COMSOL 6.4 - Tubular Centrifuge

A tubular centrifuge is a type of centrifugal separation device that is often used for separation of very fine solid particles from a liquid. It can be run in both a continuous and batch type configurations. The device usually consists of a cylindrical rotating bowl with a large aspect ratio where its length is often many times larger than the radius. The bowl has an inlet feed and an outlet.

The particle–liquid mixture enters the rotating bowl through the inlet feed. The large rotation speed of the bowl induces centrifugal forces on the particles which cause them to sediment near the inner walls of the bowl as long as the particle has a larger density compared to the fluid. The sedimentation rate depends on the particle densities and the rotation speeds. Larger rotation speeds enhance the sedimentation rate. Further, the particles with larger densities tend to sediment faster. This dependence on the particle densities can lead to a preferential sedimentation along the axial direction with the most dense particles sedimenting close to the inlet, while the less dense particles sediment at a larger axial distance from the inlet.

This model also demonstrates the process of restarting particle tracing simulations whereby the information of the particles from one study step is used as the initial conditions for a subsequent study step.

Model Definition

The geometry consists of a cylinder of radius 20 mm and a height of 225 mm. There is a circular inlet of radius 5 mm on end of the cylinder through which the particle-fluid mixture enters the cylinder. The mixture exits the cylinder through a circular outlet of radius 5 mm on the face opposite to the inlet. The geometry as presented in Figure 1 shows the cylinder and the outlet.

The particle-fluid mixture sample used in this model is water containing spherical particles of a fixed diameter of 20 μm. The sample contains three types of particles with densities of 1200, 1500, and 2200 kg/m3. The sample enters the inlet at a flow rate of 2.5 l/min. The fluid entering the cylinder faces a sudden expansion and the corresponding Reynolds number suggests that the flow is turbulent. This model solves the Reynolds-Averaged Navier–Stokes equations and uses the standard k-ω turbulence model to account for the turbulent fluid flow.

The cylinder rotates with an angular speed of 1000 rad/s. The effect of rotation on the fluid flow is modeled using a feature. It is assumed that fluid velocities are stationary and thus the study step is utilized to account for the rotational effects via the inclusion centrifugal and Coriolis forces. Similarly, the rotational effects on the particle motion are included via the feature which adds these fictitious forces to the particles. The feature is used to apply gravitational force in the −z direction. Drag forces are applied in entire domain using the feature. Inertial effects due to the virtual mass and pressure gradient contributions are also included in the drag force calculations. A study step is used to solve the particle trajectories for a total simulation time of 2 s.

Figure 1: Model geometry of a tubular centrifuge device.

Notes on COMSOL Implementation

The model is solved in two steps. First, the fluid velocity and pressure are solved using the Turbulent Flow, k-ω interface and study step. Then the particle trajectories are solved for using the Particle Tracing for Fluid Flow interface and study step.

A total of 3000 particles are released having a fixed diameter of 20 μm. The particle densities are equally distributed over 1200, 1500 and 2200 kg/m3. The particles are released along the inlet boundary with number density proportional to the fluid velocity along the axial direction. The wall condition is applied at the all the remaining boundaries to represent the sedimentation of the particles on the cylinder walls.

The rotational effects on the fluid and the particles can be accounted for by solving the equations of motion in either the lab-fixed (inertial) frame of reference or a (noninertial) reference frame attached to the cylinder. The analysis in the inertial reference frame would include solving the appropriate equations along with a moving mesh, which can be achieved via the feature. However, the moving mesh computations can be computationally expensive in 3D, especially at large rotational speeds such as those of interest in this example. This can be avoided in situations where the geometry is rotationally invariant about the rotation axis, and the fluid flow is expected to be stationary in the spatial frame. In such situations, the model analysis is best suited in the noninertial reference frame, whereby the rotational effects are included via the addition of fictitious forces such as the centrifugal and the Coriolis forces.

For the fluid flow, this is easily achieved using the study step. For the particle motion, the Particle Tracing Module contains the feature that transforms the coordinate system from the lab-fixed inertial frame to the rotating noninertial frame and includes the fictitious force contributions to the forces on the particles. Since the coordinate system is now attached to a rotating frame, the mesh movement has to be disabled. This is achieved by disabling the feature via the option in the study step used to compute the particle trajectories.

The particle trajectories are computed for a total simulation time of 2 seconds. This is split into two study steps, with the particle data from the final time step of the first study step being used as the initial conditions for the second study step. In order to accomplish this, it is essential that the checkbox is selected in the physics interface settings. The solutions from the two studies are then combined into one solution using the study step.

Note that the particle tracing for an overall simulation time of 2 seconds can also be achieved directly in one study step. This is split into two study steps in this model to demonstrate the technique of restarting the particle tracing simulations and combining the solutions from two studies.

Results and Discussion

Figure 2 shows the velocity field (z direction) and the streamlines. The streamlines are colored based on the z-component of the fluid velocity. As the flow is fully developed, the velocity at center of the inlet is higher. The sudden expansion in the geometry causes the fluid velocity in the axial direction to drop immediately after the inlet. The rotational effects are clearly evident in the streamlines.

Figure 2: Velocity field and streamlines for the fluid flow.

