On Danckwerts’ Boundary Conditions for the Plug-Flow with Dispersion/Reaction Model

Abstract

Accurate predictions of oxygen utilization with position in plug-flow like (aka serpentine) reactors employed for aerobic activated-sludge treatment of municipal and industrial wastewaters are desired. Three non-ideal reactor models—completely-mixed flow reactors (CMFRs) in series, segregated flow, and plug-flow with dispersion (PFD)—and the ideal plug-flow model were investigated considering typical reaction kinetics and a typical reactor configuration to perform such predictions. Significant differences arose between predictions from the PFD model relative to those from the CMFR in series and segregated flow models. These differences led to re-examination of the PFD model and the inlet/outlet boundary conditions postulated more than six decades ago by P. V. Danckwerts. Major criticisms are found in the literature addressing the PFD model arising from the concentration “jump” at the inlet boundary necessitated by specification of conservation of reactant flux at the expense of continuity of target reactant concentration. A fresh theoretical examination of the dispersive/reactive processes within a closed reactor with particular attention paid to the inlet and outlet boundary planes led to alternative statements for the inlet and outlet boundary conditions. Revisions proposed herein to Danckwerts’ boundary conditions for the PFD/reaction model preserve both continuity of reactant concentration and conservation of reactant flux at both inlet and outlet boundaries. Hypothetical computations based on pseudo-first-order kinetics for a reaction carried out within a serpentine reactor of typical configuration illustrate the differences between concentration profiles through the reactor predicted using the proposed and Danckwerts’ boundary conditions.