Tutorial: Modeling Bubble Breakup and Coalescence ...

Tutorial: Modeling Bubble Breakup and Coalescence in a Bubble Column Reactor

Introduction The purpose of this tutorial is to provide guidelines for solving the flow break-up, and coalescence of gas bubbles in a gas-liquid bubble column reactor using a population balance approach coupled with the Eulerian multiphase model in FLUENT 6.3. The population balance approach is used to solve for bubble flow and size distribution in an axisymmetric bubble column, for a population of six different bubble sizes. This tutorial demonstrates how to do the following: • Set up a two-phase, unsteady bubble column problem for an air-water bubble column using the Eulerian multiphase model. • Activate and setup a population balance model with six bubble sizes. • Solve the case using appropriate solver settings and solution monitors. • Postprocess the resulting data for bubble size distribution.

Prerequisites This tutorial assumes that you are familiar with the FLUENT interface, basic setup, solution procedures, and the use of the Eulerian multiphase mixture model. This tutorial does not cover the mechanics of using this model, but focuses on setting up the population balance problem for bubble size distribution and solving it. The population balance module is provided as an add-on module with the standard FLUENT licensed software. A special license is required to use the population balance module. If you have not used the Eulerian multiphase model before, refer to the FLUENT User’s Guide and the FLUENT Tutorial Guide. Also, refer the Population Balance Model Manual [2] for a comprehensive overview of the FLUENT population balance model and its application in solving multiphase flows involving a secondary phase with a size distribution.

Problem Description Figure 1 shows the schematic representation of the air-water bubble column of diameter of 0.29 m and height of 2 m. Air is injected into the water column through an inlet at the bottom, which has a diameter of 0.23 m, with a constant velocity of 0.02 m/s. The

c Fluent Inc. January 10, 2007

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Modeling Bubble Breakup and Coalescence in a Bubble Column Reactor

initial diameter of the injected air bubbles is 3 mm. You will model this column as a 2D, axisymmetric column. The injection of air causes the development of a turbulent flow pattern in the liquid column, which transports the bubbles throughout the column. Due to the effects of turbulence and collisions between individual bubbles, the bubbles breakup and coalesce with each other. As a result, bubbles with a range of sizes are formed in the bubble column. The size distribution of the bubbles, plays a critical role in any mass transfer and reactions that may occur between the air and the liquid, as in a Fischer-Tropsch synthesis process. Hence resolving the bubble size distribution is an important task in the CFD analysis of bubble column reactors. This can be accomplished using the population balance model in FLUENT.

Figure 1: Problem Schematic

Solution Strategy 1. In this tutorial, you will set up the two phase flow problem using the Eulerian mixture mutiphase model. The population balance model will be activated using TUI commands. The specialized panel for this model will be used to define the size distribution problem. you will select the discrete method with six size bins to represent the the bubble size distribution. The volume ratio will be set to 4 with a minimum size of 0.001911 m or 1.911 mm. The six size bins correspond to the bubble diameters 0.012, 0.00756, 0.004762, 0.003, 0.00189, and 0.001191 metres respectively. The size bins will be chosen such that the inlet bubble size of 3 mm, i.e. 0.003 m, lies in the middle of the bin sizes. You will also activate the aggregation and breakage kernels and choose the Luo model. The flow and population balance problem will be setup and solved in transient mode until an equilibrium solution is reached. Finally, you will use the postprocessing capabilities to analyze the flow and resulting size distribution.

