(67j) Single-Field Numerical Modeling of Gas-Liquid Mass Transfer

Mass transport and its transfer at interfaces in multiphase systems is a key aspect of numerous natural phenomena and industrial processes, ranging from geochemical reactions to chemical engineering applications, such as bubble column reactors, waste-water treatment facilities, chemical reactors and many others. These complex processes include viscous and capillary effects, chemical reactions and the coupling between mass transfer and fluid dynamics. This study focuses on modeling diffusion-driven mass transfer in an incompressible two-phase system consisting of a pure gas phase and a multicomponent liquid phase. Specifically, we examine systems where the gas species is diluted into the liquid phase referred to as the solution. Depending on whether the solution is undersaturated or supersaturated, the gas phase volume decreases or increases, generating dissolution or precipitation phenomena, respectively. As solute transport limits the kinetics, the interface is at thermodynamic equilibrium and is characterized by a concentration jump. Experimental measurements are generally expensive and limited by the available measuring techniques, which usually provide global quantities and do not give information about local details, such as local mass transfer rates. Numerical simulation is therefore a powerful tool to investigate these processes.

Studies in the literature initially addressed the numerical modeling of mass transfer without accounting for phase volume change through various methodologies. These approaches are based on conventional multiphase flow techniques such as : (i) the Level Set (LS) method (1), (ii) the Front Tracking (FT) approach (2) and (iii) the Volume-of-Fluid (VOF) approach (3) which is the most commonly used method. Indeed, the VOF has been applied successfully to a variety of multiphase problems with complex interfaces and one of its major strengths is the capability to conserve mass.

In comparison, few studies in the literature have addressed the numerical modeling of mass transfer while accounting for phase volume change. The complexity arises from two major challenges : first to account for phase volume variations induced by mass transfer while maintaining the concentration discontinuity at interfaces, and second, to deal with the mass volume changes that occur when species transfer between liquid and gas phases. In diffusion-driven mass transfer problems, the analysis of phase volume change induced by the mass transferred between two phases has been performed using the LS methods ((4), in a conservative manner) and the VOF methods through two approaches : the single-field (or one-fluid) approach (5; 6; 7; 8) and the two-field approach (9; 10; 11). These two VOF methods differ in their conceptualization. Rather than solving separate species transport equations in each phase and then closing the system through interface conditions, as employed in two-field approaches, the single-field formulation utilises a unified methodology. This approach incorporates the principles of species flux conservation, both in the bulk phases and at the interface, into one comprehensive set of equations. This approach is referred to in the literature as the Continuous Species Transfer (CST) model (12; 13) and, more recently, the Compressive Continuous Species Transfer (C-CST) model (14; 5). The concentration jump at the interface is addressed through a supplementary flux that incorporates the concentration jump effect as a source term. This source is distributed across mesh cells in proximity to the interface, following an approach similar to the Continuum Surface Force (CSF) model (15), which distributes surface tension effects as volumetric sources in regions near the interface.

In this work, the C-CST single-field formulation is implemented in the Notus open-source CFD software using a VOF approach and the jump in species concentration across the interface is modeled by applying Henry’s Law. In the VOF approach, an indicator function is used to track the presence of two fluids. VOF methods fall into two categories : algebraic and geometric. Algebraic VOF methods employ compressive differencing schemes to discretize the advective transport equation for the volume fraction. These schemes maintain the boundedness of the phase volume fraction while preventing interface smearing, thus keeping it as sharp as possible. Geometric VOF methods, on the other hand, capture the interface using geometrical reconstruction techniques. The interface is explicitly reconstructed from the volume fraction field, and the reconstructed interface segments are advected, often in a Lagrangian manner. The implementation of the C-CST model is based on an algebraic VOF method through the Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM) (16) for both volume fraction and species transport equations, ensuring consistency between these equation (13). The objective is to remove the use of an artificial numerical compressive term typically used within the C-CST single-field framework (7) with diffusive advection scheme.

Among the studies addressing mass transfer phenomena with phase volume change, it is noteworthy that every study examines dissolution processes, whereas the precipitation problem remains largely unexplored in the literature. While the two-field formulation has recently demonstrated accurate predictions of precipitation phenomena (10), comparable results have not yet been achieved within the single-field framework. However, we show that state-of-the-art numerical modeling of single field C-CST method fails to reproduce precipitation phenomena. To address this issue, we propose to use a two-field numerical methodology, for source terms involving mass transfer rate, developed by (10) for diffusion-driven problems and build upon the work made in temperature-driven problems. This original discretization of mass transfer rate aims to improve the computational capabilities of the single-field C-CST model to match those of the two-field formulation in diffusion-driven mass transfer problems for both dissolution and precipitation phenomena.

Additionally, our contribution extends beyond implementation to theoretical foundation, presenting new analytical solutions for both one-dimensional and two-dimensional static precipitation phenomena based on the three-dimensional solution developed by (17). This comprehensive approach establishes complete dimensional (1D, 2D and 3D) benchmarks that account for phase volume changes during precipitation, addressing a significant gap in the existing literature. Comprehensive validation across multiple scenarios demonstrates the robustness of our approach. For bubble with static barycentre cases, the method shows excellent agreement with established analytical solutions across multiple dimensions : the one-dimensional solutions of (18), the three-dimensional approximative solution of (19) for dissolution phenomena, and (17) three-dimensional solution for precipitation. Additionally, our work validates successfully against newly developed analytical solutions for both one-dimensional and two- dimensional static precipitation scenarios. This study achieves a significant step by bringing single-field approaches to parity with two-field methods in modeling both dissolution and precipitation phenomena.

In dynamic scenarios involving gas bubbles in creeping flow, our numerical results demonstrate strong alignment with (9) semi-analytical solutions across various configurations. However, an important limitation emerged when the Péclet number is higher than a threshold value : maintaining thermodynamic equilibrium at the interface required the implementation of an additional velocity term used in the advection part of the species conservation equation. This velocity only maintains thermodynamic equilibrium at the interface. While this modification proves unnecessary for cases with Péclet numbers lower than the threshold value, it raises important questions about concentration jump at the interface within the single-field framework for high Péclet numbers. This particular aspect of the methodology requires further investigation, as our analysis remains inconclusive and opens future research in competition between advection and diffusion velocities in the context of single-field mass transfer with phase volume change.

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