Consequences of the partial-equilibrium approximation for chemical reaction and transport

Abstract

This paper is concerned with the relation between various mathematical methods for formulating the equations of convective diffusion in the presence of infinitely-rapid reaction pathways. The notions of general composition variables, reaction invariance and differential geometry provide useful conceptual tools for comparing various methods. In particular, the method of ‘composite fluxes’ is shown to represent a mixed-coordinate description, with separate composition coordinates for spatial fluxes and concentration fields. While this method avoids a coordinate degeneracy sometimes associated with fast-reaction invariants, it is shown that such degeneracy should never occur in chemical systems having stable equilibria for the fast reactions. An elementary reaction scheme, involving mobile intermediates or adsorbed surface species, is treated to illustrate the coupled and facilitated diffusion effects resulting from the partial-equilibrium approximation. Also, the relation of the latter to the well-known quasi steady-state approximation (QSSA) of chemical kinetics is established. It is shown that, apart from certain exceptional circumstances, the usual ab initio invocation of the QSSA has no unforeseen consequences for diffusive transport. On the other hand, strong spatial fluxes can cause breakdown of such equilibrium approximations, in much the same way as strong unsteadiness is known to do.

Dissipation Potentials for Reaction-Diffusion Systems

“…The above results are rendered more evident by means of the standard decomposition of the reaction rates into r stoichiometrically independent reaction pathways with rates ω j , j = 1:r ≤ n: (12) where N is the rank-r, n×r matrix of stoichiometric coefficients ν i j (representing the number of moles of species i produced by reaction j, with negative values distinguishing reactants from products 11 ). With this decomposition, the dissipation takes on a standard form: 1 …”

“…The vector λ = [λ i ] is in essence the set of Lagrange multipliers or "reactive" forces associated with the conservation of atoms or, more precisely, any independent set of n − r reaction invariants. 12 Accordingly, it seems proper to replace eq 9 by …”

“…The underlying mathematical techniques are subsumed in numerous previous analyses. 9,12,21,24 In 9,21 assume that the forces denoted here by α,λ lie in the same Euclidean space as the fluxes, whereas technically they should be regarded as dual spaces, distinguished here by superscripts and subscripts.) The vector λ = [λ i ] is in essence the set of Lagrange multipliers or "reactive" forces associated with the conservation of atoms or, more precisely, any independent set of n − r reaction invariants.…”