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2. The population balance model is used for solving multiphase flow problems where the secondary phase has a size distribution such as droplets, bubbles or crystals, which evolves and changes with the flow due to phenomena like nucleation, growth, aggregation or coalescence, and breakage. The population balance model uses a balance equation, similar to the mass, energy and momentum balance, to track the changes in the size distribution. The size distribution can be determined using one of the three approaches: the discrete method, the standard method of moments and the quadrature method of moments. [2] In this tutorial, you will use the discrete method to compute the bubble size distribution. Here, the range of particle sizes in the particle size distribution is divided into a finite number of intervals or discrete “bins”. The bubble sizes chosen for the bins are required to be in geometric progression with the ratio of bubbles volumes of adjacent size bins, or volume ratio, set to an integer power of 2. Thus the bubble diameters are in geometric progression with a size ratio which is the cube root of an integer power of 2. A transport equation is solved for each bin with a corresponding scalar, which represents the volume fraction of gas in that bin. Thus, the sum of the scalars for all the discrete bins is equal to the gas phase volume fraction. Source terms in the transport equation account for the “birth” and “death” of bubbles in each size bin, when they enter or leave the bin due to breakup and coalescence. These terms are computed using specific models, or “kernels”, which are published in the scientific literature. In this tutorial, you will use the breakup and coalescence kernels for bubble columns developed by Luo et.al. [3] After the transport equations for the scalars have been solved, the value of the number density function for each size bin is calculated. This is simply the volume fraction of each bin i.e., the scalar value, divided by the volume of a single bubble, yielding the number of bubbles per unit volume or number density. The values of the number density function for all size bins gives the bubble size distribution. The transport equations from the population balance model and the momentum equations are coupled due to user-defined drag based on Sauter mean diameter computed from the obtained size distribution. Both the number density function and the Sauter diameter are available in FLUENT for postprocessing. Specialized postprocessing functions for the population balance model ˙ have been added to FLUENTYou will report and plot volume and surface averages of the size distribution. You will also compute the statistical moments of the size distribution, which represent aggregate quantities such as the total number of bubbles or the total bubble surface area per unit volume. Please refer to [2], [3], and [4] for details regarding the population balance model and its application to bubble column reactors.


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Preparation 1. Copy the file bubcol new2.msh.gz into your working folder. 2. Start the 2D double precision (2ddp) version of FLUENT.

Setup and Solution Step 1: Grid 1. Read the mesh file bubcol new2.msh. 2. Check the grid. Grid −→Check 3. Display the grid. Display −→Grid... 4. Rotate the grid display. Display −→Views... (a) Select axis from the Mirror Planes selection list to enable the symmetry. (b) Click Camera... to open the Camera Parameters panel. i. Drag the indicator of the dial with the left mouse button in the counterclockwise direction until the upright view is displayed (Figure 2).

Grid

FLUENT 6.3 (2d, dp, pbns, lam)

Figure 2: Grid Display ii. Click Apply and close the Camera Parameters panel.

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(c) Click Apply and close the Views panel. 5. Close the Grid Display panel. Step 2: Models 1. Define the solver parameters. Define −→ Models −→Solver... (a) Select Axisymmetric from the Space list. (b) Select Unsteady from the Time list. (c) Click OK to close the Solver panel. 2. Enable the Eulerian multiphase model. Define −→ Models −→Multiphase... (a) Select Eulerian from the Model list. (b) Click OK to close the Multiphase Model panel. 3. Enable turbulence model. Define −→ Model −→Viscous... (a) Select standard k-epsilon (2 eqn) from the Model list. (b) Select Mixture from the k-epsilon Multiphase Model list. (c) Click OK to close the Viscous Model panel.


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Step 3: Materials 1. Copy a new material from the materials database. Define −→Materials... (a) Click Fluent Database... to open the Fluent Database Materials panel. (b) Select water-liquid (h2o) from the Fluent Fluid Material list. (c) Click Copy and close the Fluent Database Materials and Materials panel. Step 4: Phases 1. Define new phases. Define −→Phases... (a) Set water-liquid as the primary-phase. (b) Set air as the secondary-phase. Step 5: Operating Conditions 1. Specify the following operating conditions Define −→Operating Conditions... (a) Enable Gravity and set the Gravitational Acceleration to a value of -9.81 m/s2 in the X direction. (b) Enable Specified Operating Density and specify a value of 1.225 for Operating Density (kg/m3). (c) Click OK to close the Operating Conditions panel. Step 6: Setup the Population Balance Model 1. Enable the population balance model. (a) Type the TUI command define models addon-module in the console. (b) Enter 5 for the module number to enable the Population Balance model. The GUI now changes and an item is added to the Models menu.

