# How to Simulate Particle Tracing in a Laminar Static Mixer

October 1, 2013

Laminar static mixers are used for the accurate mixing of fluids (both liquid and gas). Unlike a mixer containing moving blades, a static mixer contains twisted stationary blades that are positioned at different angles throughout the cylindrical flow channel of the mixer. When a fluid is pumped through the channel, the alternating directions of the cross-sectional blades cause the fluid to become mixed as it passes along the length of the channel. This mixing technique allows for precise control over the amount of mixing that takes place throughout the process — even in the mixing of very small amounts of fluid. In a recent video, we demonstrated how you can use COMSOL Multiphysics along with the CFD Module and Particle Tracing Module to evaluate the performance of a laminar static mixer.

### Simulating CFD and Particle Tracing in a Laminar Static Mixer

In the instructional video below, the basic modeling steps for setting up and solving a laminar static mixer model are shown. In the video, you will learn how to model both the fluid flow through the mixer, as well as the particle trajectories through the mixing channel, and finally how to analyze the mixer’s performance.

In the video, you will see how to model the laminar static mixer in two steps. The first step involves the CFD Module in performing a stationary study to measure the velocity and pressure within the mixer. The second step leverages the Particle Tracing Module and a time-dependent study to track the movement of the particles. In the second step, you evaluate the transmission probability, or transmission coefficient, of the mixer. A transmission coefficient is the ratio of the number of particles that reach the outlet of the channel divided by the number of particles released into the channel, and it is important when evaluating the success of the mixer. In this simulation, 18% of the particles get stuck to the walls of the channel and remain stuck in the mixer. In order to visualize the way the particles mix and flow through the channel, a Poincaré map (shown below) can be used to see how the position of the particles change over time. In this model geometry, the particles are still not completely mixed after reaching the end of the channel, as pockets of red and blue color remain.

Poincaré maps of the particle trajectories at different points in time. The color indicates the particles’ initial positions.

If you want to learn how to set up such a map by defining the color of the particle in relation to their initial position within the channel, follow along with the video above.

### Video Transcription

In this example you will learn how to use COMSOL’s Particle Tracing Module, together with the CFD Module, to evaluate the mixing performance of a static mixer, comprised of a small cylindrical flow channel with stationary cross-sectional blades. When a fluid is pumped through the pipe, the mixing blades cut up the flow field, causing the particles traveling through the mixer to displace relative to each other, creating a mixing effect. This model studies the flow in a twisted-blade static mixer and helps analyze the mixing performance using Poincaré maps. We will solve the model in two stages. First, we will define a stationary study, to compute the fluid velocity and pressure. Second, we will define a time dependent study, to track the motion of the particles through the mixer.

Let’s begin by adding the physics. Select Fluid Flow, Single-Phase flow, and add Laminar Flow. Choose Stationary as your study type and click finish.

Import the geometry by right clicking on Geometry 1 and choosing import. Browse to the laminar_mixer_particle.mphbin using the file path seen above. Click Import to view the geometry. Right-click Global Definitions, select Parameters, and enter in the tube radius and mean velocity to be used later on. Define the flow material by adding an undefined Material. Type in 1000 for the density and 1×10-3 for the dynamic viscosity. This corresponds to the density and viscosity of the water.

Let’s define our flow parameters by adding an Inlet to the Laminar Flow node, and selecting boundary 23. Define the flow velocity in U0 as seen, necessary for a fully developed flow profile. Again right-click Laminar flow, add an Outlet, and select boundary 20. In order to ensure accurate particle motion, we will define a mesh that is fine on the mixing blades. Let’s do that by right-clicking on Mesh 1, choosing More Operations, and adding a Free Triangular mesh.

Click the Wireframe Rendering button to more easily see your boundary selections, then enter them using the Paste Selection tool. Right-click on Free Triangular 1 and select Size. Calibrate for Fluid dynamics and use an Extremely Fine mesh. In the Size node, choose Extremely fine again and customize the mesh to change the Resolution of Curvature to 0.15. Add another Free Triangular mesh and this time select the Inlet, boundary 23. From the Free Triangular 2 node, add a Size and Calibrate for Fluid dynamics, this time using an Extra fine mesh. Add a Free Tetrahedral mesh for the remaining geometry and Build All.

