Evaluation and linking of effective parameters in particle‐based models and continuum models for mixing‐limited bimolecular reactions

Abstract

[1] Particle-based models and continuum models have been developed to quantify mixinglimited bimolecular reactions for decades. Effective model parameters control reaction kinetics, but the relationship between the particle-based model parameter (such as the interaction radius R) and the continuum model parameter (i.e., the effective rate coefficient K f ) remains obscure. This study attempts to evaluate and link R and K f for the second-order bimolecular reaction in both the bulk and the sharp-concentration-gradient (SCG) systems. First, in the bulk system, the agent-based method reveals that R remains constant for irreversible reactions and decreases nonlinearly in time for a reversible reaction, while mathematical analysis shows that K f transitions from an exponential to a power-law function. Qualitative link between R and K f can then be built for the irreversible reaction with equal initial reactant concentrations. Second, in the SCG system with a reaction interface, numerical experiments show that when R and K f decline as t À1/2 (for example, to account for the reactant front expansion), the two models capture the transient power-law growth of product mass, and their effective parameters have the same functional form. Finally, revisiting of laboratory experiments further shows that the best fit factor in R and K f is on the same order, and both models can efficiently describe chemical kinetics observed in the SCG system. Effective model parameters used to describe reaction kinetics therefore may be linked directly, where the exact linkage may depend on the chemical and physical properties of the system. Citation: Zhang, Y., C. Papelis, P. Sun, and Z. Yu (2013), Evaluation and linking of effective parameters in particle-based models and continuum models for mixing-limited bimolecular reactions, Water Resour. Res., 49,

Upscaling of Mixing‐Limited Bimolecular Chemical Reactions in Poiseuille Flow

“…In order to study the detailed dispersion and reaction behavior, we use a numerical reactive random walk particle tracking (RWPT; Alhashmi et al, 2015;Benson & Meerschaert, 2008;Edery et al, 2009Edery et al, , 2010Zhang et al, 2013) to determine the evolution of the species concentrations and the global reactivity in terms of the total product mass. The mixing behavior is characterized in terms of effective dispersion, which measures the average width of a point injection in the channel cross section, or in other words, the Green function of the transport problem (Dentz & Carrera, 2007).…”

Modeling Bimolecular Reactive Transport With Mixing‐Limitation: Theory and Application to Column Experiments

“…Porta et al () perform comparative assessment of the effective models of Sanchez‐Vila et al () and Hochstetler and Kitanidis () with the classical fully mixed model. Zhang et al () do the same for the model of Sanchez‐Vila et al () and an exponential model (apparently unaware of the similar model by Rubio et al, ) in a comprehensive effort to bridge these continuum models with particle tracking model concepts. Barnard () suggests an empirical reaction network involving fifth‐order kinetics governing transfer of mass from unmixed A (and B ) to mixed ( A m and B m ) phases, with only the mixed phases undergoing bimolecular reaction that is kinetically controlled with a new empirical rate coefficient that is much smaller than that measured in application.…”

“…Several comprehensive reviews of the numerous subsequent modeling efforts targeting these data are available (e.g., Alhashmi et al, ; Barnard, ; Chiogna & Bellin, ; Porta et al, ; Zhang et al, ), and for brevity this material will be only superficially treated here. Lagrangian particle tracking approaches (e.g., Edery et al () using continuous time random walks; Zhang et al () extending the scheme of Benson and Meerschaert (); Ding et al () who build on the same scheme and an effective constant bimolecular kinetic rate; Alhashmi et al () who develop a particle tracking scheme for transport and reaction at the pore scale after solving the Navier‐Stokes equations for pore‐scale flow) and Eulerian continuum models (e.g., Rubio et al () using an effective exponential time‐dependent bimolecular kinetic rate; Sanchez‐Vila et al () using an effective power law time‐dependent bimolecular kinetic rate; Hochstetler and Kitanidis () using an effective constant bimolecular kinetic rate; Zhang et al () testing and extending both approaches of Rubio et al () and Sanchez‐Vila et al (); Chiogna and Bellin () using a beta‐distribution of mixing ratios following Oates () with equilibrium bimolecular kinetic rate; Barnard () using an effective constant bimolecular kinetic rate and nonlinear kinetics in an extended reaction network) have been applied to different subsets of the data of RK00 and of G02. Yet only the studies by Zhang et al () and by Ding et al () provide modeling of all the experiments of both studies.…”

“…Barnard () suggests an empirical reaction network involving fifth‐order kinetics governing transfer of mass from unmixed A (and B ) to mixed ( A m and B m ) phases, with only the mixed phases undergoing bimolecular reaction that is kinetically controlled with a new empirical rate coefficient that is much smaller than that measured in application. Barnard () also provides a qualitative evaluation of the performance of the models of Chiogna and Bellin (), Ding et al (), Sanchez‐Vila et al (), Zhang et al (), and Alhashmi et al () in application to the data of G02.…”

On the validity of effective formulations for transport through heterogeneous porous media

“…They deal with chemical reactivity either in a stochastic manner, representing reactivity with molecular analogies, or in classical approaches by means of concentrations Cirpka et al, 2012;Ding et al, 2013;Hayek et al, 2012;Knutson et al, 2007;Zhang et al, 2013). Extensions are both required for application purposes and attractive for capturing the consequences of anomalous transport to potential "anomalous" and enhanced reactivity (Battiato et al, 2009;Sadhukhan et al, 2014;Scheibe et al, 2015;Tartakovsky et al, 2009).…”