(371a) Direct Numerical Simulation of an Alternative Smx Element Orientation for Laminar Liquid-Liquid Mixing

Emulsions and liquid-liquid (L-L) mixing processes are of central importance in a broad range of industrial sectors, ranging from food and cosmetics to polymer and mineral processing, biotechnology, and pharmaceuticals (Valdes et al., 2022). Despite the fact that conventional high-energy methods, which involve the use of stirred vessels, remain the most common choice for liquid-liquid mixing, static mixers have proven to be a viable alternative; this is because they can retain, or even improve, the performance obtained through traditional methods at a lower capital and operational cost (Thakur et al., 2003). In particular, the SMX family of mixers has been successfully implemented in numerous industrial applications, handling highly viscous fluids and substantially large viscosity ratios (rvis ∼ O(10^6)) (Hirschberg et al., 2009). Many former studies have explored the impact of varying the SMX’s geometrical features in pursuit of an optimal or enhanced configuration (Liu et al., 2006; Singh et al., 2009; Heniche et al., 2005; Hirschberg et al., 2009; Meijer et al., 2012).

Nonetheless, performance metrics targeted in such studies mostly focus around pressure drop/energy consumption vs. miscible mixing criteria (e.g., concentration profiles and interfacial stretch) and flow streamline analysis, excluding relevant metrics for multiphase systems (e.g., droplet size distribution, DSD). In this sense, proposed modifications for the SMX (e.g., SMX+ design with reduced bar width) have essentially been conceived and tested for miscible mixing applications and are intrinsically expected to improve two-phase mixing performance as well. Hirschberg et al. (2009) explicitly tested a L-L dispersion scenario and verified that Sauter mean diameter correlations for the SMX are also applicable for the SMX+, but no further research followed on the dispersion performance. Furthermore, additional effects relevant in L-L mixing, such as the presence of surfactants, have been seldom included in performance studies given the extra layers of complexity that must be handled by the numerical framework. Despite this, the role of surfactants is of critical interest when dealing with L-L systems since the properties and stability of most structured goods (i.e., emulsion-based products) are heavily correlated with the manipulation of the interfacial tension between phases (Wong et al., 2015; Leal-Calderon et al., 2007; Valdes et al., 2022). In addition, previous studies have pointed out the deficiencies behind assuming a constant equilibrium interfacial tension and disregarding key interfacial and bulk mass transfer dynamics for the surfactant species (Haas, 1987; Lobry et al., 2011; Das et al., 2013). Based on the above, we consider pertinent to carry out a comprehensive mixing performance analysis with two-phase dispersion metrics as target objectives, where extra complexities (i.e., surfactants) are included in the computations to provide a more realistic, and industrially-relevant, depiction of the mixing operation.

We have utilized a high-fidelity, 3D direct numerical simulation (DNS) framework coupled with a state-of-the-art interface capturing algorithm based on a hybrid front-tracking level-set methodology, mounted on a massively parallelized computer architecture (Shin et al., 2017, 2018). Our code solves explicitly the unsteady dynamics of the free-interface, providing a wealth of information on the interfacial dynamics which are inaccessible via experiments or volume-averaged numerical approaches. In addition, our numerical framework resolves surfactant transport for both the interface and bulk phase through a set of convection-diffusion equations, and couples it to the interfacial tension via a Langmuir equation of state, as described thoroughly by Shin et al. (2018).

An alternative orientation (90â—¦ rotation) for the SMX element pair was tested (see Fig. 1), inspired by the spacer sections between elements trialed for similar static mixers (Abdolkarimi and Ganji, 2014). This rather simple modification is based on the premise that a considerable amount of daughter droplets and satellite structures are generated via hydrodynamically-driven breakage and capillary instabilities (Valdes et al., 2023). Even though geometrically-induced deformation strongly drives the formation of unstable ligaments and structures in the SMX, hydrodynamically-driven breakups frequently spawn numerous daughter droplets from a single unstable ligament, thus heavily contributing to the number of small and medium-sized droplets. These events are mostly independent of the flow field imposed by the SMX element (Valdes et al., 2023). In this study, the alternative SMX arrangement is contrasted against the dispersion performance of a standard 2-element SMX mixer, for both clean and surfactant-laden scenarios, varying relevant physicochemical parameters (elasticity and adsorption-desorption kinetics).

As expected by design, a lower pressure drop (thus lower power consumption) is obtained for the alternative configuration with a substantially different longitudinal velocity profile (due to the fewer crossbars in the second element). Results for the clean case with a multi-drop inlet set-up (see Valdes et al. (2023)) show that both configurations produce a comparable DSD but for a significantly lower pressure drop with the alternative mixer design. The latter is also associated with a slightly skewed shape towards larger sizes for the alternative SMX due to elongated but unbroken ligaments in the spacer section. In contrast, surfactant-laden simulations for a 3-drop inlet show a massive disparity in performance between the standard and alternative mixer configurations in terms of the DSD and number of drops when modelling soluble, highly desorptive surfactants. In such scenarios, the alternative SMX generates substantially wider distributions, skewed towards larger sizes, and a droplet count close to the clean case at high desorptive coefficients. Interestingly, poorly desorptive surfactants exhibit the least DSD variability for the alternative SMX, unlike the standard SMX where the least variable DSDs are achieved by the highly desorptive surfactants. Such a disparity is significantly reduced when assuming fully-insoluble surfactants, particularly at higher surfactant elasticities, where the droplet count and DSDs are seen to agree to a much higher degree.

Acknowledgements

This work is supported by the Engineering and Physical Sciences Research Council, United Kingdom, through the EPSRC PREMIERE (EP/T000414/1) Programme Grant. This work is also supported by the Colombian Ministry of Science, Technology and Innovation MINCIENCIAS, through a doctoral studentship for JV. O.K.M. acknowledges the Royal Academy of Engineering for a Research Chair in Multiphase Fluid Dynamics. We also acknowledge the HPC facilities provided by the Research Computing Service (RCS) of Imperial College London. D.J. and J.C. acknowledge support through computing time at the Institut du Developpement et des Ressources en Informatique Scientifique (IDRIS) of the Centre National de la Recherche Scientifique (CNRS), coordinated by GENCI (Grand Equipement National de Calcul Intensif) Grant 2022 A0122B06721.

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