COMSOL 6.4 - Mixing of Water in a Flat Bottom Mixer

This model simulates the flow in a flat bottom mixer agitated by a pitched four blade impeller. The mixed fluid is water, and the flow is assumed turbulent due to the significant mixer Reynolds number (5.33·104). The flow is modeled using the k-ε turbulence model, and a time-dependent simulation corresponding to 34 revolutions of the impeller is performed in order to reach the operating conditions of the mixer.

When postprocessing the results, the self-similarity of the axial flow along the baffles is analyzed. In agreement with Ref. 1, the normalized velocity profiles at different axial positions are found to be self-similar indicating that the flow in this region resembles a three dimensional wall jet.

Model Definition

The mixer simulated consists of a baffled flat-bottom vessel and a pitched four-blade impeller. The mixer geometry and the simulation conditions used in this model correspond to the ones used in the experiments of Bittorf and Kresta (Ref. 1). The height (H) and diameter (T) of the mixer vessel is 0.24 m. The diameter of the impeller, Da, is 0.33T, and the clearance, C, between the impeller and the bottom is 0.40Da. The pitch of the impeller blades is 45°. The geometry of the mixer simulated is shown in Figure 1. The fluid contained in the mixer is water, which is mixed using an impeller rotational frequency of N = 8.58 turns per second. The resulting impeller Reynolds number is:

(1)

where μ and ρ are the dynamic viscosity (SI unit: kg/(m·s)) and density (SI unit: kg/m3) of water, respectively. The impeller Reynolds number is significantly higher than one, and the resulting flow will be treated as turbulent. The turbulent flow is modeled using the k-ε turbulence model, and the time-dependent problem is solved for 34 revolutions of the impeller.

Figure 1: Mixer geometry showing the vessel, baffles, and impeller.

Wall Jet Flow

Bittorf and Kresta (Ref. 1) compared the flow at the wall of a stirred tank to that of a turbulent wall jet. They found that the flow along the baffles of the stirred tank compares well to a three-dimensional wall jet. In order to assess whether the flow in this region of the mixer has the characteristics of a wall jet, the self-similarity of the axial velocity profiles can be examined. Self-similarity implies that the mean velocity profiles can be collapsed onto a single profile when scaled with properly chosen length and velocity scales. The existence of self-similarity indicates that the turbulent time scale is small enough for the turbulence to adjust locally to the development of the jet.

For wall jets, the spreading is typically found to be characterized by the maximum jet velocity, Um, and the distance from the wall to the position where the local velocity is equal to half of the maximum velocity value. The latter is often referred to as the jet half width and denoted by y1/2. Using the velocity data in Ref. 1, a similarity profile for a wall jet in a recirculating flow was suggested by Bhattacharya and Kresta in Ref. 2:

(2)

Here η denotes the distance from the wall normalized by the half width, η = y/y1/2. When analyzing the results, the self-similarity will be assessed by plotting the normalized axial velocity along the line shown in Figure 2. This corresponds to the position used in Ref. 1.

Figure 2: Measurement position for analyzing the self-similarity of the axial velocity profiles.

Solution Procedure

In order to minimize the time required to reach a fully developed, yet transient, flow state, the model is solved in two steps. First, a Frozen Rotor study is used to reach a good initial solution without having to solve the startup of the problem. In order to converge this step, a parametric sweep is used to first solve the model with a low Reynolds number and increased dynamic viscosity, and then compute it again with the actual dynamic viscosity of the fluid.

The frozen rotor solution is then used as initial condition for the transient simulation. During the transient simulation the model is run for 4.0 s corresponding to 34 full revolutions of the impeller.

Time averaging

An Identity Mapping nonlocal coupling is used when computing time-averaged velocity profiles. This is done to make sure that the average is computed along the measurement positions shown Figure 2, regardless of the position of the rotating impeller domain. The averages are computed using the time average operator in the manner of

timeavg(3.5,4,idmap1(w),'nointerp')

where the axial velocity is averaged between t = 3.5 and 4 s. The nointerp flag is used to compute the average without applying interpolation between the stored time steps.

