Modelling Extraction in Microchannels with Stratified Flow: Channel Geometry, Flow Configuration and Marangoni Stresses

In this work, we present a hierarchy of mathematical models for extraction in microchannels with two-phase stratified flow. A flat stable inter-fluid interface is considered. We present three models for rectangular channels, of varying degrees of complexity: 2D, 1D Laminar and 1D Plug models. The predictions of these models are compared with each other and with experiments reported in the literature. Conditions under which a simpler 1D model can replace the complex 2D model are identified. Next, a detailed model for channels with a circular cross-section is developed using bipolar cylindrical coordinates. We also show how a much simpler description can be obtained by applying the 1D rectangular channel models to a circular channel. The relative performance of co-current and counter-current flow configurations is analysed next, using the 1D Plug flow model. Finally, the model is extended to account for Marangoni stresses that result from the variation of interfacial tension with solute concentration. The effect of Marangoni stresses on extraction in stable stratified flow is found to be insignificant. This is because the flow induced by Marangoni stresses is too weak to control the pressure-driven primary flow.