CFD Simulation of the Global Behavior of

Bubble Columns using VOF Models

A. Horvath, Ch. Jordan, Ch. Kuttner, S. Pohn and M. HarasekNovember 2009

Institute of Chemical EngineeringVienna University of Technology

Getreidemarkt 9/166, 1060 Wien, Austria

Introduction

Bubble columns are a well known type of reactor in the field of chemicalengineering. In a typical bubble flow reactor gas is dispersed at the bottomof a column through a distributor. The buoyancy driven two-phase flowyields excellent mixing behavior and a large inner surface area without theneed of moving parts, fixed baffles or any type of filling material in thecolumn. Bubble columns are typically used as fermenters, in gas stripping,loop-type reactors, mixers and other applications.

In this work transient ab-initio simulations of rising bubble swarms, us-ing an open source CFD (Computational Fluid Dynamics) program, arecompared to experimental results. Simulation runs were done using two dif-ferent implementations of a VOF (Volume of Fluid) model, a finite volumeapproach, that is capable of describing the phase distributions of completelyimmiscible gas and liquid phases. It is found that the computational costcan be affordable (even when using a higher grid density) to get results, ingood agreement with the experiment, for applications in chemical engineer-ing (e.g. design and improvement of unit operations).

Simulation

Geometry and grid

A laboratory scale bubble column with inner dimensions of 200x200x2000 mmwas simulated. The un-aerated column was filled with water up to 1000 mm.In the lower part of the column a vertical separator plate was installed torun the bubble column in looping configuration where only half of the gasdistributor area in the bottom is aerated.

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The superficial gas velocity was set to v=0.04 m/s. The ”wind-box”below the gas distributor was included in the simulation domain to geta physically correct inlet pressure distribution. A constant inlet pressureat the gas-distributor plate was regarded as an oversimplification of theboundary conditions. The aluminum distributor plate had 7x7 equidistantholes with a diameter of 3 mm. The orifices were completely resolved inthe finite volume grid which consists of hexahedral cells with an equi-angle-skew factor below 0.3. The effects of two different grid densities (360000and 2700000 cells) in identical geometries were tested in the simulations.

Figure 1: Geometry with installed separator showing a typical looping typeflow pattern in the bubble column

Model and numerical background

For the simulations of sharp fluid-fluid interfaces using a finite volume ap-proach the VOF (Volume of Fluid) Model is very suitable (Akhtar et.al.,2007). As an alternative to the VOF model an Euler-Euler model can beused, that describes two completely inter-penetrating phases. Two completesets of conservation equations are solved. Euler-Euler models are suitable forsimulation of fluidized solid phases or mixing liquid phases. A considerabledrawback of the Euler-Euler approach is the interaction of the phases andbubble-sizes that need to be defined by the user. Bubble coalescence anddispersion (ab-initio) are difficult to model using Euler-Euler approaches.The VOF model assumes two or more completely immiscible fluids. Here

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we will limit the number of fluids to two, as we are only considering air andwater. A scalar indicator α is used to define the volume of one fluid V1 inevery control volume (the relative content of the other fluid V2 is simply1-α) and thus α = V1/V .

The indicator introduced above is a scalar function in 3 dimensionalspace that needs to be discretized (in space and time) in order to be usablein a direct finite volume solver. Apart from the usual discretizations ofthe continuity-, Navier-Stokes- and energy-equations a few exemplary stepsare outlined below that yield a suitable mathematical setup. Below someof the inner workings of the VOF Model implemented in the OpenFOAMsolver are described (Ubbink, 1997 and Rusche, 2002). The Fluent Solverprobably works in a similar way but is regarded as a black-box, due tolacking documentation, with only a few options that can be influenced bythe user.

