bubble analysis with cfd

8/6/2019 Bubble Analysis With CFD

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Research Center

College of Engineering

King Saud University

Final Research Report No 7/424

Analysis of Bubble Column Hydrodynamics Using

Computational Fluid Dynamics

By

Dr. Waheed A. Al-Masry

Jamad Al-Awal 1427

June 2006


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Table of Contents

Page

ABSTRACT 3

ACKNOWLEDGMENT 4

LIST OF FIGURES 5

1. INTRODUCTION 7

2. ANSYS CFX CODE 10

3. GOVERNING EQUATIONS 14

4. MULTIPHASE MODELING 17

5. BACKGROUND 21

6. SIMULATIONS AND RESULTS 25

7. SCALE-UP STRATEGY 44

8. CONCLUSIONS 45

9. REFERENCES 46

APPENDIX A 48


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ABSTRACT

This report describes the steady-state and transient computation of the hydrodynamics of an

experimental bubble column reactor with internal liquid circulation for a range of

superficial gas velocities. Tetrahedral mesh is used with CFX software. Predictions

compared well with the experimental data.

. .

.

.

.


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LIST OF FIGURES

Fig. 1: Ansys CFX structure. 12

Fig. 2: Ansys CFX files. 15

Fig. 3: Schematic diagram of the experimental setup. 26

Fig. 4: Thirty-degree segment axisymmetric mesh. 28

Fig. 5: Enlargement of the thirty-degree segment axisymmetric mesh at

the top.

29

Fig. 6: Enlargement of the thirty-degree segment axisymmetric mesh at

the bottom.

29

Fig. 7: RMS for momentum and mass. 30

Fig. 8: RMS for turbulence quantities. 31

Fig. 9: RMS for volume fraction. 32

Fig. 10: Front view for the bubble column before simulation at

symmetry wall.

34

Fig. 11: Graphical representation of volume fraction in the riser and in

the downcomer.

34

Fig. 12: Vector presentation of water velocity. 35

Fig. 13: Vector presentation of water velocity at the top. 35

Fig. 14: Vector presentation of water velocity at the bottom. 36

Fig. 15: Vector presentation of air velocity at the top. 36

Fig. 16: Vector presentation of water velocity at the bottom. 37

Fig. 17: Air volume fraction in the downcomer. 37


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Fig. 18: Air volume fraction in the downcomer showing only presence

at the edges of the draft tube.

38

Fig. 19: Air volume fraction in the riser showing. 38

Fig. 20: Displaying wire-frame of the entire bubble column with 12

rotations.

39

Fig. 21: Air volume fraction in the riser of the entire bubble column. 40

Fig. 22: Air volume fraction in the downcomer of the entire bubble

column.

40

Fig. 23: Air volume fraction in the entire bubble column. 41

Fig. 24: Air volume fraction in the riser showing the sparger. 41

Fig. 25: Air volume fraction in the bubble column showing the sparger. 42

Fig. 26: Air volume fraction in the bubble column showing the sparger

and the 12 axisymmetrical segments.

42

Fig. 27: Gas holdup in the riser for a superficial gas velocity of 0.015

m/s for deifferent mesh sizes.

43

Fig. 28: Plot of gas holdup in the riser versus superficial gas velocity. 44


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1. INTRODUCTION

The use of computational fluid dynamics (CFD) to predict internal and external flows has

risen dramatically in the past decade. The wide spread availability of engineering

workstations together with efficient solution algorithms and sophisticated pre- and post-

processing facilities enabled the use of commercial CFD codes by engineers for research,

development and design tasks in industry. The codes that now on the market may be

extremely powerful, but their operation still requires a high level of skill and understanding

from the operator to obtain meaningful results in complex situations. Throughout, one of

the key messages is that CFD can not be professed adequately without continued reference

to experimental validation. The early ideas of the computational laboratory to supersede

experimentation have fortunately gone out of fashion.

