On the conditional superiority of counter-current over co-current extraction in microchannels

In liquid-liquid extraction, counter-current flow of the phases always results in an improved performance as compared to co-current flow, under similar operating conditions. However, it is challenging to implement counter-current flow in a microchannel. Therefore, the improvement in extraction performance must be significant to justify the selection of counter-current flow over co-current flow in microchannels. In this study, we identify the range of fluid properties and operating conditions for which counter-current operation exhibits significant benefits. For this, simplified mathematical models are developed for both co-current and counter-current extraction in the stratified flow regime. These models, while being simple, capture the essential physics of the extraction process and facilitate a thorough investigation of the relative extraction performance across the parameter space. An analytical solution, based on the theory of Sturm-Liouville linear operators, is obtained for the case of co-current flow. The counter-current model belongs to the class of two-way diffusion equations for which a novel semi-analytical solution is presented. The analysis of the predictions of the models shows that the relative extraction performance is governed by a general principle of maximum gain at mediocre performance. These results help identify the significantly restricted range of operating parameters for which counter-current operation is a truly attractive alternative to the co-current mode of extraction in microchannels.