  We present a porous electrode model for lithium-ion batteries using
Butler--Volmer reaction kinetics. We model lithium concentration in both the
solid and fluid phase along with solid and liquid electric potential. Through
asymptotic reduction, we show that the electric potentials are spatially
homogeneous which decouples the problem into a series of time-dependent
problems. These problems can be solved on three distinguished time scales, an
early time scale where capacitance effects in the electrode dominate, a
mid-range time scale where a spatial concentration gradient forms in the
electrolyte, and a long-time scale where each of the electrodes saturate and
deplete with lithium respectively. The solid-phase concentration profiles are
linear functions of time and the electrolyte potential is everywhere zero,
which allows the model to be reduced to a system of two uncoupled ordinary
differential equations. Analytic and numerical results are compared with full
numerical simulations and experimental discharge curves demonstrating excellent
agreement.
