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A new article by Abel Valverde (Universitat Politècnica de Catalunya), Alba Cabrera-Codony (Universitat de Girona), Marc Calvo-Schwarzwalder (Zayed University) and Timothy G. Myers (Centre de Recerca Matemàtica), published by the International Journal of Heat and Mass Transfer, proposes a mathematical formulation that, unlike standard models, focuses on the impact of adsorbent particle size on the efficiency of column sorption.

In our ongoing quest for a more sustainable future, one of the pivotal challenges we face is the removal of contaminants from our atmosphere and water. The most widely used method is a technique known as column sorption, either through absorption (where fluid molecules are dissolved or soaked up inside a solid or liquid) or adsorption (where molecules attach to a material’s surface). This process is like a molecular filter, selectively trapping and removing undesirable substances from gases or liquids as they flow through a column. Think of it as a super-efficient bouncer at a nightclub, only allowing the most unwanted party crashers to be ejected.

While this process may sound straightforward, the reality is much more intricate. Column sorption, as it stands, can be quite costly while more cost-effective alternatives do not offer the same environmental benefits. To understand why, we need to delve into the nitty-gritty of these filtration systems. Scientists typically start by testing small sorption columns about 1 to 15 centimetres long. However, real industrial-sized filters can be colossal in comparison, stretching up to 5 meters in length. Here’s the problem; what we discover from the small ones doesn’t always translate seamlessly to their bigger counterparts. A host of factors contributes to this unpredictability, including changes in flow patterns and the influence of wall proximity in small columns.

Now a team of researchers from the Universitat Politècnica de Catalunya, the Universitat de Girona, Zayed University, and the Centre de Recerca Matemàtica has developed a new mathematical model that accounts for the size of the adsorbent particles, which could help work around the scale-up problem when moving from experimental studies to working devices. This work sets up the basis to help us prepare for future research on how to make the big filters work better and be more cost-effective.


A Mathematical Model Accounting for Intra-particle Diffusion.

In the article, published by the International Journal of Heat and Mass Transfer, the research team has focused on a concept called intra-particle diffusion. This refers to how contaminants move within the adsorbent particles. As these particles grow in size, the time it takes for contaminants to reach the inner sanctum of adsorption sites also increases. Sometimes, this diffusion timescale becomes comparable to the adsorption timescale, leading to an imbalance in the governing equations. In simpler terms, it’s like a marathon where some runners have a head start and others need to catch up.

Another crucial aspect of sorption column studies is the breakthrough curve, a graphical representation of the concentration of contaminants at the column outlet over time that shows us how efficiently our filter works. To obtain a breakthrough curve, a column or adsorption bed filled with a solid adsorbent material is commonly used. In the column, a mixture containing the components to be separated is introduced. As the contaminants flow through the column, initially, our filter is at the peak of its performance, effectively capturing the unwanted components. However, as time goes on, it gets worn out and can’t catch any more.

Schematic of the experimental setup.

To validate their models, the researchers turned to experiments to compare their findings with real-world breakthrough data. When they used bagasse fly ash as the adsorbent for wastewater treatment from a sugar distillery, for instance, smaller particles behaved predictably, producing a more common S-shaped breakthrough curve. However, larger particles marched to a different drumbeat, displaying an initial linear rise followed by an exponential increase. These findings deviated from more conventional mathematical models.

Similarly, when they looked at phosphorus removal with biochar microspheres, larger spheres exhibited a similar initial linear increase, followed by a slow decay, while smaller spheres followed a different pattern. Despite variations in materials and adsorption mechanisms, the common denominator appeared to be the impact of particle size.


Exploring the Environmental Applications of Diffusion

The study analyses comprehensive analytical models with and without intra-particle diffusion, based on linear and nonlinear sinks. The researchers developed a sophisticated mathematical formulation that takes into account the size of the particles and their internal diffusion. This model introduces a rate parameter, essentially a measure of how easily adsorbent particles allow contaminants to be absorbed. A high value means easy entry and a low value indicates difficulty. This parameter’s behaviour was confirmed through experiments, showing it’s sensitive to factors like inlet concentration and flow rate.

This model is more complex than traditional models involving mass balance and kinetic equations. However, it offers the advantage of accommodating size effects and intra-particle diffusion. In certain scenarios, it aligns with traditional models. Conversely, in cases where contaminants take their time to enter or adsorption mainly occurs on the particle’s outer surface, this model exhibits clear distinctions, surpassing traditional models in predicting breakthrough trends.

This study is part of a project funded by the Spanish Ministry of Science and Innovation and provides a significant contribution to our understanding of column sorption and contaminant removal. Future research endeavours may explore even more advanced models, including those that consider a broader range of effects, such as pore blocking by large molecules. As we continue our journey toward a cleaner tomorrow, the synergy between mathematics and environmental science remains one of our most potent tools.

Referenced article:

Valverde, A., Cabrera-Codony, A., Calvo-Schwarzwalder, M., & Myers, T. G. (2024). Investigating the impact of adsorbent particle size on column adsorption kinetics through a mathematical model. International Journal of Heat and Mass Transfer, 218, 124724.

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Timothy G. Myers

Centre de Recerca Matemàtica

I am currently a Senior Researcher and head of the Industrial Maths Research Group (IMRG) at the Centre de Recerca Matematica (CRM) and also hold adjunct positions at U. Limerick and U. Politecnica de Catalunya. I have over 30 years of research experience and, since 2011, have the highest research rating with the Catalan Accreditation Agency, AQU, L’Acreditació de la Recerca Avançada. I play an active role in promoting industrial and applicable mathematics throughout the world, having helped organise meetings in the UK, Spain, South Africa, and Canada.


Alba Cabrera-Codony

Universitat de Girona

PhD in 2016 at the University of Girona, a postdoctoral fellow at Cabot Corp., Boston, USA. PI of the SILCAP project co-funded by European Union´s H2020 Marie Sklodowska-Curie Actions and ACCIÓ TecnioSpring+ programme. In 2021 I was awarded a Juan de Cierva – Incorporación fellowship which will allow me to consolidate my research profile at LEQUIA-UdG.


Abel Valverde

Universitat Politècnica de Catalunya

During the 2017/2018 academic year, I began my doctorate at the UPC under the supervision of Prof. Francesc Recasens and Prof. Juan Jesús Pérez. My main research line until 2022 has been the development and resolution of various mathematical models of chemical reactors, extractors and sorption columns. Among the main results of my research are the development of new mathematical models and the definition of physical parameters.


Marc Calvo-Schwarzwalder

Zayed University

Marc Calvo-Schwarzwalder received his Ph.D. Cum Laude in Applied Mathematics from the Universitat Politècnica de Catalunya (UPC) in 2019. He developed part of his research at the University of Limerick and the University of Oxford. His main research area is the study of mathematical models for industrial processes. His PhD focused on the mathematical description of the heat transfer at the nanoscale. He has also worked on the description of nanocrystal growth and is now focusing on providing mathematical models for carbon capture processes.

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