Lithium-ion batteries are commonly simulated using a volume-averaged formulation (porous electrode theory), using effective properties such as conductivity and diffusivity and assuming a simplified spherical geometry of the electrode particles. Microscopic scans show that the shape, size, and orientation of the particles is highly variable, which can significantly alter pathways for ionic, electronic, or thermal transport. Unlike a volume-averaged spherical formulation, a particle-scale simulation applied to a real electrode geometry is able to predict localized phenomena and enables the optimization of material morphology to maximize battery performance. The physical processes involved in lithium-ion battery simulations are electron, ion, mass, and thermal transport, and a particle-scale simulation yields the spatial variation of Li concentration, electrostatic potential, and temperature throughout time as the battery discharges. In this paper we develop a fully-coupled finite volume methodology for the simulation of the electrochemical equations in a lithium ion battery cell. An unstructured mesh of arbitrary convex polyhedral cells is used to mesh synthetic spherical particles and electrolyte. Second-order discrete conservation equations for Li ion transport and electrostatic potential are developed. Butler-Volmer kinetics are included at the electrode/electrolyte interface. A block-sparse unstructured linear system results from the discretization of the governing equations, and is solved using a BiCGSTAB solver with an algebraic multigrid pre-conditioner. A fully-implicit time stepping scheme is used to compute transients. Our methodology fully couples species, electrostatics, and Butler-Volmer kinetics in a stable and efficient computational algorithm. We demonstrate that it is more stable and takes less time to converge than a conventional sequential solution procedure.

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