Mathematical modeling of a water-in-oil emulsion droplet behavior under the microwave impact

The problem of microwave (MW) electromagnetic radiation impact on a single water-in-oil droplet is considered. The system of heat equations within the droplet and in the surrounding liquid, incompressible Navier-Stokes equations within the droplet and in the surrounding liquid, and equation of state are considered. The formulated problem is solved numerically using TDMA (Tri-diagonal-matrix algorithm), SIMPLE algorithm and VOF method (volume of fluid method for the dynamics of free boundaries) in Euler coordinates. The results in the form of the dependence of the temperature within the droplet and in the surrounding liquid on the time of microwave impact and streamlines thermal convection are represented; dependence of the velocity of droplet's moving on the power of the of the microwave impact is shown. The obtained results can help to establish criteria for the efficient applicable of the microwave method for the water-in-oil emulsions destruction.


Introduction
The formation of high stability of water-in-oil emulsions is one of the negative factors in extracting and processing of oil, their preparation and transportation, as well as liquidation/recycling of oilsludge barns. The use of conventional techniques to destroy the emulsions yields no positive results. The use of microwave radiation is one of the perspective method for the destruction of water-in-oil emulsions [1][2][3].
The emulsions are regarded as a heterogeneous system comprising two immiscible liquids such as oil and water, with one dispersed in the other in the presence of surfactants [4]. Besides, tiny solid rock particles (clay, quartz, salts, etc.) suspended in that system can also act as emulsion stabilizers [5]. High stability of water-in-oil emulsions is conditioned by the oil high-molecular polar components which cover water droplets forming the so called armor envelope, and that prevents the coalescence of the droplets [6,7].
To study the MW actions on the water droplet surrounded by a dielectric liquid the system of heat and Navier-Stokes equations within the droplet and in the surrounding liquid is considered. These equations are based on fundamental laws of motion of multi-phase medium. Formulation of the problem. It is assumed that the water droplet with radius r 0 surrounded by a hydrocarbon liquid in a gravitational and MW fields. Spherical droplet is in the center of a cylindrical vessel with the height h and the radius r 1 ( fig.1.). The surface tension effect is modeling. In addition, it is considered, that the infinitely thin of armor envelope is formed on the surface of the droplet. It is assumed, that droplet doesn't deform and keeps spherical shape in all time of influence. The inertial and ponderomotive forces are neglected. It's taken into account that the dissipation of the microwave energy is within a water droplet only. The motion of each component of the system is described by a system of thermal convection equations in linear Boussinesq approximation [8]. The law of conservation of energy and the conditions of dynamic equilibrium are executed on the interface. TDMA (Tri-diagonal-matrix algorithm), SIMPLE algorithm and VOF method (volume of fluid method for the dynamics of free boundaries) were used for the numerical solution of this problem [9]. The surface tension converts to the bulk force using continuum surface force (CSF) model. The model interprets surface tension as a continuous, three-dimensional effect across an interface, rather than as a boundary value condition on the interface [10]. The direction and magnitude of bulk force are determined by the density gradient and curvature of the surface. In view of the assumptions made about the safety of a spherical droplet bulk forces are always directed to the center of the droplet and the curvature of the surface of the droplet is constant and equal to 2/r 0 . Electromagnetic field energy is converted on heating of the water, and is modeled as a distributed heat source [2], magnitude of which is determined from the expression: where ω is the frequency of electromagnetic radiation; 0 ε is the dielectric constant; ε ′ is the dielectric permittivity; δ tan is the dielectric loss tangent; E is the electrical intensity. Then the dimensionless system of equations of thermal convection is considered in the following dimensionless form: where ρ , c , µ , k are density, specific heat, dynamic viscosity and thermal conductivity of the medium; u and v are the velocity vector; p is the pressure; T is the temperature; β is the coefficient of  Accordingly the boundary conditions can be written as follows:

Calculation results
Profiles of temperature along the equator and the poles in the droplet and surrounding liquid in case q=5*10 7 W/m 3 and temperature coefficient Ɣ=1.5 1/K are shown in fig. 2-3.  In the case of the MW influence the field energy dissipates directly within the droplet. Along the equator the droplet heats irregularly, one can observe two characteristic peaks. This is due to the fact that the convective flow ( fig. 4) carries heating, thereby the center of droplet is heated slower than the edge ( fig. 5). Also, both poles of the droplet are heated differently too. Surrounding liquid is heated primarily from the top this is due to the fact that the droplet falls down ( fig. 5). Graph of droplet velocity depending on the time is shown in fig. 6.    The droplet of water is heated by the previous case, but over time the heated region on top of the drop is increased significantly ( fig. 7-8). This is due to the fact that in this case, the droplet doesn't fall, so the convective flow ( fig. 9) carries the heated liquid from bottom to top. This is clearly seen in fig. 10.  It should also be noted that the maximum temperature doesn't exceed 75 degrees and doesn't change with time due to temperature stabilization.

Conclusions
The numerical results show, that the singularity heating of the droplet and droplet movement in the liquid are different at various temperature coefficients. In case of decrease of the temperature coefficient one can see a more intense heating of the surrounding liquid with motionless droplet. When Ɣ=1.5 1/K the droplet up down with established velocity V=0.00271 m/sec. The influence of the temperature coefficient is caused by the fact that the increasing of the temperature coefficient leads to more intense heating the medium, reducing its viscosity, as a result of which droplets fall downwards under the action of gravity. Streamlines in both cases are the same, but fluid moves clockwise to the right and counter-clockwise to the left.