Results and Discussion

The resulting flow field in the mixer after 4.0 s of transient simulation is shown in Figure 3. The streamlines show how the impeller pumps fluid downward along the shaft and ejects it toward the bottom of the vessel. The maximum velocity magnitude occurs at the tip of the impeller blades.

The velocity magnitude in a single cross section including the velocity vectors is shown in Figure 4. From this figure the main flow structure of the mixer can be further examined. The fluid flows with high speed along the bottom of the mixer. Upon reaching the wall, vertical high speed streaks form along the outer walls, and the fluid velocity decreases toward the top of the vessel. Fluid at the top of the vessel travel toward the impeller shaft prior to becoming pumped down toward the impeller.

Figure 3: Fluid flow at t = 4.0 s. The color of the slice plots and the width of the streamlines indicate the velocity magnitude.

Figure 4: Fluid flow magnitude and direction, at t = 4.0 s, in a cross section of the mixer.

The characteristics of the upward flow along the outer walls is studied in Figure 5 to analyze the wall jet. The averaged axial velocity, measured along the line shown in Figure 2, is plotted at different heights from the vessel bottom. The lines correspond to z/T equal to 0.458, 0.563, 0.667, and 0.771. The fluid velocity profiles are scaled by their individual maximum velocity, and the distance from the wall is normalized by the jet half width as defined in the section Wall Jet Flow. The similarity solution of Bhattacharya and Kresta (Equation 2) is also plotted.

The scaled velocity profiles are found to collapse on top of the similarity solution, at least for the first three positions from the bottom. The collapse occurs for η < 1.5; further out from the wall, the velocity is directed in the opposite direction, corresponding to the downward flow along the impeller shaft as seen in Figure 3 and Figure 4. The collapse in the region with upward flow indicates that the flow along the outer walls does indeed correspond to a wall jet flow. The results in Figure 5 compare well with the experimental results in Ref. 1.

Figure 5: Normalized axial velocity profiles measured at increasing mixer heights.

References

1. K.J. Bittorf and S.M. Kresta, “Three-dimensional wall jets: axial flow in a stirred tank,” AIChE Journal, vol. 47, no. 6, pp. 1277–1284, 2001.

2. S. Bhattacharya and S.M. Kresta, “CFD Simulations of Three-dimensional Wall Jets in a Stirred Tank,” Can. J. Chem. Eng., vol. 80, no. 4, pp. 1–15, 2002.

Mixer_Module/Benchmarks/wall_jet_4PB_mixer_flat

Modeling Instructions

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Geometry 1

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Browse to the model’s Application Libraries folder and double-click the file wall_jet_4PB_mixer_flat_geom_sequence.mph.

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Global Definitions

Parameters 1

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In the table, enter the following settings:


Name	Expression	Value	Description
N0	8.58[Hz]	8.58 Hz	Rotational speed of impeller
visc_fact	1	1	Viscosity factor

Moving Mesh

Rotating Domain 1

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Definitions

Identity Mapping 1 (idmap1)

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Materials

Water, liquid (mat1)

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Property	Variable	Value	Unit	Property group
Dynamic viscosity	mu	eta(T[1/K])[Pas]visc_fact	Pa·s	Basic

Turbulent Flow, k-ε (spf)

Pressure Point Constraint 1

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Symmetry 1

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Interior Wall 1

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Interior Wall 2

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Free Tetrahedral 1

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Study 1

Step 1: Frozen Rotor

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Parameter name	Parameter value list	Parameter unit
visc_fact (Viscosity factor)	10 1

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Definitions

Identity Boundary Pair 1 (ap1)

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Study 2

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Results

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The following instructions recreate Figure 3. Start by creating a selection for all walls in the node. The geometry sequence already contains some selections that you can use.

Geometry 1

Union Selection 1 (unisel1)

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