Page 152 - Lindens Handbook of Batteries

MATHEMATICAL MODELING OF BATTERIES 6.5

cable is made of, as long as one can measure the conductivity of the material experimentally and
prove independently that each of the above laws holds true under the operating conditions of interest.
The utility of the model is versatile. For example, one can choose the material of which the weld is
made, by knowing the amount of heat that will be generated for a given discharge rate and duration.
Alternatively, one can determine the maximum amount of current that can be passed safely without
damaging the welds. Note that the values for I and t are specified depending on the situation of
interest, in this case, the discharge rate and the duration of discharge, respectively. These variables
specify the operating conditions. The variables V and ∆H represent the measured physical quantities.
A parameter like the conductivity of the metal, on the other hand, remains constant for different
durations or rates of discharge—it is an inherent property of the material. Every mathematical model
thus comprises input variables, measured variables, and physical parameters.
The next task in the development of a mathematical model for a battery is to build such compo-
nent models for other physical phenomena that take place within the battery during its operation.
The complexity of a mathematical model depends on the number of physical processes we wish
to describe and the accuracy with which these must be described. For example, if we had chosen
to ignore the heat-up at the weld-joints, one may adequately describe the flow of current across
the bus-bar using Eq. (6.1) only. On the other hand, for some materials, the resistance of the cable
changes with temperature. If we wish to capture these effects very accurately, we need to use addi-
tional equations that describe the change in the parameter R shown in Eq. (6.2). An efficient model
should thus strike a balance between the complexity associated with describing the data set and the
need for insight gained from the results. The following sections describe complementary approaches
used to develop such models. Finally, the various components are integrated to study the interaction
among the different parts of the battery in response to the mutually interactive physical phenomena
of interest. Based on the degree of comprehension the models provide, they are usually classified
into empirical and mechanistic models.

6.3 BUILDING EMPIRICAL MODELS

An empirical model often assumes the form of an expression that relates the operating conditions
such as the rate of discharge or the load across the battery with the measured quantities such as the
temperature or voltage of the battery. Such an expression is often obtained by prior knowledge of
the battery’s behavior or by trial and error. However, a basic understanding of the various limita-
tions within the cell serves as a guideline. For example, the equivalent circuit shown in Fig.6.3 was
designed to account for the deviation of the cell voltage from the open circuit voltage due to several
factors as described in Sec. 6.1.1. A popular objective of a battery model is to predict the cell voltage
as a function of the state-of-charge (SOC) of the battery. This section demonstrates the development
of such a relationship for the circuit shown in Fig. 6.3.
The voltage drop across each of the resistors R and R follows Ohm’s law [see Eq. (6.1)]. The
ct
rate of charge buildup in the capacitor C equals the current that flows through the capacitor. This is
D
expressed mathematically as follows:
dq
I = (6.3)
2
dt
Kirchoff’s node-and-loop rules relate the currents that flow across the different branches of the cir-
cuit and the voltage across each branch. These rules state that the voltage across any branch is the
sum of all voltage drops along the branch and that the sum of all currents that enter or exit a node on
the circuit is zero. For example, the total current I that branches out into I and I as shown in Fig. 6.3.
2
3
Hence according to Kirchoff’s laws we have
I = I + 2 I (6.4)
3