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2. Set the parameters for the population balance model. Define −→ Models −→Population Balance...

(a) Select Discrete from the Method list. (b) Select Geometric Ratio from the Definition list. (c) Select air from the Phase drop-down list. (d) Enter 6 for Number of Bins, 4 for Ratio Exponent and 0.001911 m for Min. (e) Click the Print Bins button to print the discrete bubble sizes for each bin. (f) Enable Aggregation Kernel and Breakage Kernel from the Phenomena group box. (g) Select luo-model from the Aggregation Kernel and Breakage Kernel drop-down lists. (h) Click OK to close the Population Balance Model panel. In the Phases panel for air, the Diameter property changes to Sauter-mean i.e. the Population Balance model is automatically set to calculate the Sauter diameter for the mean bubble size.


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Step 7: Boundary Conditions Define −→Boundary Conditions... 1. Set boundary conditions for inlet. (a) Select vinlet from the Zone selection list, and air from the Phase drop-down list and click Set..... i. Click the Mometum tab: A. Select Magnitude, Normal to Boundary from the Velocity Specification Method drop-down list. B. Enter 0.02 m/s for the Velocity Magnitude drop-down list. ii. Click the Multiphase tab. A. Enter 1 for Volume Fraction. B. Make sure that Specified Value is selected from the Boundary Condition drop-down lists for the population balance variables (air Bin-0 through air Bin-5). C. Enter 1 for Boundary Value for air Bin-3 and retain the default value of 0 for the other variables. D. Click OK to close the Velocity Inlet panel. 2. Set turbulence boundary conditions at the inlet. (a) Select vinlet from the Zone selection list, and mixture from the Phase drop-down list, and click Set.... i. Click the Momentum tab and select Intensity and Hydraulic Diameter from the Specification Method drop-down list. ii. Enter 5 % for Turbulence Intensity and 0.145 m for Hydraulic Diameter. iii. Click OK to close the Velocity Inlet panel. 3. Set the boundary conditions for the outlet. (a) Select outlet from the Zone selection list, and air from the Phase drop-down list, and click Set.... i. Click the Multiphase tab. A. Enter 1 for Backflow Volume Fraction. B. Set the value of air Bin-3 to 1 and retain 0 for the other variables in the Population Balance Boundary Value group box.

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4. Set turbulence boundary conditions at the outlet. (a) Select outlet from the Zone selection list, and mixture from the Phase drop-down list, and click Set.... i. Select Intensity and Hydraulic Diameter from the Turbulence Specification Method drop-down list. ii. Enter 5 % for Backflow Turbulence Intensity and 0.145 m for Backflow Hydraulic Diameter. iii. Click OK to close the Pressure Outlet panel. 5. Close the Boundary Conditions panel. Step 8: Set Solver Controls Solve −→ Controls −→Solution... 1. Set the Under-Relaxation Factors as follows: Under-Relaxation Factors Pressure Momentum Volume Fraction Turbulence Kinetic Energy Turbulence Dissipation Rate air Bin

Value 0.3 0.2 0.2 0.8 0.8 0.8

2. Select Phase Coupled SIMPLE from the Pressure-Velocity Coupling drop-down list. 3. Retain the default settings for the Discretization parameters. Step 9: Initialize the Solution and Apply a Patch 1. Mark the region for adaption. Adapt −→Region... (a) Select Inside from the Options list and Quad from the Shapes list. (b) Enter the values for the coordinates as shown in the table. Parameter X Min X Max Y Min Y Max

Value 1.8 2.0 0 0.145

(c) Click Mark to select the region for adaption. (d) Close the Region Adaption panel.


