Sponsored feature, words by the University of Portsmouth


Dr William Lee - Department of Mathematics - University of Portsmouth


Every coffee drinker remembers the last time they had a really horrible cup of coffee. For me it was at a conference I went to last summer. The coffee was too strong and far too bitter, but it was early in the morning and I knew I needed coffee to stay awake during the talks so I drank it anyway.

While I certainly blamed the conference organisers for making me drink such an awful brew, it wasn't really their fault. Given that more than two billion cups of coffee are drunk every day, we know surprisingly little about how to reliably make good coffee.

I became interested in applying mathematics to coffee brewing at an event called a 'European Study Group with Industry' I helped to organise at the University of Limerick in 2012. Study groups are like hackathons for mathematicians, except that we've been doing this since the 1960s. A study group is a week-long conference at which about eighty academic mathematicians, from undergraduates to professors, work on six to eight problems brought by industry. Mathematics can be used to address all sorts of challenges so there are an amazing variety of problems that have been brought to study groups. In study groups I've helped to organise we have applied mathematical, statistical or operations research methods to analyse bubbles in beer, blowing microscopic fuses, acid polishing lead crystal glass, packing cars onto transporters, and of course the brewing of filter coffee.

In the 2012 study group held in Limerick, one company who participated were Philips, who were interested in whether mathematics could be applied to the brewing of filter coffee and thus ultimately to helping to design coffee machines. During the week we were able to make enough progress to demonstrate to Philips that mathematical modelling of coffee brewing was worth further investigation. While some of the models we used involved fairly advanced mathematics we also looked at models simple enough to be taught in A-Level classes in mathematics, chemistry and physics: first order differential equations, chemical reaction rates, and exponential decay. Engineers from Philips, myself and colleagues from the University of Limerick have collaborated on developing mathematical models of brewing coffee since then.

We hope to develop a complete theory of coffee brewing that could be used to inform the design of filter coffee machines in the same way that industry uses the theories of fluid and solid mechanics to design aeroplanes and racing cars. Right now coffee machines are designed by a trial and error process in which to test a new design you must build it and taste the coffee it brews.

We are well aware that there are many factors which influence the brewing of coffee. The one we have chosen to focus on first is the influence of coffee grain size. To develop mathematical models to address this we used macroscopic averaging: a process by which the complicated equations describing the system on the scale of the coffee machine can be derived from simple microscopic equations. Essentially we are integrating the equations over a volume containing many coffee grains which results in equations in terms of averaged quantities.

These equations are then either solved on a computer using numerical methods or approximately using asymptotic analysis methods. (These advanced techniques are typically covered in the second or third year of a mathematics degree.)

One surprising result we found was that there are two processes responsible for coffee extraction. Initially there is a rapid extraction of coffee from the surface of the coffee grains followed by a slower extraction of coffee from the interior of the coffee. The size of the coffee grains is important in controlling this. Increasing the size of coffee grains would mean less overall surface area (for a fixed amount of coffee) and so reduce the amount of coffee available for rapid extraction. Increasing the size of coffee grains also slows down the rate of extraction from the interior of grains since the coffee has further to travel.

While we have made substantial progress towards our goal of taking mathematical models of coffee brewing to the point where we could design coffee machines by computer but more work is still needed. The next step of the journey is to look at the way the coffee bed deforms in the filter as the coffee is brewed. Initially, of course the coffee lies at the base of the filter paper, but as brewing proceeds the inflow of water deforms the coffee bed and eventually it ends up coating the sides of the filter paper. Experiments suggest that the final shape of the coffee bed is correlated with the taste of the coffee.

Dr William LeeWilliam Lee joined the Department of Mathematics at the University of Portsmouth earlier this year. He is currently organising an Undergraduate Study Group that will see all second year students of the mathematics degree have a chance to apply their mathematical modelling skills to industrial problems. In addition William is developing an MMath Course for Portsmouth and continuing his research on applying mathematical modelling to coffee and many other industrial activities.


He will be responding to your comments while representing the UK at the EU-MATHS-IN meeting on Industrial Mathematics in Rome.