Figure 3 show the particle trajectories at the end of 2 s. The particles are colored by their densities. The fluid carries the particles toward the outlet, while the centrifugal forces push the particles toward the cylinder walls. Denser particles are carried by the water for shorter distances and end up sedimenting closer to the inlet, while the less dense particles are carried further along. A large portion of the particles with the lowest density are found to sediment close to the outlet.

This preferential sedimentation is visualized in Figure 4, where the distribution of particle densities as a function of their z-coordinate is plotted at the end of 2 s. Finally, Figure 5 shows the number of particles sedimented on the walls as a function of time for each particle type. This illustrates the dependence of the sedimentation rates on the particle density.

Figure 3: Particle trajectories colored by particle diameter.

Figure 4: 1D histogram showing the distribution of particle densities collected at the wall as a function of the z-coordinate.

Figure 5: Number of sedimented particles as a function of time.

Particle_Tracing_Module/Fluid_Flow/tubular_centrifuge

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > > > ω (spf).

Click .

In the tree, select > > > .

Click .

In the tree, select > > .

Click .

Click

In the tree, select > .

Click

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
omega	1000[rad/s]	1000 rad/s	Angular speed
L_tube	225[mm]	0.225 m	Tube length
R_tube	20[mm]	0.02 m	Tube radius
R_feed	5[mm]	0.005 m	Feed radius
Vol_f	2.5[l/min]	4.1667E-5 m³/s	Fluid volumetric flow rate
dia_part	20[um]	2E-5 m	Particle diameter
num_rel	1000	1000	Number of particles per type

Geometry 1

Cylinder 1 (cyl1)

In the toolbar, click

In the window for , locate the section.

In the text field, type R_tube.

In the text field, type L_tube.

Click

Work Plane 1 (wp1)

In the toolbar, click

Work Plane 1 (wp1) > Plane Geometry

In the window, click .

Work Plane 1 (wp1) > Circle 1 (c1)

In the toolbar, click

In the window for , locate the section.

In the text field, type R_feed.

Click

Work Plane 2 (wp2)

In the window, under > right-click and choose .

In the window for , locate the section.

In the text field, type L_tube.

Form Union (fin)

In the toolbar, click

Click the

button in the toolbar.

Definitions

Create explicit selections for the inlet, outlet and the walls of the cylinder.

Inlet

In the toolbar, click

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

Locate the section. From the list, choose .

Select Boundary 5 only.

Outlet

In the toolbar, click

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

Locate the section. From the list, choose .

Select Boundary 6 only.

Walls

In the toolbar, click

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

Locate the section. From the list, choose .

Select Boundaries 1–4, 7, and 8 only.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Now specify the settings for the feature which usually controls the moving mesh. Even though this model solves the governing equations (for both the fluid and the particles) in a noninertial reference frame and thus avoids the need for a moving mesh, this node is needed for the study step.

Moving Mesh

Rotating Domain 1

In the window for , locate the section.

From the list, choose .

In the ω text field, type omega.

Turbulent Flow, k-ω (spf)

In the window, under click ω.

In the window for ω, locate the section.

Select the checkbox.

Inlet 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Locate the section. Click the button.

In the V0 text field, type Vol_f.

Outlet 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. Select the checkbox.

Particle Tracing for Fluid Flow (fpt)

In the window, under click .

In the window for , locate the section.

Select the checkbox. This is necessary to use the particle trajectories from one study step as initial conditions for a subsequent study step.

Particle Properties 1

In the window, under > click .

In the window for , locate the section.

From the ρp list, choose . In the associated text field, type 1200[kg/m^3].

In the dp text field, type dia_part.

Particle Properties 2

In the toolbar, click

and choose .

In the window for , locate the section.

From the ρp list, choose . In the associated text field, type 1500[kg/m^3].

In the dp text field, type dia_part.

Particle Properties 3

Right-click and choose .

In the window for , locate the section.

In the ρp text field, type 2200[kg/m^3].

Rotating Frame 1

In the toolbar, click

and choose .

In the window for , locate the section.

In the Ω text field, type omega.

Drag Force 1

In the toolbar, click

and choose .

Select Domain 1 only.

In the window for , locate the section.

From the list, choose .

From the u list, choose .

Locate the section. Select the checkbox. This is important to capture the inertial effects of the particle.

Gravity Force 1

In the toolbar, click

and choose .

Select Domain 1 only.

Inlet 1

Release the particles such that number density of the particles is proportional to the axial component of the fluid density at the inlet. The fluid velocity at the inlet is used to initialize the particle velocities. Note that since the fluid velocity is defined in an inertial reference frame and the particle velocities are defined in the rotating (noninertial) reference frame, the moving frame velocity must be subtracted from the fluid velocity.

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the N text field, type num_rel.

In the ρ text field, type w.

Locate the section. From the u list, choose .

Click to expand the section. Select the checkbox.

Inlet 2

Right-click and choose .

In the window for , click to expand the section.

From the list, choose .

Inlet 3

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Outlet 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Particle Counter 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

From the list, choose .

Particle Counter 2

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Particle Counter 3

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Add a study step to follow the study step. This study step only solves for the Particle Tracing for Fluid Flow interface. Further, since the usage of the feature removes the need for a moving mesh, the feature must be disabled for this study step.

Study 1

Step 3: Time Dependent

In the toolbar, click