Lithium-ion batteries: A case study of mathematics and manufacturing

By Dr Iain Moyles

Lithium-ion batteries are everywhere in our society.  They power the things that entertain us (phones, computers), the things that transport us (cars, planes, etc.), and even the things that save our lives (pacemakers and other implantable devices)!  They are so ubiquitous that in 2012, over 660 million cylindrical lithium-ion cells were produced and will claim a market size of over €24 billion by 2020.  This market will continue to grow as we require more energy storage options and as we reduce our dependence on non-alternative energy.  The United Kingdom, for example, has already planned to ban petrol and diesel cars by 2040 with electric cars filling the void.  However, there are technical challenges to achieving these goals.  Currently power densities range from 250-340 W/kg while in 2015 costs for lithium-ion batteries ranged from 240-400 €/kWh.  In order to be viable to market needs, it is expected that power densities need to reach 700 W/kg and cost below 75 €/kWh by 2030.

Aside from technical challenges, there are concerns with battery lifetime and safety, the latter being featured in the news prominently in recent years due to the overheating batteries in cellphones.  For example, at the beginning of March 2018, an LG phone overheated on a plane in Toronto’s Pearson airport causing burns to the user and delaying the flight.  Battery lifetime has plagued Apple recently as they announced they have been reducing the performance of older products to minimise the consumer impact from battery fatigue.  This claim has been met with skepticism as users suspect these performance reductions are being used as a tool to spur new product sales.

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Figure 1: Like any good Canadian living abroad, I was checking CBC news before heading to bed on March 1, 2018 and came across two important battery related news stories.

Research into lithium-ion batteries is of immediate economic and societal concern and one aspect of that research which I’m involved in is mathematical modelling.  Mathematical modelling is useful in better understanding the theory behind design and operation of batteries while also improving simulation methods and speeds to reduce experimental costs.  Over the last year, I formed a consortium of mathematical modellers around the world (myself, Dr Matthew Hennessy, Prof. Tim Myers, and Prof. Brian Wetton) to address some of these mathematical modelling challenges supported by funding from the Irish Research Council and the Royal Irish Academy.  Throughout the year, we analysed state-of-the-art mathematical models of lithium-ion battery electrochemistry.  Using dimensional analysis, we were able to reduce the models to entirely analytic forms and reasonably reproduce a discharge curve from a real battery.  Charge/discharge curves are standard metrics associated to a given battery as they determine the operating battery voltage throughout the charge/discharge of the battery.  We are currently preparing a manuscript based on our results.

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Figure 2:Comparison of analytically reduced model (black) and actual battery data (red) for a battery discharge curve

Figure 2 shows the important role that a mathematician can play in scientific research as we are often able to simplify problems and extract important information.  This tends to direct new avenues of scientific inquest, improves the understanding of theory and application of the scientific process, and reduces computational and experimental cost as simulations are made to be insightful and efficient.

The next goal for the consortium is to apply similar techniques to thermal management in lithium-ion batteries and have a better understanding of the dominant heat-generation processes.  We will also be expanding our work into battery material and lifetime as new collaborations with experimental partners blossom.

Biography: Iain Moyles is a postdoctoral researcher in the Mathematics and Applications Consortium for Science and Industry (MACSI) at the University of Limerick his research is funded by Science Foundation Ireland grant 13/IA/1923.  Aside from lithium-ion batteries, some of his industrial mathematics pursuits include nutrient transport and dynamics in soil, groundwater quality due to septic tanks, and physical properties of hard-to-boil sugars.  He regularly attends study groups with industry all around the world.

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