Porous electrode theory coupled with transport and reaction mechanisms is a widely used technique to model Li-ion batteries employing an appropriate discretization or approximation for solid phase diffusion with electrode particles. One of the major difficulties in simulating Li-ion battery models is the need to account for solid phase diffusion in a second-radial-dimension r, which increases the computation time/cost to a great extent. Various methods that reduce the computational cost have been introduced to treat this phenomenon, but most of them do not guarantee mass conservation. The aim of this paper is to introduce an inherently mass conserving yet computationally efficient method for solid phase diffusion based on Lobatto III A quadrature. This paper also presents coupling of the new solid phase reformulation scheme with a macro-homogeneous porous electrode theory based pseudo 2D model for Li-ion battery. Lithium-ion chemistry has been identified as a good candidate for high-power/high-energy secondary batteries which are expected to play a vital role in the future of automobile, power storage, military, mobile, and space applications. Significant efforts have been made and reported in literature regarding the modeling and understanding of Lithium-ion batteries using physics based first-principles models. The most widely used first principles model for the lithium-ion battery is the porous electrode pseudo two dimensional (P2D) model, 1 which is based on the fundamentals of electrochemistry and transport phenomena. These models are represented by coupled nonlinear PDEs in 1-2 dimensions, are typically solved numerically and require few minutes to hours to simulate.For the P2D model, the diffusion of Lithium ion into the solid electrode particles is solved in a pseudo dimension r, which is coupled to the macro-homogenous model at the surface of the intercalation particles. This pseudo 2 dimensional approximation avoids the need for a solution of a full 2 dimensional model and hence the name. Accurate predictions of the concentration at the surface of the particle are therefore important, as it contributes to the exchange current density for the reaction at the particle-electrolyte interface. Typically, solid phase diffusion in the micro-scale is modeled using Fick's law of diffusion. More detailed schemes involving pressure induced diffusion along with solid phase diffusion have also been reported in literature.
2These models are important especially for high capacity materials where the stress developed affects the concentration profile inside the intercalation particle. For phase changing materials, the shrinking core model 3 has been used and approximate solutions have been proposed. 4 Cahn-Hilliard models 5 have also been employed to track the phase boundary within active material particles during Lithium intercalation.6,7 One of the major difficulties in the electrochemical engineering models is that even the inclusion of a simple Fickian model for solid phase diffusion in a second dimension r increa...