# Environmental Mathematics in Barcelona

The Industrial Mathematics group at the Centre de Recerca Matematica are currently involved in two research areas related to current environmental challenges, namely carbon capture and green roofs.

Carbon Capture

The carbon capture involves modelling the flow of a gas through a porous media contained in a column. This represents the industrial process of contaminant removal via adsorption. Mathematically the process is described by an advection-diffusion equation coupled to a sink term, representing the mass removal. Temperature may also play a role. For most of the process all of the carbon is removed, so the advection-diffusion equation holds over a finite, growing domain. The problem is therefore analogous to a Stefan problem with a flowing liquid. Early results show excellent agreement with experiment. In Figure 1 we show a schematic of the experiment and also comparison of the mathematical prediction (both travelling wave approximation and numerical solutions) and experimental measurements of the CO2 content at the outlet of an adsorption column.

Fig 1 a) Schematic of column adsorption experiment, b) comparison of CO2 concentrations at column outlet: red dots are experimental, solid line approximate mathematical model, dashed line numerical result.

The model has also been successfully applied to the removal of contaminants from a liquid. In Figure 2 we compare the prediction of our approximate analytical solution with experimental data for the removal of dye and amoxicillin from water. Also shown are the predictions of earlier, standard approximate models.

Fig. 2 a) Concentration of congo red dye at column outlet, filtered by soil b) concentration of amoxicillin at column outlet, filtered by activated carbon. Red dots are experimental results, solid line approximate analytical solution of [2]. Other curves from standard previous models.

Green Roofs

Green roofs, green spaces and green walls are becoming increasingly popular in cities around the world. There are many reasons for this interest, such as air  purification,  runoff  reduction,  heat  island  reduction,  roof  life extension, acoustic noise reduction, biodiversity preservation, energy saving in building. The IM group started this project during a study group meeting in South Africa in January (University of Zululand, MISGSA 2020).

The mathematical model was essentially a heat equation with a rather complex boundary condition at the surface, to account for a plant layer. Preliminary models showed that light coloured roofs absorbed less energy than dark (e.g. painted green) or soil roofs. However, once plants and evaporation or transpiration were included in the model the green roofs, as expected, absorbed significantly less energy. In Figure 3 we compare the energy absorbed by a concrete roof (black line) compared a soil roof (blue line) with evaporation, for typical Barcelona summer conditions.

In the future it is hoped to put this work into practice with local town halls.

Fig. 3 Comparison of energy absorbed in a concrete roof (black line) and soil roof with evaporation of 4mm/day (blue line), subject to typical Barcelona summer conditions.

Relevant Literature

1. T.G. Myers, F. Font, M.G. Hennessy. Mathematical modelling of carbon capture in a packed column by adsorption. Applied Energy, 278, 2020.
2. T.G. Myers, F. Font. Mass transfer from a fluid flowing through a porous media. Int. J. Heat Mass Trans., 163, 2020.
3. T. Myers, N. Fowkes, A. G. Fareo and S. Goqo. Estimating the effect of green roofs on a city’s energy footprint. Proceedings of the MISG South Africa, 2020