Low-dimensional models for describing mixing effects in laminar flow tubular reactors

Abstract

The Liapunov–Schmidt (LS) technique of bifurcation theory is used to average the convective-diffusion equation in the transverse direction and obtain low-dimensional two-mode models that describe mixing effects in laminar flow tubular reactors. For the isothermal case, these models are described by a pair of equations involving two modes, namely, the spatially averaged (〈 C 〉) and the mixing-cup ( C m) concentration vectors. The first equation traces the evolution of C m with residence time, while the second is a local balance equation that describes local mixing as an exchange between the reaction scale (represented by 〈 C 〉) and the convection scale (represented by C m ) in terms of the local mixing time. The LS method also shows that such low-dimensional description is possible only if the local Damköhler number (ratio of local mixing time to reaction time) satisfies the convergence criteria of being less than 0.858. It is shown that the two-mode models have the same accuracy as the infinite (radial) mode convection model, within the range of validity of the latter. The two-mode models for homogeneous reactors have many similarities with the classical two-phase models for heterogeneous catalytic reactors, with the transfer coefficient concept (between surface and mixing cup concentrations, C S and C m , respectively) being replaced by that of an exchange coefficient (between 〈 C 〉 and C m ). Examples are presented to illustrate the usefulness of the two-mode models in predicting the effects of non-identical local mixing times, non-uniform reactant feeding and non-linear kinetics on conversion and yields of products for single and multiple reactions in laminar flow tubular reactors.