COMSOL 6.4 - Separation Through Electrocoalescence

A mixture of immiscible liquids is known as an emulsion. Many chemical processes result in emulsions consisting of the desired product and a solvent. Droplets of most emulsions will coalesce given enough time, but it is often desirable to speed up the separation process.

If the liquids have different electric permittivities, an electric field can be applied across the emulsion to stimulate coalescence. This method, known as electrocoalescence, has important applications, for instance, in the separation of oil from water.

To model electrocoalescence, you need to solve the Navier–Stokes equations, describing the fluid motion, and track the interfaces between the immiscible fluids. In order to include the electric forces acting on the fluids, you also have to solve for the local electric field. This complex multiphysics process can readily be set up and solved with COMSOL Multiphysics.

Model Definition

Geometry

Two droplets of water with radii of 1.6 and 1.2 mm, respectively, are transported in an oil phase flowing between two parallel plates. The average velocity of the multiphase flow is 5 cm/s. An electric potential of 5 kV is applied over the plates.

Figure 1: Two water droplets are transported in an oil phase flowing between two plates.

The Two-Phase Flow, Phase Field Interface

The Laminar Two-Phase Flow, Phase Field interface sets up the equations for the fluid motion according to the Navier–Stokes equations:

where u denotes velocity (SI unit: m/s), ρ the density (SI unit: kg/m3), μ dynamic viscosity (SI unit: Pas), p pressure (SI unit: Pa), g gravity (SI unit: m/s2). Fst is the surface tension force (SI unit: N/m3), and F is any additional volume force (SI unit: N/m3).

To track the fluid interface, the Laminar Two-Phase Flow, Phase Field interface uses a phase-field method:

The phase-field variable ϕ is −1 in water and 1 in oil. The density and viscosity, which is different for oil and water, is automatically calculated from the phase-field variable ϕ, as well as the surface tension force. σ is the surface tension coefficient (SI unit: N/m), ε is a numerical parameter (m) that determines the thickness of the fluid interface, that is, the region where the phase-field variable varies smoothly from −1 to 1. χ controls the mobility of the interface.

The Electrostatics interface

The Electrostatics interface sets up the following equations for the electric potential V:

Here, ε0 is the permittivity of vacuum, and εr is the relative permittivity.

The Coupling of the Two Physics

The software automatically sets up the equations described in the two previous sections. You only have to specify how they are coupled. For the Two Phase Flow interface, you need to specify the electric force. The electric force is given by the divergence of the Maxwell stress tensor:

(1)

The Maxwell stress tensor is given by

where E is the electric field and D is the electric displacement field:

The present example is in 2D, so the stress tensor is

The components of the electric field are calculated by the Electrostatics interface. Their predefined variable names, along with the variable names of the permeabilities can be used directly to set up expressions calculating the component of the stress tensor.

Figure 2: Use the Variables feature to define expressions. Predefined variables and operators can be typed in directly.

The components of the volume force are given by Equation 1. Once again, these can also be entered directly as expressions in the graphical user interface. Table 1 shows the syntax for the partial derivatives of the stress tensor that express the volume force in the x and y directions.

Table 1: User Defined Variables.
Name	Expression
Tem11	-epsilon0_constes.epsilonr_iso/2(es.Ex^2+es.Ey^2)+epsilon0_constes.epsilonr_isoes.Ex^2
Tem22	-epsilon0_constes.epsilonr_iso/2(es.Ex^2+es.Ey^2)+epsilon0_constes.epsilonr_isoes.Ey^2
Tem12	epsilon0_constes.epsilonr_isoes.Ex*es.Ey
Tem21	epsilon0_constes.epsilonr_isoes.Ex*es.Ey
Fx	d(Tem11,x)+d(Tem12,y)
Fy	d(Tem21,x)+d(Tem22,y)

Finally, you also need to specify the relative permittivity, which is constant, but different, for each fluid. Define it from the internally defined volume fractions of each fluid, Vf1 and Vf2:

Here, εr1 and εr2 denote the relative permittivity of oil and water, respectively. Instead of using a user-defined variable, this model uses the Multiphase Material to define the effective material properties. This way, the variable es.epsilonr_iso in the electrostatics interface is already the correct averaged material property.

