Entropy‐based critical reaction time for mixing‐controlled reactive transport

Abstract

Entropy‐based metrics, such as the dilution index, have been proposed to quantify dilution and reactive mixing in solute transport problems. In this work, we derive the transient advection dispersion equation for the entropy density of a reactive plume. We restrict our analysis to the case where the concentration distribution of the transported species is Gaussian and we observe that, even in case of an instantaneous complete bimolecular reaction, dilution caused by dispersive processes dominates the entropy balance at early times and results in the net increase of the entropy density of a reactive species. Successively, the entropy of the reactant decreases until it vanishes. We show the existence of a unique critical value of dilution, which corresponds to the complete consumption of one of the reactants. This critical dilution index is independent of advective and dispersive processes, and depends only on the dimensionality of the problem, on the stoichiometry of the reaction and on the initial concentrations of the reactants. Furthermore, we provide simple analytical expressions to compute the critical reaction time, i.e., the time at which the critical dilution index is reached, for selected flow configurations. Our results show that, differently from the critical dilution index, the critical reaction time depends on solute transport processes such as advection and hydrodynamic dispersion.

Mixing and Reactive Fronts in the Subsurface

“…Interesting insights on the interplay between mixing dynamics and reactive processes are provided by entropy balances that allow the connection of the concept of the dilution index with reactive mixing (Chiogna et al 2012;Chiogna and Rolle 2017;Ye et al 2018). Focusing on steady-state transport and continuous release of dissolved plumes, Chiogna et al (2012) derived the transport equation for the entropy density of a reactive species:…”

Preferential Pathways for Fluid and Solutes in Heterogeneous Groundwater Systems: Self-Organization, Entropy, Work

“…While the second law of thermodynamic refers to physical entropy (introduced by Clausius (1857), section 3.1), information entropy (introduced by Shannon (1948)) is closely related and well suited for diagnosing spatial organization (section 3.3). The concepts of information and Shannon entropy haven been used widely to characterize irreversible mixing and reaction processes and their predictability (Chiogna and Rolle, 2017), the emergence of order in distributed time series (Malicke et al, 2020), information in multiscale permeability data (Dell`Oca et al, 2020) and the role of spatial variability of rainfall and topography in distributed hydrological modelling (Loritz et al, 2018(Loritz et al, , 2021. Woodbury and Ulrych (1993) and Kitanidis (1994) used the Shannon entropy to describe the spatial-time development and dilution of tracer plumes in groundwater systems.…”

Plume deformation, mixing, and reaction kinetics: An analysis of interacting helical flows in three-dimensional porous media

“…For conservative transport and considering a continuously injected plume, E Q is a monotonically increasing function with respect to the travel distance and quantifies the increased dilution of the plume due to the transverse diffusive/dispersive fluxes. This metric was later applied to study reactive transport [70,71] and its value measures the interplay between dilution and reactive mixing. In fact, the development of E Q is determined by the balance between a positive source term due to dilution and a negative sink term due to reaction in the governing transport equation of the entropy density [70]:…”