COMSOL 6.4 - Flow in an Internal Mixer

When mixing highly viscous non-Newtonian polymers and rubbers, it is important to keep track of both the temperature and the torque that is applied to the rotors in the vessel. The process must be tuned so that the temperature during mixing is below the disassociation temperature, but high enough so that mixing can occur. In this example, a HDPE polymer is stirred in an internal mixer with nonisothermal flow and a non-Newtonian Carreau model for the apparent viscosity of the fluid. Thermal effects are included in the form of an exponential law.

Model Definition

The geometry is a chamber with two counter-rotating impellers as shown in Figure 1.

Figure 1: Mixer with two counter-rotating impellers.

The viscous fluid is a HDPE (high density polyethylene) with a non-Newtonian Carreau model for the apparent viscosity. The equation for the shear, and temperature, dependent viscosity is

(1)

where

is the thermal contribution to the viscosity defined by thermal coefficients b and T0. Moreover μ∞, μ0, λ, and n are model parameters while

is the strain rate in the fluid. The outside of the cavity has a convective heat flux condition, while the other boundaries are thermally insulated. Viscous heating is a major contribution for such viscous fluids and is included as a source in the heat transfer equation as Qvd = τ : ∇u, where τ is the viscous tensor and u is the velocity vector.

The model is solved in two different studies. The first part uses a frozen-rotor approach where the rotation of the impellers is modeled in a stationary manner. This gives a stationary solution for the temperature, velocity and pressure. In order to analyze the start up and mixing behavior, a time dependent study is included. Transient results for torque on the rotors and the temperature at specific locations in the mixing cavity are presented.

Results and Discussion

The velocity together with velocity arrows obtained from the Frozen Rotor study can be seen in Figure 2.

Figure 2: Velocity and velocity arrows for the Frozen Rotor study.

Figure 3 shows the pressure.

Figure 3: Pressure distribution for the Frozen Rotor study.

Figure 4 shows the shear rate.

Figure 4: Shear rate for the Frozen Rotor study.

Figure 5 shows the viscosity and Figure 6 shows the temperature for the Frozen Rotor study.

Figure 5: Dynamic viscosity for the Frozen Rotor study.

Figure 6: Temperature and velocity arrows for the Frozen Rotor study.

For the time-dependent study, it is interesting to investigate the torque from the rotors onto the fluid (which is the reciprocal of the torque on the rotors). The rotors have a forced rotation velocity which increases slowly from 0 to 20 RPM during the first second. The torque as a function of time from the two rotors is shown in Figure 7. Moreover, the temperature after two seconds is shown in Figure 8 and the temperature as a function of time at a couple of points located at the center of the cavity is shown in Figure 9. Note that a stationary result is not achieved for the transient simulations as we have focused on investigating only the first couple of revolutions.

Figure 7: Impeller torque as a function of time during startup.

Figure 8: Temperature after two seconds.

Figure 9: Temperature at a couple of points in the middle of the cavity as a function of time.

Polymer_Flow_Module/Tutorials/internal_mixer

Notes About the COMSOL Implementation

The geometry is made as an assembly to make it easier to go from Frozen Rotor to a Time Dependent study. Note that to run the frozen rotor simulation, the rotational velocity must be set to 20 RPM.

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > > > .

Click .

Click

In the tree, select > .

Click

Geometry 1

Disable the analysis of the geometry as the remaining small geometric details can be kept.

In the window, under click .

In the window for , locate the section.

Clear the checkbox.

Import the geometry from a file.

Import 1 (imp1)

In the toolbar, click

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file internal_mixer.mphbin.

Click

Add a centroid measurement node to find the rotation axis of the two rotors. The geometry parameters are later used in the definition of the rotating domains.

Centroid Measurement 1 (cm1)

In the toolbar, click

and choose .

On the object , select Points 4, 6, 10, and 12 only.

In the window for , click

Centroid Measurement 2 (cm2)

Right-click and choose .

In the window for , locate the section.

Click the

button for .

On the object , select Points 22, 24, 28, and 30 only.

Click

Form Union (fin)

In the window, under > click .

In the window for , locate the section.

From the list, choose .

Click

Create a selection for the boundaries of the rotors that you can use later when postprocessing.

Definitions

Rotors

In the window, expand the > node.

Right-click and choose > .

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

Locate the section. From the list, choose .

Select the checkbox.

In the text field, type 50.

Select Boundaries 19–35, 39–59, 61–63, 65–86, 91–114, and 116–118 only.

Moving Mesh

Rotating Domain 1

In the window, under > click .

Select Domain 2 only.

In the window for , locate the section.

In the f text field, type -1/3.

