# Multiscale Modeling of Battery Systems: Electrochemical Transport and Thermal Runaway

Battery modules performance is strongly controlled by heat transfer and mass transport limitation at the pore-scale. Yet, accurately capturing nonlinear pore-scale effects in macroscale models with a priori error guarantees can be a daunting task. At the same time, accuracy and predictivity of macroscopic thermo-chemical models are critical safety factors that cannot be overlooked, e.g. during thermal runaway.

We employ rigorous homogenization theory, symbolic computing and algorithm refinement techniques to derive continuum and hybrid formulations to model thermal runaway and concentration field in battery modules in a computationally efficient manner, while rigorously preserving macroscopic models accuracy. First, through novel symbolic computing capabilities, we cut down the cost of analytical model development from months to seconds, since the mathematical calculations previously allocated to humans can now be completely performed symbolically by a computer.

Once the macroscopic equations are derived by Symbolica, numerical validation of the models derived is performed by calculating the absolute error between the average cell-scale solution and its continuum counterpart (Figure 2): when the error overcomes what is a priori prescribed by homogenization theory, hybridization strategies must be deployed.

Hybrid (multi-algorithm, multi-faceted) models combine descriptions at different spatial and/or temporal scales (e.g. cell- and module-scale) fashion in the same computational domain in an efficient computational. The hybrid coupling algorithms we develop guarantee that the error introduced by the coupling conditions is bounded by the upscaling error. Figure 3 shows a conceptualization of a hybrid domain and a comparison between cell scale, continuum scale and hybrid formulations (Videos 1 and 2).

Effective parameters in macroscale models can be rigorously calculated from the solution of a PDE on a representative elementary volume at the pore-scale. Using large datasets of porous media images and these PDE solutions, we have developed CNN methods to accurately determine effective parameters (diffusivity, permeability and dispersivity) of porous media, including batteries from 2D pore-scale images. Figure 4 shows how the CNN prediction significantly outperforms classical equations used to determine such parameters from electrode porosity, such as the Bruggeman relationship. The CNN also dramatically reduces the computational cost of calculating effective parameters compared with the PDE solution.

The CNN is developed for generated images but can be applied to real battery electrodes as well – both 2D and 3D. We have shown this by analyzing images from a LiCoO_{2} cathode, which is the most commonly used material for personal electronics, and Ni-YSZ anode, which is often used in fuel cells. Figure 5 shows a comparison of the CNN prediction for effective diffusivity to the reference values provided for each image.

Additionally, though the CNN is designed for a 2D input image, 3D electrodes can be processed by sampling 2D subsections. We demonstrate this by computing the permeability from five 3D images – 3 real sandstone samples and 2 generate sphere packs. The CNN prediction is taken by averaging the prediction of 100 randomly sampled 2D slices of the images. Figure 6 shows that the CNN can accurately compute the permeability of 3D samples with varying porosity and topological properties.

**References**

[1] Y. Yao, K. Pietrzyk, P. Harabin, M. Behandish, I. Battiato, `Non-intrusive Adaptive Hybrid Method for Multiscale Heat Transfer: Application to Thermal Runaway in Battery Modules', to be submitted to J. Comput. Science (2023).

[2] Pietrzyk, K., Bucci, G., Korneev, S., Behandish, M., Battiato, I., ’Automated Upscaling via Symbolic Computing for Thermal Runaway Analysis in Li-ion Battery Modules’, to be submitted to J. Comput. Science (2023).

[3] R. Weber, S. Korneev, Battiato, I., ‘Estimation of Li-Ion Battery Effective Properties through Convolutional Neural Networks’ Transport Porous Med., **145**, pages 527–548 (2022).

[4] Weber, R.; Korneev, S.; Battiato, I., “Labeled Image Dataset of Generated Porous Electrode Microstructures and Calculated Transport Parameters for Neural Network Training”, Mendeley Data, V1, doi: 10.17632/mgmxv5tjt2.1, (2022).

[5] S. Korneev, H. Arunachalam, S. Onori, Battiato, I., ‘A data-driven multiscale framework to estimate effective properties of lithium-ion batteries from microstructure images’,Transport Porous Med., 134:173–194, (2020).

[6] H. Arunachalam, S. Onori, Battiato, I., ‘ On veracity of Lithium-ion battery macroscopic models’, J. Electrochem. Soc. 162, 9, A1-A12 (2015).