Right-click on Study 1 and Compute the model. Two default plots are computed. A velocity slice plot and a contour pressure plot. Now that we have computed the flow velocity in the mixer, we will use it to define the drag force on the particles.

Add the particle tracing interface by right clicking on Model 1 and choosing Add Physics. Select Fluid Flow, then add Particle Tracing for Fluid Flow. Choose the Time Dependent preset for this study, uncheck the Solve for laminar flow check box, and click Finish.

Right-click Particle Tracing for Fluid Flow, to add a Drag Force and add domain 1 to the selection. From the Velocity field list, choose Velocity Field and from the Dynamic Viscosity list choose Dynamic Viscosity. This means that the previously computed velocity field and previously specified fluid viscosity will be used to compute the drag force on the particles.

Define an Inlet for the particles and select boundary 23. From the Initial position list choose Density, then define the Number of particles per release as 3000, and assign the density to be proportional to the laminar flow velocity. This will release more particles where the velocity magnitude is higher and fewer where it is lower.

From the Initial Velocity field list, choose Velocity Field. Now, add an Outlet for the particles and select boundary 20. Click on Particle properties, and define the Particle diameter by changing the diameter to .5 micrometers. In order for COMSOL to use the previously computed velocity field when solving for the particle trajectories, click on Step 1 Time dependent and check the Values of variables not solved for check box. From the Method list choose Solution, and from the Study list choose Study 1 stationary. Click the Range button, and in the dialog box type in 0.2 as the step size and 5 as the end time then click replace.You are now ready to Compute study 2.

Create a new data set to evaluate the transmission probability of the mixer. Right-click on the Particle 1 solution and duplicate it, then right-click on Particle 2 and add a Selection. Choose boundary as the geometric entity level and select the outlet as the boundary. From the Derived values node, add a Global evaluation. For the data set, choose Particle 2 and choose Last as the time selection. Click Replace Expression, then go to Particle Tracing for Fluid Flow, Particle statistics, and select Transmission probability.

Finally, click the Evaluate button. Under the graphics window, in Table 1, you will see that the Transmission probability at the outlet at time 5 is about 80%, meaning that about 20% of the particles remained trapped in the mixer after 5 seconds. Right-click Data set and add a Cut plane. From the Data set list, choose Particle 1, and from the Plane list, choose xz-planes. Type in 0.006 as the y-coordinate and select the Additional parallel planes check box. In the Distances field type in the following distances, and click plot. In the Distances field set the planes to be at 6, 16, 26, 36, and 42 millimeters, and click plot. Now right-click on Results and add a 3D Plot group. Change the data set to Particle 1, and the Legend position to the Bottom and select the Titletype as None. Right-click on 3D Plot Group 4 and add a Poincaré Map.

Now, from the Cut Plane list choose Cut Plane 1, then select the Radius scale factor check box and type in 6×10-5. You can plot the map to preview the distribution. To change the color, right-click on Poincaré Map and choose Color Expression. Type in the following expression to make half the map blue, and half red, and clear the Color legend check box.

Now, right-click on 3D Plot Group 4 and add a Surface. From the Data set, list choose Cut Plane 1 and type 1 into the Expression field. From the Coloring list, choose Uniform and from the Color list, choose Gray.

To visualize the Poincaré Maps individually in a 2D graph, right-click on Results and add a 2D Plot group. Clear the Plot data set edges check box and from the Data set list, choose Particle 1. Right-click on 2D Plot Group 5 and under More plots, add a Phase Portrait.

For the x-axis, manually type in the expression for the x-axis particle position. For the y-axis, manually type in the expression for the y-axis particle position. In the Coloring and style section, select the Radius scale factor check box and type in 3×10-5. Right-click on Phase Portrait 1 and choose Color Expression. Disable the Color legend and in the Expression field type in the following, then click Plot.

To better visualize the mixing performance of this static mixer, right-click on Export and add a player, choose 2D Plot Group 5 as the subject, then click the play button. You can see that at the inlet, the maroon and blue particles are evenly distributed in half. As the particles travel through the twisted mixer, they become more uniformly spread throughout the map.