Analytical Solution of the Advection-Diffusion-Reaction Problem in a Multistage Chemical Reactor

Abstract

Transport and kinetics phenomena such as advection, dispersion and reaction play a key role in determining the performance and safety of chemical reactors. While advection-dispersion-reaction (ADR) processes have been analyzed in the past for single-stage chemical reactors, this work presents a solution for the ADR equation in a multistage chemical reactor, in which, each stage may have distinct transport and kinetics properties. A series solution for the concentration distribution is written, and the coefficients are determined using boundary and interface conditions. A quasi-orthogonality relationship between eigenfunctions of the problem is derived and used for completing the derivation of the solution. It is shown that, except at very early times, the use of just a few eigenvalues is sufficient for reasonable accuracy. Trade-offs related to computational time and accuracy for a one-term approximation are analyzed. Under special conditions, results from the general multistage analysis are shown to reduce to results for a single-stage reactor, with good agreement with past work. Based on the theoretical model derived here, the evolution of the concentration field in the multistage reactor over time is analyzed. The impact of key non-dimensional parameters on the concentration field is analyzed in detail. It is shown that the species concentration in the reactor and the reactor time constant both depend strongly on the reaction rate constant, while there is only a weak dependence on the Péclet number. The theoretical derivation presented here offers a significant generalization of ADR analysis for chemical reactors, making it possible to analyze a multistage chemical reactor. In addition to this theoretical novelty, results from this work may also help improve the design and optimization of practical multistage chemical reactors.