For incompressible flow the (conservative) transport equation of α canbe written as:

∂α

∂t+∇α~u = 0 (1)

A discretization of the Gauss theorem is derived, that is applicable in ageneral way for finite volumes bounded by n flat faces (and is suitable forpolyhedral cells in unstructured grids):

∫V

∇~ϕdV =∮dV

d~S · ~ϕ =n∑f=1

∫f

d~S~ϕ

=n∑f=1

~Af · ~ϕf . (2)

The fluxes through the faces of a finite Volume cell are given by:∮∂V

αd~S~u ≈n∑f=1

αf · ~Af · ~uf =n∑f=1

αfFf . (3)

To get a bounded solution (i.e. 0<α<1) and sharp interfaces a so calledcompressive differencing scheme (see Ubbink 1997 and Olsson et.al. 2004)is used, that introduces a weighting factor (β),

αf = β · αP + (1− β) · αN . (4)

A form of the indicator function that is suitable for finite volume solverscan be given by:

aP · αt+δtP =n∑

nb=1

aN · αt+δtN + SαP . (5)

To minimize the numerical diffusion in the convection equations the tem-poral discretization is done using the Crank-Nicholson scheme that is second

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order accurate in time. More precisely: time stepping is weighted by 0.9Crank-Nicholson scheme and 0.1 Euler to avoid numerical instabilities. Im-plicit time stepping may cause numerical diffusion in direction of the flow,explicit time stepping introduces diffusion normal to the direction of flow.

Discretization methods and solver setting

Following table shows the methods chosen for the discretization of the scalarVOF indicator α and the temporal discretization in the used solvers. Allother field-variables were at least discretized with 2nd order accuracy.

Variable OpenFOAM-1.4.1 Fluent 6.3.26

α interfaceCompression Geo-Reconstructt Crank-Nicholson (w=0.9) 1st/2nd order implicit

Table 1: Discretization settings of two important field variables

The explicit time stepping scheme in Fluent proved to be unstable incombination with the used models. The size of the biggest time step inFluent was limited by the stability and number of sub-iterations of theGeo-reconstruct scheme. In OpenFOAM a dynamic limiter was adjustedto control the time step in order to keep the maximum Courant numberin the computational domain below 0.9. No turbulence model was usedin the simulations. Using a turbulence model caused an unphysically slowrise of the air bubbles due to the turbulent viscosity limiting the terminalrising velocity of the bubbles. Using a turbulence model gives unphysicalresults for both VOF and Euler-Euler models (see also Chen, 2001). Surfacetension (between air and water: η=0.07N/m) and contact angles (betweenwater and all walls: Θ=30°) were set. All simulation runs were started att=0 with an initially stationary water column of 1m height. Compressibilityeffects were neglected in the simulations. The inlet was defined as a veloc-ity inlet in Fluent and as a fixed U-value in OpenFOAM. The outlet wasdefined as a pressure outlet in Fluent and as a zero gradient boundary con-dition of pressure in OpenFOAM. The walls and the separator walls wereset to no slip in Fluent and a fixed U-value of (0 0 0) in OpenFOAM. Thematerial properties of water and air used in the simulations correspond to atemperature of 298 K.

Investigation of influential parameters on simulation results

The influence of various parameters on the simulation results is presented.In figure 2 cross sections through the middle of the bubble column are shownfor different cases. The bright areas show values of the phase indicator αbetween 0.4 and 0.6 (i.e. the interface between gas and liquid).

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Explanation for figure 2 Left : Influence of used solver (left: Open-FOAM, right: Fluent, both at t=1.5s, superficial gas velocity: 0.04m/s,both meshes: 360000 cells).

Middle: Influence of contact angle using OpenFOAM (left: not active,right: active at Θ=30°, both at t=1.5s, superficial gas velocity: 0.04 m/s,both meshes: 360000 cells).

Right : Influence of temporal discretization in the solver of Fluent (left:1st Order implicit, right: 2nd Order implicit, both at t=0.8s, looping-typebubble column with internal separator, superficial gas velocity: 0.01 m/s,both meshes: 2700000 cells).

