Transient solute transport with sorption in Poiseuille flow

Previous work on solute transport with sorption in Poiseuille flow has reached contradictory conclusions. Some have concluded that sorption increases mean solute transport velocity and decreases dispersion relative to a tracer, while others have concluded the opposite. Here we resolve this contradiction by deriving a series solution for the transient evolution that recovers previous results in the appropriate limits. This solution shows a transition in solute transport behaviour from early to late time that is captured by the first- and zeroth-order terms. Mean solute transport velocity is increased at early times and reduced at late times, while solute dispersion is initially reduced, but shows a complex dependence on the partition coefficient $k$ at late times. In the equilibrium sorption model, the time scale of the early regime and the duration of the transition to the late regime both increase with $\ln k$ for large $k$. The early regime is pronounced in strongly sorbing systems ($k\gg 1$). The kinetic sorption model shows a similar transition from the early to the late transport regime and recovers the equilibrium results when adsorption and desorption rates are large. As the reaction rates slow down, the duration of the early regime increases, but the changes in transport velocity and dispersion relative to a tracer diminish. In general, if the partition coefficient $k$ is large, the early regime is well developed and the behaviour is well characterized by the analysis of the limiting case without desorption.