ISSN 2658–5782

DOI 10.21662/mfs

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2025. Vol. 20. Issue 4, Pp. 188–196
URL: https://multiphasesystems.online/mfs2025.4.025,en
DOI: https://doi.org/10.21662/mfs2025.4.025
Mathematical modeling of the gravitational sedimentation process of emulsion particles
A.A. Mukhutdinova 🖂
Mavlyutov Institute of Mechanics UFRC RAS, Ufa, Russia

Abstract

The development of new and optimization of existing methods for separating emulsions is a pressing issue for various sectors of the oil and gas and chemical industries, as well as in medicine. Studying the gravitational sedimentation of emulsions allows us to identify and explore important patterns and insights into the properties of the interface between two liquids, which are essential for solving experimental and current technical issues. Droplet size and dispersed phase concentration are key parameters that affect the rheological characteristics and stability of an emulsion. In this paper, we investigate the gravitational sedimentation of mono- and polydisperse water-in-oil emulsions. The primary objective is to determine the patterns of gravitational sedimentation of mono- and polydisperse emulsions using numerical modeling in OpenFoam. The modeling is performed under the assumptions that the viscosity, density, and temperature of the media are constant; the liquids are immiscible, incompressible, and Newtonian; and no chemical reactions occur between the two liquids. The problem is solved numerically in the interFoam solver for two incompressible, isothermal, immiscible fluids using the volume-of-fluid method with the Navier-Stokes equations. Numerical modeling revealed the following patterns of sedimentation for a polydisperse emulsion compared to a monodisperse emulsion with equivalent droplet diameters. At low emulsion concentrations (one percent) and small diameters (400 µm and less), the time it takes for a polydisperse emulsion to completely separate is the same as for a monodisperse emulsion. At emulsion concentrations from one to ten percent and droplet diameters greater than 400 µm, a polydisperse emulsion separates more slowly than the corresponding monodisperse emulsion. As the concentration of a polydisperse emulsion increases, the separation time increases and then decreases once a certain concentration is reached.

Citation

Mukhutdinova AA. Mathematical modeling of the gravitational sedimentation process of emulsion particles. Multiphase Systems. 2025;20(4):188–196 (in Russian).

Keywords

mathematical modeling;
gravity sedimentation;
monodisperse emulsion;
polydisperse emulsion;
droplet;
OpenFOAM

Article outline

This paper presents a numerical study of the gravitational sedimentation of water-in-oil emulsions, taking into account the effects of droplet size and dispersed phase concentration. The relevance of this study lies in the widespread use of water-in-oil emulsions in the oil and gas, chemical, and medical industries, where it is important to effectively separate mixtures and predict their behavior under gravitational sedimentation conditions. Studying the gravitational sedimentation of emulsions allows us to identify and explore important patterns and insights into the properties of the interface between two liquids, which are essential for solving experimental and current technical issues. Droplet size and dispersed phase concentration are key parameters that affect the rheological characteristics and stability of an emulsion.

The primary objective of the study is to identify sedimentation patterns of mono- and polydisperse emulsions using mathematical modeling in the OpenFOAM package. It is assumed that the viscosity, density, and temperature of the media are constant; the liquids are immiscible, incompressible, and Newtonian; no chemical reactions occur between the two liquids. The problem is solved numerically in the interFoam solver for two incompressible, isothermal, immiscible fluids using the volume of fluid (VOF) method with the Navier–Stokes equations. This means that the material properties are constant in the region filled with one of the two fluids, with the exception of the interphase boundary. The damBreak solver is used to set up the two-dimensional model. The problem is characterized by unsteady flow of the two fluids. The two-phase algorithm in interFoam is based on the VOF method, which uses the solids transport equation to determine the relative volume fraction of the two phases, or phase fraction α, in each computational cell. To ensure a correct numerical solution, the computational grid parameters and convergence criteria were specified, and a discretization sensitivity analysis of the results was performed. The computational domain was discretized using a uniform grid of 800 × 2000 cells, which corresponds to a minimum cell size of approximately 50 µm and ensures at least 20 cells per minimum droplet diameter. Near the interface, the dynamicRefineFvMesh algorithm for local refinement to two levels was used, subject to the condition |∇α| > 0.1. The minimum cell size after refinement was approximately 25 µm. The modeling is performed for droplet sizes ranging from 400 to 800 μm and dispersed phase concentrations from 1 to 10%. To quickly prepare the model for calculation, a program was written in MATLAB that randomly distributes the positions of drops in a given area using the built-in randi function and writes their coordinates to a file.

The numerical results reveal significant differences in the behavior of mono- and polydisperse systems. It was found that at small droplet diameters (up to 400 µm) and low concentrations (approximately 1%), the complete separation time for both systems is virtually identical. However, with increasing concentration to 3–10% and droplet sizes above 400 µm, polydisperse emulsions settle more slowly due to the presence of small droplets, which affect the overall sedimentation rate. At droplet diameters of 800 µm, a nonlinear pattern of sedimentation time was observed: initially, it increases with increasing concentration and then decreases.

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