Page 156 - Lindens Handbook of Batteries

MATHEMATICAL MODELING OF BATTERIES 6.9

6.4 MECHANISTIC MODELS

Mechanistic models relate the battery characteristics to physical properties of the constituent materials.
Such properties are usually measurable in independent experiments.
For example, a mechanistic model for the Ohm’s law equation shown in Eq. (6.1) can be built
by describing the resistance parameter R in terms of physically measurable properties of the copper
bus-bar. In this case, the properties of interest are the electronic conductivity of the metal (σ ), the
c
cross-sectional area (A ), and the length of the bus-bar (L). Each of these properties is characteristic
c
of the bus-bar. The resistance R is related to these parameters as follows:
L
R = (6.10)
σ cc
A
and hence Eq. (6.1) can be rewritten as
L
V = I (6.11)
σ cc
A
Note that Eq. (6.11) can be used for a cable of any given dimension, made up of any material whose
electronic conductivity is known. The use of Eq. (6.1), however, requires that we measure the resis-
tance parameter R every time the bus-bar is replaced. Also, the use of a mechanistic model enables one
to identify better materials (e.g., a bus-bar with higher electronic conductivity) suited for an applica-
tion. We now proceed to developing mechanistic models to describe the other physical processes that
take place within the battery. A few processes commonly encountered in batteries are movement of
ions in the electrolyte, movement of electrons within the electrodes, and chemical and electrochemi-
cal reactions. See Chap. 1 for background information on the basic equations governing each of these
processes. In this section, we employ these concepts to build a mechanistic model for a battery.

6.4.1 Charge Transport by Electrons

The total voltage across a cell (V) can be approximated as the sum of the potential drop across the elec-
trodes, across the electrolyte, and other losses arising from contact resistances. In the following sections,
subscript 1 will be used to denote the properties/variables in the electrodes and subscript 2 to represent
the corresponding variables in the electrolyte. We already considered voltage drop due to the flow of
electrons across metal cables in Eq. (6.11) above. An equivalent of this equation can be used to represent
contact resistances. The voltage drop across the electrodes (∇φ , 1 j ) is also governed by Ohm’s law
i 1
=
∇ 1, j =φ - eff , jn or p (6.12)
σ
j
eff
where i is the current per unit area (called the current density) and σ is the effective electronic
j
1
conductivity of the electrode material within electrode j ( j = n refers to the negative electrode or
the anode and j = p refers to the positive electrode or the cathode). Usually a battery electrode is
comprised of several components such as solid solutions of different metals, or composites of active
material, binders, and other components. The effective conductivity corrects for the additional com-
ponents within the electrode. Usually the effective conductivity is calculated as the sum of the con-
ductivities of the individual components, scaled in proportion to the composition of the electrode
σ eff = ∑ x k σ k (6.13)
j
k
Here x is the proportion of the individual components k that constitute the electrode, and σ refers
k
k
eff
to the electronic conductivities of the pure components. Alternatively, σ can be measured directly
j
after the electrode is assembled.