Marc Calvo-Schwarzwalder (PDEs Group, IMTech)
Column sorption is a practical and widely used method for removing contaminants from carrier fluids and has attracted considerable attention in recent years. The process consists of passing a fluid through a column packed with a material capable of selectively capturing specific components. The mathematical modelling of this process, together with the underlying chemical and physical mechanisms, has been a central focus of an ongoing interdisciplinary collaboration involving multiple research centres and universities across Europe.
This work extends beyond mathematics alone, relying on close collaboration with chemical and environmental engineers as well as experimental scientists. Participating institutions include UPC, with researchers from the Departments of Mathematics, Fluid Mechanics, and Chemical Engineering; the Universitat de Girona; LEQUIA (Laboratori d’Enginyeria Química i Ambiental); IQS (Institut Químic de Sarrià); ICIQ (Institut Català d’Investigació Química); the University of Leeds; the University of Oxford; and, more recently, the University of British Columbia (Vancouver, Canada). Interdisciplinary collaborations of this kind are essential for the development of new technologies and the advancement of existing ones.
Mathematically, the removal of trace-amounts of a single contaminant in a column of length \(L\) typically consists of one diffusion-advection-reaction equation of the form
\begin{equation}\label{single:advection}
\frac{\partial c}{\partial t}+u\frac{\partial c}{\partial x}=D\frac{\partial^2 c}{\partial x^2}-\alpha\frac{\partial q}{\partial t}\, \;\;\;\;\; (9)
\end{equation}
with \(c(x,t)\) being the cross-sectional average of the molar concentration of contaminant at time \(t>0\) and at some point \(x\in(0,L)\) of the column; \(u\) is the interstitial velocity of the mixture flowing through the column, which remains approximately constant in the case of trace amounts; \(D\) is a dispersion coefficient and \(\alpha \dot q\) is a sink term describing the rate at which the contaminant is being removed. The quantity \(q(x,t)\) quantifies the amount of contaminant that is being captured per unit mass of adsorbent material. In addition, one must consider the so-called kinetic equation
\begin{equation*}
\frac{\partial q}{\partial t}=\Phi(c,q)\, ,
\end{equation*}
where the form of \(\Phi\) depends on the physical and/or chemical mechanisms. A common model used at this point is the one developed by Langmuir [1], which allows a contaminant to adsorb and desorb from the adsorbent at rated \(k_a\) and \(k_d\),
\begin{equation}
\Phi_{L}(c,q)=k_ac(q_{m}-q)-k_dq\, \;\;\;\;\; (10)
\end{equation}
with \(q_{m}\) being the maximal amount of contaminant that can be removed by the adsorbent. This mathematical formulation has been widely studied for the case when both trace or large amounts of one contaminant are being removed [2-8].
In practise, a carrier fluid typically contains more than one contaminant, which results in significantly more complicated kinetics. Mathematically, the process is also much more complicated since competition among the different species must be accounted for. When multiple contaminants are being removed, the different species compete among them to occupy the available sites on the surface of the adsorbent. For instance, a free molecule of one species can remove an adsorbed molecule of another species and occupy the site instead, see Figure 18. Besides the individual adsorption and desorption processes, which are described similarly to the single contaminant case, these interactions must be included in the mathematical formulation.
Recently [9, 10], we have extended the single-contaminant model to a system where two species compete against each other to be adsorbed, eventually replacing each other on occupied adsorption sites. The molar concentration \(c_i\) of each species satisfies an equation that is similar to Eq. (9), with the particularity that the removed amounts are now quantified by \(\theta_i\) rather than \(q_i\), which we refer to as the fractional coverage of each species. Both quantities are related via \(\theta_i=q_i/q_{m,i}\). For each contaminant, the advection-diffusion equation now reads
\begin{equation*}
\frac{\partial c_i}{\partial t}+u\frac{\partial c_i}{\partial x}=D\frac{\partial^2 c_i}{\partial x^2}-\alpha_i\frac{\partial \theta_i}{\partial t}\, ,
\end{equation*}
whereas the kinetic equations take the form
\begin{align}\label{multi:q}
\frac{\partial \theta_i}{\partial t}=& \underbrace{k_{ad,i}c_{i}\left(1-\sum_{j}\theta_j\right)-k_{de,i}\theta_i}_{\Phi_i}\nonumber\\
& +\underbrace{\left(\sum_{i\neq j} k_{i,j}\theta_j\right)c_i-\left(\sum_{i\neq j} k_{j,i}c_j\right)\theta_i}_{\Psi_i}\, \;\;\;\;\; (11)
\end{align}
where \(\Phi_i\) and \(\Psi_i\) respectively account for individual adsorption mechanisms and interactions among species. In Eq. (11), we have used the Langmuir model to describe individual adsorption. This general form allows generalising the model to any number of contaminants \(N\), which we have done in a follow-up paper that is currently being reviewed [11].
