Mixing and reactions: Evolution from the Initial Condition

The Taylor dispersion process has been used for decades as the archetype for describing the fundamentals of dispersion processes. Primarily this is because it contains all of the necessary complexities (the interplay between longitudinal convection and transverse diffusion being the most important among the processes involved), but is simple enough to allow careful analytical study. We extend recent work by our group that examines dispersion near the initial condition (and consistently converges to the appropriate asymptotic-in-time solution) to the case of mixing and reaction. A particular strength of the approach is that it can handle any initial spatial configuration of reactants, and thus is able to predict the solute evolution in systems that are relevant to the field (where early time behavior is frequently of primary interest, and the initial conditions are often complex).

The approach yields and upscaled result that is of the classical form of the macroscopic convection-dispersion equation; importantly, however, there is an additional, non-conventional source term that arises in the analysis. This term is accomplishes two important functions in the macroscale equation: (1) It separates the early-time configuration from the definition of the dispersion coefficient, so that the value of the dispersion coefficient does not depend upon the initial condition, and (2) it captures the important features of the microscope ``mixedness" of the system, so that the effective reaction rate in the macroscale balance equation does not over-predict the rate and extent of reaction.