Locate the section. Specify the rax vector as


geom1.cm1.x	X
geom1.cm1.y	Y
geom1.cm1.z	Z

Rotating Domain 2

Right-click > > and choose .

Select Domain 3 only.

In the window for , locate the section.

In the f text field, type 1/3.

Locate the section. Specify the rax vector as


geom1.cm2.x	X
geom1.cm2.y	Y
geom1.cm2.z	Z

Laminar Flow (spf)

Fluid Properties 1

In the window, under > click .

In the window for , locate the section.

Find the subsection. From the list, choose .

From the list, choose .

Materials

HDPE

In the window, under right-click and choose .

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

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


Property	Variable	Value	Unit	Property group
Zero shear rate viscosity	mu0	1130	Pa·s	Carreau model
Infinite shear rate viscosity	mu_inf	50	Pa·s	Carreau model
Relaxation time	lam_car	0.95	s	Carreau model
Power index	n_car	0.85	1	Carreau model
Heat capacity at constant pressure	Cp	250	J/(kg·K)	Basic
Thermal conductivity	k_iso ; kii = k_iso, kij = 0	0.2	W/(m·K)	Basic
Density	rho	760	kg/m³	Basic

Laminar Flow (spf)

Fluid Properties 1

In the window, under > click .

In the window for , click to expand the section.

From the list, choose .

In the T0 text field, type 453.

In the b text field, type 0.02.

Pressure Point Constraint 1

In the toolbar, click

and choose .

Select Point 17 only.

Heat Transfer in Fluids (ht)

Initial Values 1

In the window, under > click .

In the window for , locate the section.

In the T text field, type 313.

Heat Flux 1

In the toolbar, click

and choose .

Add a convective heat flux to all boundaries.

Select Boundary 8 only.

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the h text field, type 40.

In the Text text field, type 313.

Now apply "Thermal insulation" boundary condition to the boundaries of the rotors. This overrides the previous boundary condition on these boundaries.

Thermal Insulation 2

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Mesh 1

In the window, under click .

In the window for , locate the section.

In the table, select the checkbox for .

From the list, choose .

Locate the section. From the list, choose .

Boundary Layer Properties 1

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type 5.

In the text field, type 1.15.

In the text field, type 1.5.

Click

Click the

button in the toolbar.

Study 1

In the toolbar, click

Results

Multislice 1

In the window, expand the node, then click .

In the window for , locate the section.

Find the subsection. In the text field, type 0.

Find the subsection. In the text field, type 3.

Locate the section. From the list, choose .

In the toolbar, click

Arrow Volume 1

In the window, right-click and choose .

In the window for , locate the section.

Find the subsection. In the text field, type 60.

Find the subsection. In the text field, type 40.

Find the subsection. In the text field, type 3.

In the toolbar, click

Viscosity

In the toolbar, click

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

Surface 1

Right-click and choose .

In the window for , locate the section.

In the text field, type 1.

Material Appearance 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Selection 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Slice 1

In the window, right-click and choose .

In the window for , locate the section.

In the text field, type spf.mu.

Locate the section. From the list, choose .

In the text field, type 3.

Locate the section. From the list, choose .

From the list, choose .

Viscosity

In the window, click .

In the window for , locate the section.

Clear the checkbox.

In the toolbar, click

Shear Rate

Right-click and choose .

In the window for , type Shear Rate in the text field.

Slice 1

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type spf.sr.

In the toolbar, click

Temperature

In the window, right-click and choose .

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

Multislice 1

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type T.

Arrow Volume 1

In the window, click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

Define a step function so that you can specify a smooth startup of the rotation velocity for the rotors.

Definitions

Step 1 (step1)

In the toolbar, click

and choose .

In the window for , click to expand the section.

In the text field, type 0.5.

Click

Locate the section. In the text field, type .5.

Locate the section. In the text field, type 1.

Click

Moving Mesh

Rotating Domain 1

In the window, under > click .

In the window for , locate the section.

In the f text field, type -1/3*step1(t).

Rotating Domain 2

In the window, click .

In the window for , locate the section.

In the f text field, type 1/3*step1(t).

Add Study

In the toolbar, click

and choose .

Go to the window.

Find the subsection. In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Study 2

Step 1: Time Dependent

In the window for , locate the section.

In the text field, type range(0,0.1,2).

Laminar Flow (spf)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Click the

button in the toolbar.

In the dialog, select > in the tree.

In the tree, select the checkbox for the node > .

Click .

In the window for , click to expand the section.

Find the subsection. Select the checkbox.

Study 2

In the window, click .

In the window for , locate the section.

Clear the checkbox to disable the generation of default plots for this study. Instead, the plots from the first study will be copied and adapted for this study.

Solution 2 (sol2)

In the toolbar, click