Page 148 - Lindens Handbook of Batteries

CHAPTER 6

MATHEMATICAL MODELING
OF BATTERIES

Shriram Santhanagopalan and Ralph E. White

6.1 INTRODUCTION

Mathematical modeling of batteries can be described as a process of developing an equation or a set
of equations to describe the performance of a battery. For example, a simple, single equation model
can be used to predict the capacity of a battery as a function of the discharge current obtained from
that battery. More complicated models can be developed based on equations used to describe the
phenomena that occur between the current collectors of a single pair of electrodes. For example, a
model for a lithium-ion cell with one spatial coordinate (from the anode to the cathode, say) could
consist of the anode current collector (e.g., copper metal foil), an anode electrode coating made
of carbon, a separator, a cathode electrode coating (e.g., made of LiCoO ), and finally the cathode cur-
2
rent collector (e.g., aluminum foil). Such a model would be based on the spatial coordinate between
the current collectors and a unit of projected electrode coating area for the other two spatial coordi-
nates. This model can be used, for example, to predict the performance of a lithium-ion jelly roll by
appropriately accounting for the current collectors that are coated on both sides, the actual projected
electrode area to form the cell, etc. A mathematical model of a battery with multiple electrode pairs
could then be formed by internal or external connections between the cells in a series or parallel
arrangement as needed for the voltage and capacity requirements.
The level of detail of a mathematical model depends on its intended use. For example, complex
three-dimensional models (three spatial coordinates and time) with multiple electrode pairs have
been developed to study the thermal characteristics of the electrode pairs and cells made from them.
This chapter begins with a description of the evolution of battery models.

6.1.1 Evolution of Battery Models
The earliest mathematical models for batteries were simply empirical relationships between mea-
sured parameters, such as the battery voltage, overall resistance, density of the electrolyte, pressure
within the can or temperature of the cell, versus the remaining capacity under different operating
conditions. These models are still used today, and perhaps the best known example is Peukert’s rela-
1
tionship. This equation has been used to represent the discharge capacity as a function of discharge
current for a lead-acid cell, for example, as shown in Fig. 6.1, which shows a comparison between
the capacity predicted by Peukert’s equation and the experimentally measured value for the cell
capacity. This simple relationship has been used under a variety of different scenarios for several
decades in the battery industry.

6.1