Page 158 - Lindens Handbook of Batteries
P. 158
MATHEMATICAL MODELING OF BATTERIES 6.11
where D is the diffusion coefficient of ions k. In the case of flow batteries, there is a convective
k
velocity in addition to v in Eq. (6.20). Thus the modified flux is now given by
k
N = k c v + k ( k - ) v D ∇ k c (6.21)
k
Here v is the velocity of electrolyte flow. Combining Eqs. (6.17) and (6.21) gives 7
φ
N =- z uFc ∇- D ∇ k c + k c v (6.22)
k k
k
k
k
The equation presented above represents the case of dilute electrolytic solutions. More sophisticated
models that consider the mutual interaction of ions within the electrolyte and the effects of tempera-
ture on the conductivity of the electrolyte are available. 8
6.4.3 Driving Forces for Charge Transfer Across the Interface
The unique component of storing charge in batteries is the conversion of chemical energy into
electrical energy or vice versa. Faraday’s law dictates the maximum amount of charge generated for
a given amount of active material. At equilibrium (when no current flows across the plates of the
battery), the driving force is referred to as the open circuit potential and is related to the free energy
of the system by Faraday’s law 9
∆ G
0
E =- (6.23)
nF
The negative sign implies that the free energy is reduced when the battery is discharged. In practice,
the generation of electrical energy from chemicals depends on the temperature and the concentration
of the chemical species taking part in the reaction generating the electrical energy. The open circuit
voltage (E) under practical conditions where the battery operates is related to the equilibrium value
(E ) for changes to temperature and concentrations is given by the Nernst equation
0
=
EE + 0 RT ln c Oxd (6.24)
c
nF Red
where c Oxd refers to the concentration at the electrode surface of the species that release the electrons
to the external circuit of the battery for the current to flow, and c Red refers to the surface concentra-
tion of the ions that complete the electric path by moving across the electrolyte from one electrode
plate to another. More complicated models exist that relate the surface concentration of the reacting
species to the open circuit voltage of the battery 10
An alternative to rigorous relationships between the open circuit voltage and the surface con-
centrations is the use of empirical expressions. This is particularly true of intercalation electrodes
used in the lithium-insertion batteries, which are claimed to exhibit a non-Nernstian behavior. The
most popular approach to model the open circuit potential in such cases is to measure the voltage
of the individual electrodes with respect to a standard reference, at a very slow charge or discharge
rate. The concentration of the reacting species in this case is assumed to be uniform throughout the
reference cell and is calculated by counting the coulombs and using Faraday’s law. Figure 6.6 shows
examples of such measurements.
6.4.4 Rate of Charge Transfer
Like any chemical reaction, the efficiency of charge transfer also depends on how fast the reaction
can take place. The rate of reaction is related to the local overpotential at the reacting interface j, by
the Butler-Volmer expression 7
α nF η s j -α nF η s,,j
,
,
i = i 0, j exp aj j , - exp c jj (6.25)
j
RT RT 
