Analysis of Traveling-Wave Equations in Sorption Processes

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 and analyzed using a traveling-wave approach, which reduces the system to a second-order ordinary differential equation depending on the inverse Péclet 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 traveling wave. Using analytical continuation, we conclude that, at least for small values of the inverse Péclet number, the concentration profile transitions from a clean downstream state of the adsorbent matrix to a 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 traveling-wave approximation remains remarkably robust for moderate values of the inverse Péclet number.