# Multi-scale modelling of Li-ion batteries

My name is Zénó Farkas, 22 year old student from Hungary. Currently I’m studying at Eötvös Loránd University (ELTE) of Budapest, Faculty of Sience as analytical mathematician of Bsc. This is my last semester and I’m writing my Bsc thesis about multi scale modelling of Li-ion batteries. This theme has been offered by my current internal consultant, Dr. István Faragó, and by my external consultant, Ákos Kriston of the European Commission, Joint Research Center (JRC) in the Netherlands. I also applied for a trainee position at the JRC and finally I got this job. As a result, I’ll spend 5 months in Petten where I’m going to deal with modelling of Li-ion battery energy storage systems for automotive applications. After completion of the Bsc I’d like to continue this topic at the Applied Mathematician Msc of ELTE.

I would like briefly to describe, that I’m working on.

Li-ion batteries are a power sources which have been developed for portable and automotive applications.

Li-ion battery (LIB) is key enabling technology to store electricity, in a form of chemical energy, for an electric vehicle (EV) with desirable driving range and power. In spite of the current success of EVs the technology needs to be improved significantly. Transfer of molecular-scale understanding and discoveries into new and improved products and processes requires integration of LIB behavior across a range of length and time scales. Therefore, new mathematical-modeling approaches are needed that couple continuum equations (diffusion, heat transfer, migration) that describe current, potential, and concentration distributions with molecular-level events in order to describe, design, and control complex electrochemical systems accurately.

The continuous mathematical description is based on a non-linear parabolic system of partial differential equations where the non-linearity is in the source term. This type of system is difficult to solve therefore methods are developed for these models. One of the applied is the operator splitting method. This approach has different types, like Sequential Splitting, Symmetrical Splitting, Strang-Marchuk Splitting, and they have many computational benefits.

These models have the possibility to integrate different time and length scales, without the following partial differential equation system (For shake of simplicity, we give only the equations, without detailed desciption of the initial and boundary conditions) are going to solve.

where cs (solid Li+ concentration in the particle on the surface), c (concentration of Li+-ions in the electrolyte phase), u (overpotential) are time-, space-dependent functions, J is the electrochemical reaction which depends on cs, c, and u and the other symbols are dimensionless parameters.

After using the operator splitting method we discretize our model on the mesh:

which means that we approximate the values of the unknown functions on the mesh-points.

In spite of that, it doesn’t give an explicit formula for the mesh-function, however, based on these solutions, we can observe its behaviour more easiliy. We also note that these equations have simplified form therefore they don’t reflect the full complexity of a real battery. In the future we have to solve similar equations coupling different time and length scales.

I hope, I aroused your interest in Li-ion batteries!

Best regards,

László Zénó Farkas