The Industrial Mathematics Research Group at the Centre de Recerca Matemàtica has slowly been evolving into the Environmental Mathematics Research Group and grown to encompass a truly interdisciplinary team: mathematicians from CRM and the University of Girona, Fluid Dynamicists and Chemical Engineers from U. Politecnica de Catalunya and Environmental Chemists from Girona. We also collaborate with local groups who are developing pilot plants. Recent research topics include energy absorption by green roofs, improving Direct Absorption Solar Cells, modelling Li-ion batteries and the capture of environmental contaminants by adsorption [1,2,3,4,5]. The latter topic forms the primary focus of the group’s present work.
A recent study estimated that “there is only a 0.1% chance of limiting global average temperature change to the Paris Climate Agreement aspirational goal of 1.5 °C by 2100” [6]. If global warming exceeds a certain level the Intergovernmental Panel on Climate Change (IPCC) suggest it could be reduced through net negative emissions through, for example, greater deployment of greenhouse gas removal [IPCC 6th Assessment Report, 2021]. The UN Sustainable Goal of a “toxic free environment” already requires the removal of a multitude of existing contaminants. One of the most promising removal techniques is known as column adsorption. Column adsorption involves passing a fluid through a tube filled with a porous material capable of removing certain components of the fluid. The process continues until the material becomes saturated, then no more removal occurs, and the fluid passes through unchanged, see Figure 1. It is employed in a wide range of remediation processes, including greenhouse gas capture, water and groundwater treatment, biogas and flue gas cleansing and many more.
Figure 1: Schematic of adsorption column. Fluid enters the column at x=0, passes by the adsorbent material and, contaminant attaches to the internal and external surfaces.
There are two main attachment mechanisms: physisorption occurs when a contaminant molecule is held by van der Waals forces; chemisorption involves a chemical reaction leading to new bonds. A standard laboratory experiment would involve a column of the order 5cm long and radius 5mm with a steady flow and contaminant escaping after around 15 minutes. Industrial columns are of the order 5m tall and may run continuously for months with a varying gas intake. Figure 2 shows a laboratory set-up used by group members at the Laboratory of Chemical and Environmental Engineering, University of Girona.
Figure 2: Experimental set-up showing a series of adsorption columns with varying amounts of activated carbon. Water contaminated with toluene is slowly being forced through the columns.
To date the group have developed models to describe the removal of contaminants through physical and chemical adsorption, the removal of large quantities of fluid and simultaneous removal of multiple contaminants. The basic mathematical description involves a mass balance describing the movement of carrier fluid and contaminant through a column coupled to a kinetic equation describing the attachment process, this acts as a sink in the mass balance. All model results are verified against experimental data from the literature and data produced in Girona. By taking a more rigorous mathematical approach using non-dimensionalisation, asymptotics and travelling wave techniques as well as numerics we have been able to produce significantly more accurate results than any published previously and, significantly, usually with a single fitting parameter (the adsorption rate). And we’re pretty pleased with ourselves!
The most common way to analyse results from an adsorption experiment is through the breakthrough curve, this represents the concentration of contaminant escaping at the column outlet (normalised by the inlet value). In Figure 3 we show a comparison of breakthrough data against the group’s model for chemisorption (solid line) and physisorption (dashed line). This is for the adsorption of CO2 on a polyethyleneimine coated substrate. In this case it is known that a chemical reaction occurs and so the chemisorption model is more accurate.
Figure 3: Comparison of breakthrough curves for chemisorption (solid line) and physisorption (dashed line) models with experimental data (circles) for the capture of carbon dioxide on polyethyleneimine.
In Figure 4 we present recent data for the simultaneous adsorption of two siloxanes, L2 and D4. Siloxanes are used in the home in antiperspirants, skin-care creams, hair conditioners, and color cosmetics and in industry in automotive polishes, fuel additives and antifoaming agents. They easily enter the water cycle. Some are toxic, such as D4, and must be removed when treating drinking water. In Figure 4 we see how initially both D4 and L2 are adsorbed but later the D4 displaces L2, such that for a period a higher concentration of L2 exits the column than enters. This is not a desirable feature!
Figure 4: Comparison of multicontaminant model with experimental breakthrough data for the capture of L2 (red squares), D4 (blue circles).
The method developed by the group has been converted into a free web-based app MathCol, try it if you happen to have any breakthrough data https://mathcol.crm.cat/. For further information on the IMRG/EMRG’s work on contaminant capture, feel free to contact Prof. Tim Myers (tmyers@crm.cat).
[1] A mathematical model for the energy stored in green roofs. M Aguareles, M Calvo-Schwarzwalder, F Font, TG Myers. Applied Mathematical Modelling, 2023.
[2] Time-dependent modelling of nanofluid-based direct absorption parabolic trough solar collectors. GJ O’Keeffe, SL Mitchell, TG Myers, V Cregan. Solar Energy 2018.
[3] Asymptotic reduction of a porous electrode model for lithium-ion batteries. IR Moyles, MG Hennessy, TG Myers, BR Wetton. SIAM Journal on Applied Mathematics, 2019.
[4] On the development of a consistent mathematical model for adsorption in a packed column (and why standard models fail). TG Myers, A Cabrera-Codony, A Valverde. International Journal of Heat and Mass Transfer, 2023.
[5] Mathematical analysis of a Sips-based model for column adsorption. M Aguareles, E Barrabés, T Myers, A Valverde. Physica D: Nonlinear Phenomena, 2023.
[6] L.R. Vargas Zeppetello, A.E. Raftery, D.S. Battisti. Probabilistic projections of increased heat stress driven by climate change. Communications Earth & Environment, 2022.