  In mathematical models of lithium-ion batteries, the highly heterogeneous
porous electrodes are frequently approximated as comprising spherical particles
of uniform size, leading to the commonly-used single-particle model (SPM) when
transport in the electrolyte is assumed to be fast. Here electrode
heterogeneity is modelled by extending this to a distribution of particle
sizes. Unimodal and bimodal particle-size distributions (PSD) are considered.
For a unimodal PSD, the effect of the spread of the distribution on the cell
dynamics is investigated, and choice of effective particle radius when
approximating by an SPM assessed. Asymptotic techniques are used to derive a
correction to the SPM valid for narrow, but realistic, PSDs. In addition, it is
shown that the heterogeneous internal states of all particles (relevant when
modelling degradation, for example) can be efficiently computed after-the-fact.
For a bimodal PSD, the results are well approximated by a double-particle model
(DPM), with one size representing each mode. Results for lithium iron phosphate
with a bimodal PSD show that the DPM captures an experimentally-observed
double-plateau in the discharge curve, suggesting it is entirely due to
bimodality.