Figure 2: Left: influence of solver (configuration without vertical separator),Middle: Influence of contact angle (no vertical separator), Right: Influenceof temporal discretization

Comparison and conclusion

The numerical experiments clearly show that the choice of solver and dis-cretization methods (both spatial and temporal) have the biggest influenceon the results. Lower order discretizations yield ”smeared” interfaces andphysically incorrect results. For the comparisons of experimental and simula-tion results the loop-type configuration is presented. The superficial gas ve-locity is set to v=0.04 m/s. The same finite volume grid containing 2700000cells was used for Fluent and OpenFOAM. The simulation results are dis-played in figure 3 and 4. The cross sectional area is located in the middleof the column. The left part of each figure shows the simulation result of

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Fluent, in the middle is a synchronized black and white picture captured bya Redlake Motion Pro high speed camera (at 2000 fps), the right part is thesimulation result of OpenFOAM. The bright areas correspond to values ofthe VOF indicator α between 0.4 and 0.6. The cumulated simulation timefor t=0.8 s using Fluent is 480 CPU days and 13.6 CPU days for Opoen-FOAM. Both simulations were run on the same hardware: Dual 1.9GHzPower5+ processors. The operating system used for Fluent was AIX 5.3and Linux 2.6.18 was used for OpenFOAM. The simulation runs in Flu-ent were parallelized on four CPUs while the OpenFOAM simulations wereparallelized on two CPUs.

The comparisons show that both solvers produce a good prediction ofthe gas hold-up in the bubble column, which can be measured by the heightof the liquid level. It can be seen that both solvers over-predict the rising ve-locity of the leading bubbles. Due to time constraints numerical experimentswith Fluent were stopped.

t=0.2 s t=0.4 s

Figure 3: Comparison of two Solvers and the experiment. Left: Fluent,middle: experiment, right: OpenFOAM

Using OpenFOAM five seconds of real time for the looping-type bubblecolumn at two different gas flow rates were simulated yielding results in verygood agreement with the experiments. Gas hold up and gas distribution inthe down-comer are predicted correctly (see figure 5, left: simulation resultof OpenFOAM: iso-surface for α=0.75, right: synchronous still image fromhigh speed camera), yet the simulation of the smaller bubbles would needan even higher grid density and adaptive refinement around interfaces (seeTheodorakakos, 2004).

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t=0.6 s t=0.8 s

Figure 4: Comparison of two Solvers and the experiment. Left: Fluent,middle: experiment, right: OpenFOAM

t=1.3 s t=4.9 s

Figure 5: Comparison of OpenFOAM (left) with experiment (right)

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Nomenclature

aP,N General matrix coefficients of cell (P) and neighbours (N) (1)f Center of the cell face~Af Face area vector (m)~uf Velocity vector at face center (m/s)n Number of faces of the control volume (1)Ff Volumetric flux rate (m3/s)V Total volume of the control volume (m3)Vi Volume of fluid i in the control volume (m3)SαP Source termα Volume of fluid indicator function (1)αP VOF of Cell (1)αN VOF of the nearest neighbor cells (1)δt Time step size (s)~ϕ Arbitrary vector functiond~S Surface area vector (m2)∂V Surface area of the control volume (m2)

References

[1] Akhtar, A. and Pareek, V. and Tade M., (2007), CFD Simulations for Contin-uous Flow of Bubbles through Gas-Liquid Columns: Application of VOF Method,Chemical Product and Process Modeling, 2, Issue 1, Article 9

[2] Chen L. and Manasseh R., (2001), Numerical Study of Air Release from anUnderwater Opening, Proc. 4th Int. Conf. on Multiphase Flow, New Orleans, LA,USA, 27 May to 1 June 2001, ed. E.E. Michaelides, paper 348

[3] Olsson, E. and Kreiss, G., (2005), A conservative level set method for two phaseflow, Journal of Computational Physics, 210, 225-246

[4] Rusche H., (2002), Computational Fluid Dynamics of dispersed two-phase flowsat high phase fractions, Thesis, Dpt. Of Mech.Eng., Imperial College, London

[5] Theodorakakos A. and Bergles G., (2004), Simulation of sharp gas-liquid inter-face using VOF method and adaptive grid local refinement around the interface,Int. Jou. For numerical methods in fluids, 45, 421-439

[6] Ubbink O., (1997), Numerical prediction of two fluid systems with sharp inter-faces, Thesis, Dpt. Of Mech.Eng., Imperial College, London

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