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2. Initialize the solution using the following settings. Solve −→ Initialize −→Initialize... (a) Enter 0.1 (m2/s2) for Turbulence Kinetic Energy and 0.25 (m2/s3) for Turbulence Dissipation Rate in the Initial Values group box. (b) Click Init and close the Solution Initialization panel. 3. Patch the selected region. Solve −→ Initialize −→Patch... (a) Select air from the Phase drop-down list. (b) Select Bin 3 from the Variable selection list and enter 1 for Value. (c) Select hexahedron-r0 for Registers to Patch. (d) Click Patch. (e) Select Volume Fraction from the Variable selection list and set Value to 1. (f) Click Patch and close the Patch panel. Step 10: Solution Monitors 1. Set residual monitors. Solve −→ Monitors −→Residual... (a) Enable Plot from the Options list. (b) Retain the default Convergence Criteria for all parameters. (c) Click OK to close the Residual Monitors panel. 2. Set surface monitors. Surface −→Point...

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(a) Enter x0 (m) = 1.5 and y0 (m) = 0. (b) Keep New Surface Name as pointF-5 and click Create. 3. Create surface monitors for three of the bubble sizes. Solve −→ Monitors −→Surface... (a) Increase the number of Surface Monitors to 3. (b) Enable Plot, Print, and Write for each monitor. (c) Select Time Step from the When drop-down list for each monitor. (d) Click Define... for monitor-1 and specify the following parameters. i. Select Vertex Average from the Report Type drop-down list. ii. Set Plot Window to 1. iii. Select Population Balance Variables... and Bin-0 fraction from the Report of drop-down list. iv. Select air from the Phase drop-down list. v. Select point-5 from the Surfaces selection list. vi. Click OK to close the Define Surface Monitor panel. (e) Click Define... for monitor-2 and specify the following parameters. i. Set Plot Window to 2. ii. Select Bin-3 fraction from the Report of drop-down list. iii. Click OK to close the Define Surface Monitor panel. (f) Click Define... for monitor-3 and specify the following parameters. i. Set Plot Window to 3. ii. Select Bin-5 fraction from the Report of drop-down list. iii. Click OK to close the Define Surface Monitor panel. (g) Click OK to close the Surface Monitors panel. 4. Save the initial case file (bubcol new2-initial.cas.gz). Step 11: Solution 1. Start the calculations using the following settings. Solve −→Iterate... (a) Enter 0.01 s for Time Step Size, 5000 for Number of Time Steps, and 100 for Max Iterations per Time Step. (b) Click Iterate to start the calculations (Figure 3). Note: This will take several hours to converge. 2. Save the case and data files (bubcol new2.cas.gz and bubcol new2.dat.gz).


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Residuals continuity u-water u-air v-water v-air k epsilon bin-0-fraction bin-1-fraction bin-2-fraction bin-3-fraction bin-4-fraction bin-5-fraction vf-air

1e-02

1e-03

1e-04

1e-05

1e-06 5500 6000 6500 7000 7500 8000 8500 9000 9500 1000010500

Iterations

Scaled Residuals (Time=5.0000e+01) FLUENT 6.3 (axi, dp, pbns, eulerian, ske, unsteady)

Figure 3: Scaled Residuals

3.00e-03 2.50e-03 2.00e-03

Vertex Average Bin-0 fraction

1.50e-03 1.00e-03 5.00e-04 0.00e+00 0

500 1e+031.5e+032e+032.5e+033e+033.5e+034e+034.5e+035e+03

Time Step

Convergence history of Bin-0 fraction on point-5 (in SI units) (Time=5.0000e+01) FLUENT 6.3 (axi, dp, pbns, eulerian, ske, unsteady)

Figure 4: Convergence History of air Bin-0

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6.00e-01 5.00e-01 4.00e-01


3.00e-01 2.00e-01 1.00e-01 0.00e+00 0

500 1e+031.5e+032e+032.5e+033e+033.5e+034e+034.5e+035e+03

Time Step



5.00e-05 4.50e-05 4.00e-05 3.50e-05 3.00e-05


2.50e-05 2.00e-05 1.50e-05 1.00e-05 5.00e-06 0.00e+00 0

500 1e+031.5e+032e+032.5e+033e+033.5e+034e+034.5e+035e+03

Time Step




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Step 12: Postprocessing the Results 1. Display filled contours of air volume fraction (Figure 7). Display −→Contours.. (a) Select Phases... and Volume Fraction from the Contours of drop-down list. (b) Select air from the Phase drop-down list, disable Auto Range from the Options list and enter 0 for Min and 0.1 for Max. (c) Click Display.

