Efficient Mass Conserving Reformulation Schemes for Nonlinear Solid-Phase Spherical Diffusion Equation for Lithium Intercalation

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.

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