This model demonstrates how to obtain the temperature distribution in a simplified tabletop lab mixer using the Rotating Machinery, Nonisothermal Flow branch in the Mixer Module. The key instructive element is a demonstration of the Frozen Rotor method, which substantially reduces the computational time for a mixing study.

The model geometry is shown in Figure 1-3. It represents a cross-section of a tabletop lab mixer. Small mixers are not always baffled. Instead, one or several rods are inserted through the mixer lid and function as baffles. The rods typically contain equipment to measure, for example, the temperature or pH level. In this model, the mixer is baffled by a simplified immersion heater, which has a constant surface temperature of 60°C.

The mixer tank is filled with water which is agitated by the impeller rotating clockwise (as indicated in Figure 1-3) with 20 revolutions per minute (rpm). In this model the thickness of the impeller blades is significantly smaller than the diameter of the tank, and the impeller blades are therefore modeled as infinitely thin.

where N is the number of rotations per second, Da the impeller diameter, and ν the kinematic viscosity. A high Reynolds number means that the flow has a tendency to become turbulent. Evaluating Equation 1-1 using ν at 60 °C gives Re = 6944. This Reynolds number indicates that the flow is at least partly turbulent. For simplicity in this instructional model, the flow is assumed to be two-dimensional and no turbulence model is used. Possible extensions of the model includes to resolve it using the full three dimensional geometry, and also to apply a turbulence model to investigate the effect of turbulent structures occurring in the flow.

A computationally more efficient method is to first simulate the flow using the frozen-rotor approach. The frozen-rotor approach is a modeling concept that treats the rotor as fixed, or frozen in space. The flow in the rotating domain is assumed to be stationary in terms of a rotating coordinate system. The effect of the rotation is then accounted for by Coriolis and centrifugal forces. The flow in the nonrotating parts is also assumed to be stationary, but in a nonrotating coordinate system (see Frozen Rotor in the CFD Module User’s Guide for more information). The result of a frozen-rotor simulation is an approximation to the flow induced by the impeller. The result depends on the angular position of the impeller and cannot represent transient effects. However, it is still a very good starting point to reach operating conditions.

Figure 1-4 shows the velocity distribution obtained from the frozen-rotor simulation. As expected, the highest velocity magnitude is found at the tip of the mixer blades. Three recirculation zones can be identified: one downstream of the immersion heater, one along the top wall, and one along the bottom wall.

Figure 1-5 shows the temperature distribution obtained from the frozen-rotor simulation. Streamlines are also included to visualize the flow field. The temperature is relatively homogeneous throughout the mixer. There are some cold spots in connection to the recirculation zones adjacent to the outer wall. This is expected because the fluid there has a longer residence time close to the solid wall, and therefore has less contact with the heated fluid closer to the center of the mixer.

The progress of a solution can be monitored using probes (see Probes in the COMSOL Multiphysics Reference Manual). The velocity magnitude and temperature are probed at (x,y) = (−0.05,0.065). The location is indicated in Figure 1-5, just outside the recirculation zone along the top wall.

The probe plots produced during the time-dependent simulation are shown in Figure 1-6. The velocity probe plot shows that the flow pattern, after an initial transient, oscillates around the frozen-rotor result with an amplitude of about 10%. The deviations in temperature are much smaller.

The velocity probe plot exhibits quasi-periodic structure. A discrete Fourier analysis produces a peak at a frequency 1.33 Hz which corresponds to the passing of the blades. Several other frequencies can be identified, the most pronounced peaks are at 0.125 Hz, 0.25 Hz, 0.42 Hz and 1.07 Hz. Evaluating the Strouhal frequency for the immersion heater (for a typical value of the Strouhal number, St = 0.2) gives a value of about 1.48 Hz. Hence, oscillations in the mixer, which include the intermittent boundary-layer separation, cannot be completely explained as being driven by the frequency of the rotating blades and the main frequency of the Kármán vortex street behind the heater. Probably, some more intricate mechanisms are involved. The temperature variations are within the tolerance set in the default solver sequence.

A more complete picture of the progress from the frozen-rotor solution toward the operating conditions can be seen through an animation. Figure 1-7 shows four snapshots from such an animation. The time runs from top left to lower-right. The most notable changes occur in the recirculation zones. The recirculation zone behind the immersion heater has two vortices which appear to be oscillating. This has no effect on the local temperature, but it significantly influences the size of the recirculation zones adjacent to the outer wall.

Looking at Figure 1-5, it can be seen that the recirculation zone along the top wall contains a single, large vortex. As the simulation progresses (t = 20 s to t = 40 s), the size and strength and position of the vortices along the top wall varies as a result of the interaction between the disturbance, produced by the immersion heater, and the outer wall.

In the Home toolbar, click Add Material  to open the Add Material window.

There are many ways to select geometric entities. When you know the domain to add, such as in this exercise, you can click the Paste Selection button  and enter the information in the Selection text field. In this example, enter 1 in the Paste Selection window. For more information about selecting geometric entities in the Graphics window, see the COMSOL Multiphysics Reference Manual.

In the Model Builder window, under Component 1 (comp1) click Heat Transfer in Fluids (ht).

To avoid unnecessarily small elements in the mixer vessel wall, add a separate Size node with reduced resolution in narrow regions.

In the Home toolbar, click Compute .

Re-create Figure 1-4 using the following steps.

Figure 1-5 can be created by the following steps.

Add a Time Dependent study in order to perform a transient simulation, using the previous solution as initial condition.

The following steps create an animation that contains the plots in Figure 1-7.