After introducing sensible non-dimensional variables and discussing the relative importance of the relevant adsorption mechanisms, we have been able to develop analytical solutions for the breakthrough concentration – the concentration that is being measured at the column outlet – of each species in a 2-component model [9, 10] and, in \(N\)-contaminant system with arbitrary \(N\). This had not been done before and represents an important step forward in understanding competitive adsorption of multiple contaminants.
Our current research focuses on several interconnected directions: investigating the role of humidity in adsorption processes, which can be viewed as a particular class of multi-component systems; developing more general analytical solutions for multi-component models, for example by relaxing restrictive assumptions on model parameters; and exploring adsorption mechanisms beyond the Langmuir model. The latter is of particular interest for systems involving the removal of metallic species from liquid phases, among other applications.
References
[1] I. Langmuir, The adsorption of gases on plane surfaces of glass, mica and platinum, Journal of the American Chemical Society 40 (9) (1918) 1361–1403. doi:10.1021/ja02242a004.
[2] T. G. Myers, F. Font, Mass transfer from a fluid flowing through a porous media, International Journal of Heat and Mass Transfer 163 (2020) 120374. doi:10.1016/j.ijheatmasstransfer.2020.120374.
[3] T. G. Myers, F. Font, M. G. Hennessy, Mathematical modelling of carbon capture in a packed column by adsorption, Appl. Energ. 278 (2020) 115565. doi:10.1016/j.apenergy.2020.115565.
[4] M. Aguareles, E. Barrabés, T. G. Myers, A. Valverde, Mathematical analysis of a Sips-based model for column adsorption, Phys. D: Nonlinear Phenom. 448 (2023) 133690. doi:10.1016/j.physd.2023.133690.
[5] A. Valverde, A. Cabrera-Codony, M. Calvo-Schwarzwalder, T. G. Myers, Investigating the impact of adsorbent particle size on column adsorption kinetics through a mathematical model, International Journal of Heat and Mass Transfer 218 (2024) 124724. doi:10.1016/j.ijheatmasstransfer.2023.124724.
[6] T. Myers, M. Calvo-Schwarzwalder, F. Font, A. Valverde, Modelling large mass removal in adsorption columns, International Communications in Heat and Mass Transfer 163 (2025) 108652. doi:10.1016/j.icheatmasstransfer.2025.108652.
[7] T. G. Myers, A. Cabrera-Codony, A. Valverde, On the development of a consistent mathematical model for adsorption in a packed column (and why standard models fail), International Journal of Heat and Mass Transfer 202 (2023) 123660. doi:10.1016/j.ijheatmasstransfer.2022.123660.
[8] L. C. Auton, M. Aguareles, A. Valverde, T. G. Myers, M. CalvoSchwarzwalder, An analytical investigation into solute transport and sorption via intra-particle diffusion in the dual-porosity limit, Applied Mathematical Modelling 130 (2024) 827–851. doi:10.1016/j.apm.2024.03.023.
[9] M. Calvo-Schwarzwalder, T. G. Myers, A. Cabrera-Codony, A. Valverde, An analytical breakthrough model for adsorption systems with two competing contaminant species, International Journal of Heat and Mass Transfer 245 (2025) 127004. doi:10.1016/j.ijheatmasstransfer.2025.127004.
[10] A. Valverde, T. G. Myers, A. Cabrera-Codony, M. Calvo-Schwarzwalder, Combining equilibrium and dynamical models to describe two-component
adsorption in a fixed bed, submitted to Journal of Hazardous Materials (2025).
[11] M. Calvo-Schwarzwalder, A. Valverde, A. Cabrera-Codony, T. G. Myers, An analytical model for competitive adsorption in a fixed bed with an arbitrary number of contaminants, To be submitted (2026).