Results and Discussion

Figure 3 shows snapshots of the velocity and water droplets at 0.05 s intervals. Contour lines of the electric potential show a dynamic behavior, clearly illustrating the bidirectional coupling in this multiphysics problem. The influence of the electric field causes the water droplets to stretch to the point where they come into contact. At this point, surface tension causes the droplets to coalesce. The surface tension forces counteract the electric forces stretching the newly formed droplet.

Figure 3: Water droplets, velocity, and the contour lines of the electric potential at 0, 0.05, 0.1, 0.015, 0.2, 0.25, 0.3 seconds.

Microfluidics_Module/Two-Phase_Flow/electrocoalescence

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > > .

Click .

In the tree, select > > > .

Click .

Click

In the tree, select > .

Click

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 30.

In the text field, type 10.

Circle 1 (c1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 1.6.

Locate the section. In the text field, type 4.

In the text field, type 6.

Circle 2 (c2)

In the toolbar, click

In the window for , locate the section.

In the text field, type 1.2.

Locate the section. In the text field, type 7.

In the text field, type 3.5.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
perm_water	80	80	Permittivity, water
perm_oil	2.2	2.2	Permittivity, oil
u_in	50[mm/s]	0.05 m/s	Average inlet velocity
u_max	3/2*u_in	0.075 m/s	Approximated maximum velocity
sigma	0.031[N/m]	0.031 N/m	Surface tension coefficient
V0	5[kV]	5000 V	Applied voltage

Definitions

Variables 1

In the toolbar, click

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
Tem11	-epsilon0_constes.epsilonr_iso/2(es.Ex^2+es.Ey^2)+epsilon0_constes.epsilonr_isoes.Ex^2	Pa	Maxwell stress tensor, 11-component
Tem22	-epsilon0_constes.epsilonr_iso/2(es.Ex^2+es.Ey^2)+epsilon0_constes.epsilonr_isoes.Ey^2	Pa	Maxwell stress tensor, 22-component
Tem12	epsilon0_constes.epsilonr_isoes.Ex*es.Ey	Pa	Maxwell stress tensor, 12-component
Tem21	epsilon0_constes.epsilonr_isoes.Ex*es.Ey	Pa	Maxwell stress tensor, 21-component
Fx	d(Tem11,x)+d(Tem12,y)	N/m³	Force, x-component
Fy	d(Tem21,x)+d(Tem22,y)	N/m³	Force, y-component

Create a step function which will be used to ramp up the inlet velocity from zero to its full value over the initial 0.01 s.

Step 1 (step1)

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type 0.005.

Click to expand the section. In the text field, type 0.01.

Click

Now, create selections to be used when setting up the boundary conditions.

Outlet

In the toolbar, click

In the window for , type Outlet in the text field.

Locate the section. From the list, choose .

Select Boundary 4 only.

Inlet

In the toolbar, click

In the window for , type Inlet in the text field.

Locate the section. From the list, choose .

Select Boundary 1 only.

Oil/Water Interface

In the toolbar, click

In the window for , type Oil/Water Interface in the text field.

Locate the section. From the list, choose .

Select Boundaries 5–12 only.

Electrostatics (es)

Initial Values 1

In the window, under > click .

In the window for , locate the section.

In the V text field, type y*V0/10[mm].

Charge Conservation in Fluids 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Electric Potential 1

In the toolbar, click

and choose .

Select Boundary 3 only (top boundary).

In the window for , locate the section.

In the V0 text field, type V0.

Ground 1

In the toolbar, click

and choose .

Select Boundary 2 only (bottom boundary).

Multiphysics

Two-Phase Flow, Phase Field 1 (tpf1)

In the window, under > click .

In the window for , locate the section.

From the list, choose . In the σ text field, type sigma.

Locate the section. Click

Materials

Phase 1 (mpmat1.phase1)

In the window, under > > click .

In the window for , locate the section.

Click

. This button is found when expanding the options next to the list.

Add Material to Phase 1 (mpmat1.phase1)

Go to the window.

In the tree, select > > .

Click .

Global Definitions

Water (mat1)

In the window, under > click .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	perm_water	1	Basic

Materials

Phase 2 (mpmat1.phase2)

In the window, under > > click .

In the window for , locate the section.

Click

. This button is found when expanding the options next to the list.

Global Definitions

Oil

In the window, under > click .

In the window for , type Oil in the text field.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	perm_oil	1	Basic
Density	rho	884[kg/m^3]	kg/m³	Basic
Dynamic viscosity	mu	0.474[Pa*s]	Pa·s	Basic