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...

Lithium ion batteries are currently used in many devices, and their demand is expected to increase in the near future. Mathematical modeling of these batteries has been done with porous electrode pseudo-2D (P2D) model [1]. One of the major difficulties in the electrochemical engineering models is the inclusion of solid phase diffusion in a pseudo second dimension, r, across the radius of the active particle which greatly increases the complexity and computation time/cost of the model as well. A common way to model the diffusion of lithium into the electrode particle is the 1-D spherical diffusion equation using Fick’s law, with the outer flux provided as a time-dependent boundary condition. This partial differential equation can be solved using the method of lines in which the solid phase is discretized to create a system of ordinary differential equations which can be solved using well-established and efficient solvers in time. Different methods such as finite difference, finite volume, and control volume have been used to discretize the spatial dependence, but with high computational cost [2]. Solid phase reformulation techniques based on optimal node spacing, parabolic profile approximations etc. to reduce computation times have been reported in literature [3-8]. In this talk, efficient reformulation techniques for solid phase diffusion will be shown. Efficient finite difference/ finite volume based methods will be demonstrated based on optimal node spacing/cell spacing in the rdomain. Moreover, a mass conserving reformulation technique based on orthogonal collocation will be shown. These reformulation techniques can predict the concentration profile within the solid particle using fewer equations than the standard methods thus reducing the CPU time required to simulate when coupled with the P2D model. In addition, efficient reformulation methods for phase-field models will also be discussed [9]. Acknowledgement The authors are thankful for the financial support by Washington University’s Chancellor’s Graduate Fellowship Program and Danforth Scholars Program, the United States Government, Advanced Research Projects Agency-Energy (ARPA-E), US Department of Energy under award # DE-AR0000275, McDonnell Academy Global Energy and Environment Partnership (MAGEEP) at Washington University in St. Louis. References [1] M. Doyle, T. F. Fuller and J. Newman, J. Electrochemic. Soc., 140,1526(1993) [2] Y. Zeng, P. Albertus, R. Klein, N. Chaturvedi, A. Kojic, M. Z. Bazant, J. Christensen, J. Electrochem. Soc., 160, A1565(2013) [3] V. Ramadesigan, V. Boovaragavan, J. C. Pirkle, Jr. and V. R. Subramanian, J. Electrochem. Soc., 157, A854 (2010) [4] V. R. Subramanian, J. A. Ritter and R. E. White, J. Electrochem. Soc., 148, E444 (2001) [5] V. R. Subramanian, V. D. Diwakar and D. Tapriyal, J. Electrochem. Soc., 152, A2002 (2005) [6] C. Wang, W. Gu and B. Liaw, J. Electrochem. Soc., 145, 3407 (1998) [7] K. Jagannathan, J. Electrochem. Soc., 156, A1028 (2009) [8] S. De, B. Suthar, D. Rife, G. Sikha, and V. R. Subramanian, J. Electrochem. Soc., 160, A1675(2013) [9] B. C. Han, A. Van der Ven, D. Morgan and G. Ceder, Electrochim. Acta., 49, 4691(2004)