On Veracity of Macroscopic Lithium-Ion Battery Models

Batteries are electrochemical energy storage devices that exhibit physico-chemical heterogeneity on a continuity of scales. As such, battery systems are amenable to mathematical descriptions on a multiplicity of scales that range from atomic to continuum. In this paper we present a new method to assess the veracity of macroscopic models of lithium-ion batteries. Macroscopic models treat the electrode as a continuum and are often employed to describe the mass and charge transfer of lithium since they are computational tractable and practical to model the system at the cell scale. Yet, they rely on a number of simplifications and assumptions that may be violated under given operating conditions. We use multiple-scale expansions to upscale the pore-scale Poisson-Nerst-Planck (PNP) equations and establish sufficient conditions under which macroscopic dual-continua mass and charge transport equations accurately represent pore-scale dynamics. We propose a new method to quantify the relative importance of three key pore-scale transport mechanisms (electromigration, diffusion and heterogenous reaction) by means of the electric Péclet (Pe) and Damköhler (Da) numbers in the electrolyte and the electrode phases. For the first time, applicability conditions of macroscopic models through a phase diagram in the (Da-Pe)-space are defined. Finally, we discuss how the new proposed tool can be used to assess the validity of macroscopic models across different battery chemistry and conditions of operation. In particular, a case study analysis is presented using commercial lithium-ion batteries that investigates the validity of Newman-type macroscopic models under temperature and current rate of charge/discharge variation.