Figure 7: Contours of Volume Fraction of Air (magnified) The changes in phase from inlet to outlet, and areas with low volume fraction as well as dead zones can be observed. 2. Create a vector plot for water velocity and observe the recirculation patterns (Figure 8). Display −→Vectors... (a) Select Velocity and water from the Vectors of and Phase drop-down lists respectively. (b) Select Phases... and Volume fraction from the Color By drop-down lists. (c) Select air from the Phase drop-down list. (d) Click Display.

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Figure 8: Water Velocity Vector Colored by Volume Fraction of Air 3. Create a contour plot of population balance for air phase (Figure 9). (a) Select Population Balance Variables... and Bin-3 fraction from the Contours of drop-down lists. (b) Select air from the Phase drop-down list. (c) Enter 0 for Min and 1 for Max and click Display. 4. Create a contour plot of number density contours for air phase (Figure 10). (a) Select Population Balance Variables... and Number Density of Bin-3 fraction from the Contours of drop-down lists. (b) Select air from the Phase drop-down list and click Display.


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Figure 9: Contours of UDS (air Bin-3) for Air Phase

Figure 10: Contours of UDS (Number Density of Bin-3) for Air Phase

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5. Calculate the moments of the bubble size distribution for the fluid region and the outlet. Report −→ Population Balance −→Moments... (a) Increase Number Of Moments to 4. (b) Select fluid from the Cell Zones selection list and click Print. The values of the moments are printed in the FLUENT window as shown: > Population Balance Moments over Surface(s) (default-interior) Moment Number Moment ------------------------- -----------------------0 59336.999 1 640.80173 2 9.2473485 3 0.18953551

Population Balance Moments over Volume(s) (fluid) Moment Number Moment ------------------------- -----------------------0 59337.55 1 639.95687 2 9.2472871 3 0.19025727

6. Plot the volume averaged discrete number density function distribution for different bubble sizes for the fluid volume. Report −→ Population Balance −→Number Density... (a) Select Volume Average from the Report Type list. (b) Select Discrete Number Density from the Fields selection list. (c) Select Histogram from the Plot Type list. Histogram is enabled only after you select Discrete Number Density from the Fields selection list. (d) Select fluid from the Cell Zones selection list. (e) Click Print to print the values in the FLUENT console. (f) Click Plot to plot the histogram of the volume averaged number density distribution with bubble diameter (Figure 11). You can also try plotting the length and volume based number density distribution. 7. Create a surface x=1 with x-coordinate equal to 1. Surface −→Iso-Surface... (a) Select Grid... and X-Coordinate from the Surface of Constant drop-down list.


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Figure 11: Volume Averaged Number Density Distribution Histogram. (b) Enter 1 for Iso-Values. (c) Enter x=1 for New Surface Name. (d) Click Create. (e) Close the Iso-Surface panel. 8. Plot the surface averaged discrete number density function distribution for different bubble sizes for the surface at x=1. Report −→ Population Balance −→Number Density... (a) Select Surface Average from the Report Type list. (b) Select Discrete Number Density from the Fields selection list. (c) Select Histogram from the Plot Type list. (d) Select x=1 from the Surfaces selection list. (e) Click Print to print the values in the FLUENT console. (f) Click Plot to plot the histogram of the surface averaged number density distribution with bubble diameter (Figure 12).

