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Model Reformulation and Efficient Simulation of Two Dimensional Electrochemical Thermal Models

^{1}using concentrated solution theory to describe the internal behavior of a lithium-ion sandwich consisting of positive and negative porous electrodes, a separator, and current collectors. This model has proven versatile enough to incorporate additional advancements allowing for the development of similar models.

^{2-7}However, even with only a single linear dimension, solving this model is difficult and computationally expensive.

Several reformulation and order reduction techniques have been presented in the past to improve the computational efficiency to allow for parameter estimation and optimization.^{8-13}However, most of these models consider only a single macro dimension in addition to the pseudo dimension for the solid electrode particles. This is sufficient in many cases. However, when temperature is considered, a second dimension (parallel to the electrodes) becomes more important. Since the height and width of a battery are often orders of magnitude larger than the thickness of the cell, a significant temperature gradient can develop parallel to the electrodes. This can affect battery health and utilization as well as safety. Therefore, an effective method to solve the battery equations in two linear dimensions would allow for better analysis of battery operation when temperature is a concern.

The model will be solved using techniques previously developed^{13}to simulate a one dimensional coupled thermal electrochemical model. That technique will be expanded to incorporate 2D effects. Since a large number of equations arise even for the 1D case, reformulation and orthogonal collocation are essential to reduce the computational cost and maintain accuracy in the two spatial dimensions when nonlinear temperature dependent parameters are used.

The reformulation will be performed using a coordinate transformation to rescale each region in the *x*-direction (perpendicular to the electrodes) to be solved in the interval from 0 to 1. This allows orthogonal collocation techniques to be used in the solution while utilizing simple trial functions. This has been used successfully in the past for 1D models. The reformulated system of equations will be solved using a fast differential algebraic equation solver, and the results will be compared with those obtained by a rigorous COMSOL simulation of the non-reformulated 2D thermal model.

**Acknowledgements**

The authors are thankful for the financial support from the United States Government, Advanced Research Projects Agency – Energy (ARPA-E), and the U.S. Department of Energy, under award number DE-AR0000275.

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* Electrochemical Society Student Member

** Electrochemical Society Active Member

z vsubramanian@seas.wustl.edu