# Mathematical modelling of washing machines and tumble dryers

By Torin Fastnedge and Chloe Bernard

Research estimates suggest that up to 35% of the microplastics found in our oceans originate from our clothing, specifically microfibres, which are released during the washing machine cycle. Subsequently, these minuscule particles enter the water and are ingested by humans through our food and water sources. Moreover, tumble dryers also generate these microfibre particles, which poses a dual challenge. Firstly, if not effectively captured, the microfibres can impede the functionality of the machine. Secondly, they have the potential to escape into the natural environment, further contributing to the microfibre issue. Therefore, it is imperative that we devise strategies to minimize the release of plastic particles from all household appliances, including washing machines and tumble dryers, in order to mitigate this problem of plastic pollution.

Two students at the Mathematical Institute University of Oxford are working in collaboration with the large appliance company Beko on the problem of microfibre release and capture in both household appliances. PhD student, Torin Fastnedge, supervised by Professor Ian Griffiths and Professor Chris Breward, is working on washing machines, modelling the release of microfibre from clothing when inside the drum. He is also looking at the filtration of these microfibres before they are dumped or recirculated. Chloe Bernard, a master’s student under the supervision of Professor Ian Griffiths and Professor Colin Please, is addressing the challenges associated with capturing lint in tumble dryers through mathematical modelling.

Torin has been creating mathematical models for fibre release. Here, he approximates the removal of material as an erosion of a layer in a channel flow. This constitutes a moving boundary problem in which the surface is subject to shear forces that remove the material. The release of individual fibres is also modelled by considering pulling forces from the shear force of the flow and frictional forces holding the fibres within the material.

Torin has also been studying a biomimetic separation technique, coined as ricochet separation [R.V. Divi et al. Science Advances, 4(9), 2018]. The idea is taken from manta-ray fish, whereby dirty water flows into a device with some ribbed or gill-like structure. As the dirty water flows over this ribbed structure (see figure below), some of the water flows down between these structures and, as the particles approach the leading edge of the structure, they experience pressure and lubrication forces, which cause the fibres to ricochet or repel upwards into the freestream flow. This device is thus able to separate some clean water from a mixture of dirty water with a higher concentration of particles.

Chloe’s project involves the study of tumble dryers with the goal of achieving two fundamental objectives: increasing the efficiency of lint capture and improving the energy efficiency of the system. To effectively address these goals, the problem is subdivided into three distinct sub-problems:

In a microscale problem, she studies the behaviour of individual fibres. Specifically, she observes and analyses the deformation of a fibre that is in contact with the filter, to determine whether the fibre will pass through the filter. This allows her to gain insights into the behaviour of fibres in the filtering process, providing valuable information for improving the capture efficiency of lint particles.

In a mesoscale problem, Chloe broadens her perspective to encompass the entire filter unit. By investigating the collective performance of the filter, she explores the interplay between various components and their impact on lint capture. This mesoscale analysis involves studying the arrangement and design of the filter, as well as assessing its efficiency in capturing lint particles of varying sizes. The findings obtained from this stage contribute to refining the filter’s structure and optimizing its effectiveness.

In a macroscale problem, Chloe shifts her focus to the overall system. Here, she examines the entirety of the tumble dryer, considering factors beyond the filter itself. The objective at this scale is to identify strategies and mechanisms that can reduce drying time, thereby enhancing energy efficiency. By analysing the drying process, heat distribution, and airflow patterns, she aims to uncover innovative approaches to streamline the operation of the system while achieving optimal drying results.

This work is of interest to Beko so that they can change their designs to their tumble dryer filters, the washing machine drum design, filter, and or change the program parameters in each appliance. Our mathematical modelling can reduce the workload, time and money that Beko needs to use, finding the most effective solutions via some optimisation and modelling results.