Lamella Mixer

At the macroscopic level, systems usually mix fluids using mechanical actuators or turbulent 3D flow. At the microscale level, however, neither of these approaches is practical or even possible. This model demonstrates the mixing of fluids using laminar-layered flow in a microfluidic mixer.

To characterize the fluid flow’s turbulent behavior scientists generally use the Reynolds number

where ρ is the fluid density, u is flow velocity, L is a characteristic length, and μ is the fluid’s dynamic viscosity. Turbulent flow takes place when the Reynolds number is high, typically when Re > 2000. At microfluidic scales, the width of a channel is in the range of 100 μm and the velocity is approximately 1 cm/s. In this case, for water-like substances Re is close to unity. The fluid flow is thus clearly laminar, so effective mixing of fluids in microfluidic devices requires other means.

Figure 1 shows a section of a component that uses layered flow to improve mixing. The mixer has several lamellae of microchannels, and the two fluids being mixed are alternated for every second layer. Pressure forces the fluid to travel in the channels from back to front. The fluid enters a larger space, the mixing chamber (visible at the front of the image). The figure does not include this chamber, but it covers the area beyond where the grid of the microchannel ends. Near the ends of the microchannels the mixing chamber has distinct lamellae of the two fluids, but this separation vanishes toward the end of the chamber.

Figure 1: Geometry of a lamella mixer (mixing chamber not visible).

Model Definition

This model analyzes the steady-state condition of the fluid flow as well as the convection and diffusion of a dissolved substance in a lamella mixer. The geometry in Figure 2 corresponds to Figure 1 except it includes only a small vertical section of the mixer with a height of 30 μm. The model starts from a plane in the middle of the channel bending to the left and ends at a plane in the middle of the channel bending to the right.

Each microchannel in the mixer has a quadratic cross section with a side of 20 μm. Because of the chosen geometry, microchannel height in the model is only 10 μm. The fluid exiting the microchannels enters a mixing chamber of length 200 μm and width 80 μm.

Figure 2: The model geometry for a lamella mixer takes advantage of symmetry so it is not necessary to model the entire height of the device.

You solve the fluid flow in the channels and in the chamber with the incompressible Navier-Stokes equations

(1)

where ρ is fluid density, u = (u, v, w) is the flow-velocity field, p is fluid pressure, I is the unit diagonal matrix, μ is the fluid’s dynamic viscosity, and F = (fx, fy, fz) is a volume force affecting the fluid. The fluid is water with ρ = 1000 kg/m3, μ = 0.001 Pa·s, and F = 0 because there are no volume forces.

The system applies a pressure of p0 = 10 Pa on all six microchannel inputs to drive the flow through the mixing chamber to where there is zero pressure. At the chamber exit the flow velocity has components only in the normal direction of the boundary.

On the microchannel and mixing-chamber walls, the no slip boundary condition applies. However, in the vertical direction, due to the geometry, you can use a symmetry boundary condition.

The following convection-diffusion equation describes the concentration of the dissolved substances in the fluid:

(2)

where c is the concentration, D is the diffusion coefficient, and R is the reaction rate. In this model, D = 10−10 m2/s, and R = 0 because the concentration is not affected by any reactions.

There is a concentration of c0 = 50 mol/m3 on the input boundaries of the channels curving to the left, whereas the channels curving to the right have zero concentration. At the output boundary of the mixing chamber the substance flows through the boundary by convection. The walls of the channels and the chamber are insulated for this dissolved substance, and on the top and bottom boundaries you use a symmetry boundary condition.

Results and Discussion

Figure 3 details fluid flow in the mixer. You can see a gradual change in the velocity magnitude across the slices, indicating a laminar parabolic flow. The streamlines do not show swirls, and there are only small changes in the flow direction. The figure also shows that the maximum flow velocity in the microchannels is at the model’s symmetry boundaries. Inside the mixing chamber, a flow velocity gradient is visible only along the x direction; the symmetry conditions on the top and bottom boundaries give a uniform velocity profile in the z direction.

The peak velocity is roughly 1.3 mm/s in the microchannels and 0.6 mm/s in the mixing chamber. Given the corresponding lengths (20 μm and 80 μm), the Reynolds numbers are Re = 0.026 and Re = 0.048, so the flow is clearly laminar.

