Mathematical model for mixing reactants in a capillary microreactor by transverse diffusion of laminar flow profiles

Transverse diffusion of laminar flow profiles (TDLFP) was recently suggested as a generic approach for mixing reactants inside a capillary microreactor. Conceptually, solutions of reactants are injected inside the capillary by high pressure as a series of consecutive plugs. Because of the laminar nature of the flow inside the capillary, the nondiffused plugs have parabolic profiles with predominantly longitudinal interfaces between the plugs. After the injection, the reactants are mixed by transverse diffusion across the longitudinal interfaces. TDLFP-based mixing is still in its infancy as only the principle was proved. Here, we develop the theory of TDLFP and introduce a dimensionless parameter, York number, which can be used in predicting the quality of TDLFP-based mixing. The theory uses a single simplifying assumption that the longitudinal diffusion is negligible; this assumption is readily satisfied. We then develop a numerical model of TDLFP and use it to simulate the concentration profiles of three reactants mixed by TDLFP in the capillary. The correlation between the York number and quality of mixing is analyzed. Two ways of improving the quality of TDLFP-based mixing are suggested and studied: (i) increasing the longitudinal interface between the plugs by a long last plug of a solvent and (ii) "shaking" the injected reactants by a series of alternating negative and positive pressure pulses. The developed theory and computational simulation of TDLFP will stimulate the practical use of capillary microreactors.