Low-Dimensional Modeling of Reactions and Transport in Stratified Microflows

Abstract

Stratified two-phase flow through microchannels has found a myriad of applications in micro-scale chemical processing and separations. These include solvent extraction (including extraction of biomolecules), treatment of toxic industrial streams and phase transfer catalysis. Mathematical models of these systems must describe transverse inter-fluid mass transfer and interfacial and bulk phase reactions, while accounting for the effects of a varying velocity profile and different phase holdups. Typically these models are composed of two partial differential equations of the convection-reaction-diffusion type- one for each phase. These models are computationally inefficient for carrying out optimization, control and parametric studies. To overcome these challenges, we rigorously average the equations by taking advantage of the separation of the time scales of transverse diffusion and axial convection.

The Lyapunov-Schmidt (LS) technique is applied to develop low dimensional models, based on Center (Slow) Manifold Theory, that describe the evolution of the transverse-average species concentration along the length of the channel. Two different models are derived. The first, called the one equation averaged (OEA) model, is based on averaging across both fluids. This model is a relatively direct application of LS reduction. However, it is unable to capture the mass transfer between the fluids near the channels inlet, where the two liquid first meet. The second model, called the two equation averaged (TEA) model, is developed to overcome this limitation. Here, the LS reduction is applied in a novel manner to average across each fluid separately. The TEA model is able to capture the behavior of the system accurately at low values of the Damkohler number and up to moderate values of the transverse Peclet number. We apply the TEA model to analyze solvent extraction, reactive extraction and a two-phase system with competitive-consecutive reactions.

Keywords: averaging, order reduction, extraction, stratified flow, microchannels

Fig 1. Comparison of the TEA and OEA models with the full PDE model for a case of reactive extraction. p: transverse Peclet number, Da: Damkohler number.