1.1 COMPUTATIONAL FLUID DYNAMICS

Computational fluid dynamics or CFD is the analysis of systems involving fluid flow, heat

transfer and associated phenomena such as chemical reactions by means of computer-based

simulation. The technique is very powerful and spans a wide range of industrial and non

industrial applications areas. Some examples are: chemical process engineering: mixing

and separation, multiphase systems, aerodynamics of aircraft and vehicles, hydrodynamics

of ships, power plants, turbo machinery, electrical and electronic engineering, external and

internal environment of buildings, marine engineering, environmental engineering,

hydrology and oceanography, meteorology, and biomedical engineering (blood flows

through arteries and veins). Increasingly CFD is becoming a vital component in the design


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of industrial products and processes. There are several unique advantages of CFD over

experimental-based approaches to fluid systems design: substantial reduction of lead times

and costs of new designs, ability to study where controlled experiments are difficult or

impossible to perform, ability to study systems under hazardous conditions at and beyond

their normal performance limits, and practically unlimited level of detail of results. The

variable cost of an experiment, in terms of facility hire and/or man-hour costs, is

proportional to the number of data points and the number of configurations tested. In

contrast CFD codes can produce extremely large volumes of results at virtually no added

expense and it is very cheap to perform parametric studies, for instance to optimize

equipment performance. CFD codes are structured around the numerical algorithms that

can tackle fluid flow problems. In order to provide easy access to their solving power all

commercial CFD packages include sophisticated user interfaces to input problem

parameters and to examine the results. Hence all codes contain three main elements: (i) a

pre-processor, (ii) a solver and (iii) a post-processor code.

Pre-processor. Pre-processor consists of the input of a flow problem to a CFD program by

means of an operator-friendly interface and the subsequent transformation of this input into

a form suitable for use by the solver. The user activities at the pre-processing stage involve:

definition of the geometry of the region of interest: the computational domain, grid

generation-the sub-division of the domain into a number of smaller, non-overlapping sub-

domains: a grid (or mesh) of cells (or control volumes or elements), selection of the

physical and chemical phenomena that need to be modeled, definition of fluid properties,

specification of appropriate boundary conditions at cells which coincide with or touch the


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domain boundary. The solution to a flow problem (velocity, pressure, temperature, etc.) is

defined at nodes inside each cell. The accuracy of a CFD solution is governed by the

number of cells in the grid. In general, the larger the number of cells the better the solution

accuracy. Both the accuracy of a solution and its cost in terms of necessary computer

hardware and calculation time are dependent on the fineness of the grid. Optimal meshes

are often non-uniform: finer in areas where large variations occur from point to point and

coarser in regions with relatively little change. Over 50% of the time spent in industry on a

CFD project is devoted to the definition of the domain geometry and grid generation.

Solver. There are three distinct streams of commercial solution techniques: finite

difference, finite element and spectral methods. In outline the numerical methods that form

the basis of the solver perform the following steps: approximation of the unknown flow

variables by means of simple functions, discretisation by substitution of the approximations

into the governing flow equations and subsequent mathematical manipulations, solution of

the algebraic equations.

Post-processor. As in pre-processing a huge amount of development work has recently

taken place in the post-processing field. Owing to the increased popularity of engineering

workstations, many of which have outstanding graphics capabilities, the leading CFD

packages are now equipped with versatile data visualization tools. These include: domain

geometry and grid display, vector plots, line and shaded contour plots, 2D and 3D surface

plots, particle tracking, view manipulation (translation, rotation, scaling etc.), color

postscript output. More recently these facilities may also include animation for dynamic


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results display and in addition to graphics all codes produce trustily alphanumeric output

and have data export facilities for further manipulation external to the code.

1.2 PROBLEM SOLVING WITH CFD

In solving fluid flow problems we need to be aware that the underlying physics is complex

and the results generated by a CFD code are at best as good as the physics (and chemistry)

embedded in it and at worst as good as its operator. Prior to the setting up and running a

CFD simulation there is a stage of identification and formulation of the flow problem in

terms of the physical and chemical phenomena that need to be considered. Typical

decisions that might be needed are whether to model a problem in two or three dimensions,

to exclude the effects of ambient temperature or pressure variations on the density of an air

flow, to choose to solve the turbulent flow equations or to neglect the effects of small air

bubbles dissolved in tap water. A good understanding of the numerical solution algorithm

is also crucial. Three mathematical concepts are useful in determining the success or

otherwise of such algorithms: convergence, consistency and ability.

