# Mathematical and Numerical Modeling of Battery (ENE210)

The course gives an introduction to the application of partial differential equations (PDEs) for modelling transport processes involved in a prototype battery, with a particular focus on Lithium ion battery. The underlying physical laws that describe the exchange of lithium between electrodes and electrolyte in discharge/charge modus is described. Numerical discretization of the resulting system of PDEs is explored after first introducing basic principles for solving basic PDEs by numerical methods.The numerical simulation of such model can be used to illustrate the dynamics of lithium concentration distribution and predict the battery performance (voltage drop with time) which is important for designing battery (internal structure and material choices).

Course description for study year 2022-2023

Facts

ENE210

1

5

Autumn

1

Autumn

Norwegian

Time table

## Content

• Fundamental mathematical equations in terms of partial differential equations (PDEs) for studying transport of lithium between electrodes via electrolyte
• Implementation of appropriate numerical method to solve the involved system of  PDEs
• Insight and experience with basic techniques for solving some fundamental PDEs (transport/diffusion)
• Use of the numerical simulation model to explore how battery performance is sensitive to various parameters

## Learning outcome

Knowledge:

The student will have an understanding of the following concepts involved in a mathematical description of battery

1) mathematical equations used to describe diffusive transport of lithium in electrodes and electrolyte;

2) electronic charge balance in electrode phase and electrolyte phase, lithium mass balance in  electrode phase and electrolyte phase and electro-chemical kinetics through Butler-Volmer equation;

3) some general understanding of numerical solution methods of basic PDEs;

4) Insight into the connection between mathematical models for transport processes and how to simulate the battery performance

5) Practical coding experience through exercises and project work.

Skills:

The student will be able to

1. understand how a battery functions during discharging/charging as described by mathematical equations based on physical laws and electrochemical equations;
2. be able to use the model to see the role played by various parameters that characterize the battery;
3. know the principles how to construct a discrete version of a PDE model and implement a code in matlab/python, compute and visualize approximate solutions;
4. have a starting point for developing new models to fit more specific battery systems

None

## Recommended prerequisites

Good knowledge in mathematics (calculus) and physics. Some experience with coding (matlab/python) will be necessary.

## Course teacher:

Helmer André Friis

Steinar Evje

Øystein Arild

## Method of work

Class room instruction, programming exercises, calculation exercises

## Open for

Energy and Petroleum Engineering - Bachelor in Engineering Computational Engineering - Master of Science Degree Programme Petroleum Engineering - Master of Science Degree Programme

## Course assessment

Form and/or discussion.

## Literature

The syllabus can be found in Leganto