Page 172 - Lindens Handbook of Batteries

MATHEMATICAL MODELING OF BATTERIES 6.25

during the first few cycles that eventually levels off. For the higher end of charge voltage (EOCV), the
anode is maintained at highly reducing potentials for longer periods of time, and as a result, the growth
rate of the film is higher in a cell cycled with a higher EOCV. Under identical conditions, a cell cycled
at a higher temperature has lower transport limitations, and hence the duration of a given cycle is lon-
ger than it would be at a lower temperature. This results in additional charging time; since the model
for the film growth depends on the amount of time taken to charge the cell [(see (6.55)]; the cell cycled
at 40°C shows higher thickness of the film during the first few cycles and hence more capacity fade.
However, after about 300 cycles, the enhancements in charging time due to the temperature are offset
by the additional resistance created by the growth of the film. As a result, the cell cycling at 25°C, but
with a higher EOCV, loses more capacity. These results indicate that if operating at a relatively higher
temperature for about 300 cycles, the cell can be programmed to have a higher EOCV (and hence
deliver higher capacity), whereas if the application demands a longer cycle life, a more conservative
EOCV should be used. Similar conclusions can be drawn from empirical models once data under
different operating conditions is available, whereas such an insight into the physical phenomena as
described above is unique to mechanistic models. Once the rate constant for the side-reaction and the
conductivity of the film are determined by independent experiments, the model allows for cell design
under a variety of different scenarios as long as the degradation mechanism proposed is valid.

6.11 DETERMINING THE RIGHT MODEL

A good mathematical model should strike a balance between the limited details associated with the input
parameters available to the end-user and the amount of insight the model can provide to improve battery
design. Limitations to the mechanistic models are attributed to the tedium involved in developing and
solving the model equations as well as the large number of parameters required by such models. Often
many of these parameters cannot be obtained from direct experimental measurements. On the other
hand, circuit analog models provide limited insight into the physical phenomena that take place within
the battery. For example, a drop in capacity at higher rates of discharge can be modeled as an increase
in the pseudocapacitance parameter C shown in Fig. 6.3. One cannot determine whether this change is
D
caused by a limitation in the diffusion of ions within the electrode or if the conductivity of the electrolyte
has been reduced over time. Consequently, making improvements to the cell design based on circuit
analog models is difficult; in this example, it is not obvious whether an increase in the porosity of the
electrode plate would resolve the issue or if the electrolyte formulation would have to be modified.
Typically, fine tuning of the parameters such as the conductivity of the electrolyte or the porosity
of the electrodes is carried out at the cell design phase. Hence, a mechanistic model is invaluable
at this design phase. In all cases, the assumptions behind a mathematical model must be carefully
explored before employing the conclusions made from simulations.

LIST OF SYMBOLS

2
a Specific area (m /m )
3
2
A Area of the electrode (m )
b Tortuosity factor
3
c Concentration of the ion k (mol/m )
k
c Volume averaged concentration of the electrolyte (mol/m )
3
c s Volume averaged concentration of the ion within the electrode (mol/m )
3
c Specific heat capacity (J/kg/K)
p
C Double layer capacitance (F)
D
D Diffusion coefficient of the ion in the electrolyte (m /s)
2