Simulating mixing limited reactions in porous media using finite differences
Jack Barnard  1, *@  
1 : Geoenvironmental Research Centre, Cardiff School of Engineering, Cardiff University  (GRC)  -  Website
Queen's Buildings, The Parade, Newport Road, Cardiff, CF24 3AA -  United Kingdom
* : Corresponding author

For systems with high second Damköhler number (reaction rate over diffusive mass transport rate) the reaction rate is limited by the mixing of the reactants. Instead of the reaction proceeding proportional to t^-1, the reaction proceeds at a rate proportional to t^-d/4, where d is the dimensionality of the system. This has been shown analytically (Toussaint & Wilczek, 1983) and numerically, using particle tracking methods (e.g. Benson & Meerschaert, 2008). This happens when reactants are initially segregated, or the forward progress of the reaction leads to the segregation of the reactants, which then diffuse back together.

 Experiments have shown that even with highly homogeneous porous media, the observed reaction rate is less than expected if the reactants were mixed completely (Gramling et al., 2002; Raje & Kapoor, 2000). The experiments performed by (Gramling et al., 2002) have been studied using numerical methods, most of which have used particle tracking methods (e.g. Ding et al., 2010, Edery et al., 2010, Zhang et al., 2013). This work presents a finite difference formulation of a model which can reproduce systems in which reactions a limited by mixing on the sub discretization scale, and where imposed mixing rates of reactants leads to reactions proceeding at t^-d/4 over long timescales. This is achieved by simulating the mixing of reactants separately from the spatial discretization of the problem by separating each reactant into a mixed and unmixed fraction and defining rate laws to describe the transfer of reactants between the mixed and unmixed states.

 The model developed is benchmarked against experimental data from (Gramling et al., 2002) and produces a fit which compares well with outputs produced by other simulation studies.

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