATHEMATICAL MODEL

Developing a mathematical model for a battery involves identifying the physical processes that
take place during the operation of the battery and how each component responds to those processes.
A systematic approach to represent the response of the components to the various processes is
by using generalized laws that describe the behavior of the materials under different scenarios.
The simplest example is the representation of current flow through a copper wire: When current
is drawn or supplied to a battery, it invariably passes through the bus-bars or tabs that connect the
electrodes to the external load (or power source). The first step is to identify the physical processes
that happen across the copper cable once the battery is connected to the load: There is a flow of
current across the cable; there is some heat-up of the cable—especially at the welded joints—
that increases with increase in the flow of current, and so on. The second step is to identify the
general rules that help us to quantify each of these phenomena. For example, the flow of current
across the copper cable follows Ohm’s law, which states that the potential drop (V) depends on
the amount of current (I) that flows across a metallic cable and the resistance the metal offers to
the flow of current:

V = IR (6.1)
where R represents the resistance of the metal. Equation (6.1) makes it clear that if the value of the
resistance R is higher, the potential drop required to pass a given amount of current out of the battery
increases. This is in line with our observation that the voltage of a battery during discharge with a
rusty plug (having a lower electronic conductivity) drops faster compared to that with proper con-
tacts. The second phenomenon of interest in this example is the heat-up of the welds. The heat-up of
a material due to passage of current was first quantified by Joule using the following relationship:

2
∆H = I R (6.2)
t
Here, ∆ H represents the amount of heat generated, R is the resistance at the weld (the heat is a func-
tion of the conductivity of the material that the weld is made up of), and t is the duration for which
the current flows across the weld. Thus we see that Eqs. (6.1) and (6.2) can be used to adequately
describe some physical observations during the passage of current through a cable; these equations
constitute the mathematical model in this example. These equations apply to any material that the