2. ANSYS CFX CODE

There are many commercial CFD codes in the market including PHOENICS, FLUENT,

FLOW3D, STAR-CD and ANSYS CFX. The work in this project will be carried out using

ANSYS CFD code. ANSYS CFX is a general purpose Computational Fluid Dynamics

(CFD) code, combining an advanced solver with powerful pre and post-processing

capabilities. The next-generation physics pre-processor, CFX-Pre, allows multiple meshes


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to be imported, allowing each section of complex geometries to use the most appropriate

mesh. ANSYS CFX-5 includes the following features:

An advanced coupled solver which is both reliable and robust.

Full integration of problem definition, analysis and results presentation.

An intuitive and interactive setup process, using menus and advanced graphics.

Detailed online help.

CFX-5 is capable of modeling:

steady-state and transient flows

laminar and turbulent flows

subsonic, transonic and supersonic flows

heat transfer and thermal radiation

buoyancy

non-Newtonian flows

transport of non-reacting scalar components

multiphase flows

combustion


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flows in multiple frames of reference

particle tracking

2.1 THE STRUCTURE OF CFX-5

CFX-5 consists of five software modules which are linked by the flow of information

required to perform a CFD analysis:

Fig. 1: Ansys CFX structure.

2.2 CFX-PRE

CFX-Pre can import mesh files produced by a range of mesh generation software packages.

Flow physics, boundary conditions, initial values and solver parameters are specified in

CFX-Pre. A full range of boundary conditions, including inlets, outlets and openings,

together with boundary conditions for heat transfer models and periodicity, are all available

in CFX-5 through CFX-Pre.


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2.3 CFX-SOLVER

CFX-Solver solves all the solution variables for the simulation for the problem

specification generated in CFX-Pre. One of the most important features of CFX-5 is its use

of a coupled solver, in which all the hydrodynamic equations are solved as a single system.

The coupled solver is faster than the traditional segregated solver and fewer iterations are

required to obtain a converged flow solution.

2.4 CFX-SOLVER MANAGER

The CFX-Solver Manager is a module that provides greater control to manage the CFD

task. Its major functions are:

Specify the input files to the CFX-Solver.

Start/stop the CFX-Solver.

Monitor the progress of the solution.

Set up the CFX-Solver for a parallel calculation.

2.5 CFX-POST

CFX-Post provides state-of-the-art interactive post-processing graphics tools to analyze and

present the CFX-5 simulation results. Important features include:

Quantitative post-processing

Command line, session file or state file input


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User-defined variables

Generation of a variety of graphical objects where Visibility, Transparency, Color and

Line/Face rendering can be controlled

Power Syntax to allow fully programmable session files

2.6 CFX-5 FILE TYPES

During the process of creating the model, running the CFX-Solver and analyzing the

results, a number of different files are created by the various modules of the software. This

section describes some of these files and their purpose. The use of these files with their

default extension is shown in the flowchart below. The standard files used and produced are

indicated with solid black lines; other possible uses are indicated with dotted lines.

3. GOVERNING EQUATIONS

The set of equations solved by Ansys CFX are the unsteady Navier-Stokes equations in

their conservation form. For all the following equations, static (thermodynamic) quantities

are given unless otherwise stated.

Transport Equations

The instantaneous equations of mass, momentum and energy conservation can be written as

follows in a stationary frame: The Continuity Equation


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( ) 0Ut

+ =

(1)

The Momentum Equations

( ) ( ( ( ) ))T MU

U U p U U St

+ = + + +

(2)

Fig. 2: Ansys CFX files.

The Energy Equation


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( ) ( )tot tot E h p

Uh T St t

+ = +

(3)

htot is defined as the Specific Total Enthalpy, which for the general case of variable

properties and compressible flow is given in terms of the Specific Static (thermodynamic)

Enthalpy, h, by:

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2toth h U= + (4)

where

( , )h h p T = (5)

If viscous work is significant, then an additional term is added to the right hand side of the

above energy equation to account for the effect of viscous shear, and the energy equation

becomes,

2( ) ( ) ( )

3

Ttottot E

h pUh T U U U U S

t t

+ = + + +

(6)

There are seven unknowns (u, v, w,p, T, , h) in the above five equations, but the set can be

closed by adding two algebraic thermodynamic equations: the Equation of State, which

relates density to pressure and temperature; and the Constitutive Equation, which relates

enthalpy to temperature and pressure.


