Dispersion and Phase Exchange Process of Chemically Reactive Solute Through Circular Tube

This article explores how a chemically reactive solute will disperse across mobile to immobile phase when injected into the fluid flowing within a long circular tube. To model this process, we utilized mathematical modeling, including advection-diffusion equations for flow of fluid within the tube and first-order chemical reaction equations to account for reversible and irreversible reactions on the tubes’ wall. We proposed a numerical method based on an explicit finite difference scheme to solve the governing equations for the dispersion of a chemically reactive solute. We used an upwind method with a conservative representation in the diffusion component to discretize the advection-diffusion equation. To ensure the stability of our proposed numerical scheme, we computed the time step constraint condition so that the maximum principle for the discrete governing equation holds. We also verified the performance of our proposed scheme through computational results that were compared with previous studies. One of our key findings was that the depletion coefficient D۰ achieved a quasi-steady state for larger absorption rates. We also observed that the advection coefficient  initially increased with an increasing absorption rate, but eventually declined due to phase exchange kinetics. The dispersion coefficient  also decreased with a rising absorption rate due to a low-velocity gradient in the middle region. Our study showed that rapid distributions are possible under certain conditions, such as a high Damköhler number (Da≥۱۰ ) and a high absorption rate (Γ>۵). Computational results show that the proposed scheme can be useful in developing an efficient pulmonary drug delivery system for periodic inhalation of drugs to determine the optimal frequency of injection.