Analysis of travelling wave equations in sorption processes

TL;DR

This paper develops and analyzes a mathematical model of an adsorption column using travelling-wave methods, simplifying the PDE system to an ODE and validating the approximation through rigorous analysis and numerical simulations.

Contribution

It introduces a singular perturbation approach to reduce the PDE model to a tractable ODE and proves the persistence of travelling wave solutions in sorption processes.

Findings

The reduced model accurately captures the concentration transition for small inverse Péclet numbers.

Numerical simulations confirm the analytical predictions and robustness of the travelling-wave approximation.

The model provides insights into the dynamics of contaminant adsorption in columns.

Abstract

This work presents a mathematical model of an adsorption column to study the evolution of contaminant concentration and adsorbed quantity along the longitudinal axis of the filter. The model is formulated as a system of partial differential equations (PDEs) and analysed using a travelling-wave approach, which reduces the system to a second-order ordinary differential equation depending on the inverse P\'eclet number, typically a small parameter. By neglecting this parameter, the model is simplified via a singular perturbation to a leading-order approximation, which can be interpreted as a slow-fast system. We rigorously justify this reduction by proving the persistence of the heteroclinic connection associated with the travelling wave. Using analytical continuation, we conclude that, at least for small values of the inverse P\'eclet number, the concentration profile transitions from a clean downstream state of the adsorbent matrix to fully upstream saturation. Numerical simulations are presented to validate the analytical results and to assess the accuracy of the reduced model. A sensitivity analysis demonstrates that the travelling-wave approximation remains remarkably robust for moderate values of the inverse P\'eclet number.