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Figure 12: Surface Averaged Number Density Distribution Histogram. 9. Plot the distribution along the central axis of the bubble column for each scalar (Figure 13). Plot −→XY Plot... (a) Select Population Balance Variables... and Bin-3 from the Y Axis Function dropdown lists. (b) Select air from the Phase drop-down list. (c) Select axis from the Surfaces selection list. (d) Click Axes... to open the Axes - Solution XY Plot panel. i. Disable Auto Range from the Options list. ii. Enter 1.8 for Maximum, and click Apply. iii. Close the Axes-Solution XY Plot panel. (e) Click Plot to see the distribution. You can see the initial bubble size distribution.


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Figure 13: Distribution of Bubble Size Along the Axis for air Bin-3 The air Bin-3 (initial bubble size) decreases from inlet to outlet. (f) Close the Solution XY Plot panel. Note: Breakup and coalescence are irrelevant in the freeboard region, which does not contain water. 10. Create and plot a custom field function that calculates the fraction of air contained in a bubble size corresponding to Bin-3. Define −→Custom Field Functions...

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(a) Select Population Balance Variables... and Bin-3 fraction from the Field Functions drop-down lists. (b) Select air from the Phase drop-down list. (c) Click the Select button to include this variable. (d) Click the multiplication sign x. (e) Select Phases... and Volume fraction from the Field Functions drop-down list and click the Select button. (f) Enter discrete-size-3-fraction for New Function Name. (g) Click Define to create the function. (h) Close the Custom Field Function Calculator panel. 11. Plot the contours of the custom field function discrete-size-3-fraction (Figure 14). Display −→Contours... (a) Select Custom Field Functions... and discrete-size-3-fraction from th Contours of drop-down list. (b) Disable Auto Range and enter 0 for Min and 0.04 for Max. (c) Click Display.

Figure 14: Contours of Custom Field Function discrete-size-3-fraction (Magnified) (d) Close the Contours panel. 12. Plot contours of the distribution of the Sauter diameter (Figure 15). Display −→Contours... (a) Select Properties...from the Contours of drop-down list.


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Figure 15: Contours of Sauter Diameter

(b) Select air from the Phase drop-down list click Display. (c) Select Diameter from the Contours of drop-down list as the fluid property for plotting. The Diameter option is available only after selecting air. 13. Similarly, plot the histogram of the Sauter diameter distribution in the fluid volume (Figure 16). Plot −→Histogram...

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(a) Select Properties... from the Histogram of drop-down list. (b) Select air from the Phase drop-down list. (c) Select Diameter from the Histogram of drop-down list as the fluid property. (d) Click Plot to get a histogram of the Sauter diameter distribution in the fluid volume. You can also click Print to print the distribution in the FLUENT console. You can now see the distribution of the length number density of bubbles with Sauter diameter.


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Figure 16: Histogram of Sauter Diameter Distribution

Suggested Exercises 1. Calculate the gas hold-up in the column using the volume integration tools in FLUENT and knowing the initial dimensions of the water column. 2. Rerun the case for a finer bubble size distribution using a geometric volume ratio of 2 around the inlet bubble diameter of 3 mm.

Summary The population balance approach is used to solve for the bubble size and flow distribution in an axisymmetric bubble column. The discrete method is chosen to directly calculate the bubble size distribution for a population of six different bubble sizes. The set up, solution process and postprocessing of gas-liquid multiphase flows with a size distribution using the population balance model in FLUENT is illustrated.

References [1] FLUENT Users Guide, Fluent Inc., 2003. [2] FLUENT 6: Population Balance Model Manual, Fluent Inc., 2004. [3] Luo, Hean; Svendsen, Hallvard F., Theoretical model for drop and bubble breakup in turbulent dispersions, AIChE Journal v. 42, no. 5, May 1996, pp. 1225-1233. [4] Sanyal, J.; Vasquez, S.; Roy, S.; Dudukovic, M.P., Numerical simulation of gas-liquid

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dynamics in cylindrical bubble column reactors, Chemical Engineering Science, v. 54, no. 21, 1999, p. 5071-5083.


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Tutorial: Modeling Bubble Breakup and Coalescence ...

Tutorial: Modeling Bubble Breakup and Coalescence ...

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