Figure 3: Fluid flow in the lamella mixer.

Figure 4 shows the concentration distribution on the model boundaries. The inflow channels see a constant concentration of 0 or 50 mol/m3 depending on the channel. Mixing starts when the fluid enters the mixing chamber. At the entrance there is a clear separation of the concentration, but this diminishes toward the end of the chamber. On the sides of the mixing chamber where the flow velocity is smaller the mixing is better than at its center. The mixing, however, is not perfect, and a reduced flow velocity, a longer mixing chamber, or some other means to increase mixing is preferable.

Figure 4: Concentration plot on the boundaries of the lamella mixer model.

To get another point of view, examine Figure 5, which shows the concentration profile at the chamber’s centerline. Near the channels the transition is very rapid, but closer to the chamber’s end the profile has a flatter sigmoid shape. On the chamber’s sides the concentration profile has the same shape, but its amplitude is between approximately 17 mol/m3 and 32 mol/m3.

If you generalize the concentration profile to cover the entire component (Figure 1), the profile would be a wave-like curve where concentration would alternate between its minimum and maximum values with a spatial frequency related to the layer thickness.

Figure 5: Concentration profile along a line in the z direction in the middle of the mixing chamber at various distances from the microchannels: 20 μm (solid), 80 μm (dotted), 140 μm (dashed), and 200 μm (dash-dotted).

Notes About the COMSOL Implementation

Because the concentration does not affect the fluid flow, it is not necessary to solve the Laminar Flow and Transport of Diluted Species interfaces simultaneously. By solving them sequentially, starting with the Navier-Stokes equations, you improve the solution’s convergence and reduce the solution time.

Microfluidics_Module/Micromixers/lamella_mixer

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

For convenience, the device geometry is inserted from an existing file. You can read the instructions for creating the geometry in the Appendix — Geometry Instructions.

In the toolbar, click and choose .

Browse to the model’s Application Libraries folder and double-click the file lamella_mixer_geom_sequence.mph.

In the toolbar, click

The dimensions of this geometry are given in micrometers, so check the length unit and change it accordingly if necessary.

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
p0	10[Pa]	10 Pa	Driving pressure
c0	50[mol/m^3]	50 mol/m³	Input concentration
D_i	1e-10[m^2/s]	1E-10 m²/s	Isotropic diffusion coefficient

Next, add a material for the fluid.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Materials

Water, liquid (mat1)

You are now ready to set up the physics.

Laminar Flow (spf)

Inlet 1

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Locate the section. In the p0 text field, type p0.

Symmetry 1

In the toolbar, click

and choose .

Click the

button in the toolbar.

In the window for , locate the section.

From the list, choose .

Click the

button in the toolbar.

Outlet 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

There are two options available for the type of crosswind diffusion used to provide numerical stabilization. The default option, is more effective at suppressing undershoots and overshoots in the concentration, whereas the second option, produces less artificial diffusion in the crosswind direction.

Transport of Diluted Species (tds)

Click the

button in the toolbar.

In the dialog box, in the tree, select the check box for the node .

Click .

In the window, under click .

In the window for , click to expand the section.

From the list, choose .

Transport Properties 1

In the window, under click .

In the window for , locate the section.

In the Dc text field, type D_i.

Note that the velocity field for the species convection is controlled by the multiphysics coupling.

Concentration 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. Select the check box.

In the c0,c text field, type c0.

Concentration 2

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. Select the check box.

Leave the concentration at its default zero value.

Outflow 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Click the

button in the toolbar.

Mesh 1

Free Tetrahedral 1

In the toolbar, click

Size

In the window, click .

In the window for , locate the section.

From the list, choose .

Click

Study 1

In the toolbar, click

Results

The following instructions show how to reproduce the plots in the Results and Discussion section.

Slice

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

Streamline 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

In the text field, type 0.01.

In the text field, type 0.025.

In the toolbar, click

Compare the result with the plot in Figure 3.

Concentration, Surface (tds)

The second of the two default concentration plots shows the concentration on the modeling domain’s boundary; compare with Figure 4.

To reproduce Figure 5, showing the concentration as a function of z at four different y positions along the mixing chamber, begin by creating four datasets. The possibility to duplicate model-tree nodes simplifies the procedure considerably.

Cut Line 3D 1

In the toolbar, click