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4. MULTIPHASE MODELING

Multiphase flow refers to the situation where more than one fluid is present. Each fluid may

possess its own flow field, or all fluids may share a common flow field. In general, the

fluids consist of different chemical species, e.g. air-water. In some applications, they may

represent different thermodynamic phases of the same species, e.g. steam-water. It is

important to distinguish between multicomponent and multiphase flow. A multicomponent

fluid is assumed to consist of a mixture of chemical species which are mixed at the

molecular level. In this case, we solve for single mean velocity and temperature fields etc.

for the fluid. Examples are gaseous mixtures, and solutes in liquids. The fluids in a

multiphase flow are assumed to be mixed at macroscopic length scales, much larger than

molecular. Examples are gas bubbles in a liquid, liquid droplets in a gas or in another

immiscible liquid etc.. In this case. it is necessary to solve for different velocity and

temperature fields etc. for each fluid. These may interact with each other by means of

interfacial forces and heat and mass transfer across the phase interfaces. For example, if

cold wet particles are injected into a fast flowing stream of hot air, the particles will be

accelerated by interphase drag, they will be heated up by heat transfer across the phase

boundary, and they will be dried by evaporation of water into water vapor at the phase

boundary.

Ansys CFX includes a variety of multiphase models to allow the simulation of multiple

fluid streams, bubbles, droplets, solid particles and free surface flows. Two distinct


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multiphase flow models are available in CFX-5, a EulerianEulerian multiphase model

and a Lagrangian Particle Tracking multiphase model.

The following are examples of multiphase flow.

Water droplets in air. Water droplets in air constitute two different fluids which are mixed

at the macroscopic level and not the microscopic level. Hence you need to use a multiphase

model and define two distinct fluids, Water and Air. Each fluid contains only one

component comprising one material (air or water), whose properties can be defined. Air is

the Continuous Fluid, and Water is the Dispersed Fluid.

Air bubbles in water. As above, you need to use a multiphase model and define two

distinct fluids: Water and Air. In this case, Air is the Dispersed Fluid, and Water is the

Continuous Fluid.

Gas-solid and liquid-solid flow. It is possible to model the motion of a large number of

solid particles in a gas or a liquid as two phase flow. Examples occur in pneumatic

conveying, sedimentation in rivers, and fluidised beds. For example, the two phases may be

Water, the Continuous Fluid, and Sand, the Dispersed Fluid. In such problems, you

should assign the solid phase a small insignificant molecular viscosity. This is permissible,

as the physics is dominated by inter-phase drag and turbulence effects. The solid phase

should be assigned free slip boundary conditions at walls. If viscosity is not set, the solid

will not be available for selection in the Fluids List in CFX-Pre.


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Three-phase flow. It is possible to have more than one dispersed phase in a continuous

phase. For example, certain regimes of water-oil-gas flow in an oil pipeline may involve

both oil droplets and gas bubbles immersed in a continuous water phase.

Poly-dispersed flow. The above dispersed flow examples assume a single mean particle

diameter for the dispersed phases. Poly-dispersed flows involve dispersed phases of

different mean diameters. One way to model such flows is to define a different phase for

each particle size, as the particle size has a strong influence on interphase transfer. For

example, to model the flow of air bubbles in water of different diameters, you need to

create three fluids, Air1, Air2, Air3 say, each with the same material properties as air. Then

set the number of fluids equal to four, and select Water, Air1, Air2 and Air3 as the four

fluids. Next, under Fluid Models, designate Water as the continuous phase, and the other

three fluids as dispersed phases of desired mean diameters.

Eulerian-Eulerian. The Eulerian-Eulerian model is one of the multiphase models that has

been implemented in Ansys CFX. Within the Eulerian-Eulerian model, the interphase

transfer terms can be modeled using either the Particle Model, the Mixture Model or the

Homogeneous Model. Inhomogeneous multiphase flow refers to the case where separate

velocity fields and other relevant fields exist for each fluid. The pressure field is shared by

all fluids. The fluids interact via interphase transfer terms. The Particle and Mixture models

are both inhomogeneous multiphase models. The homogeneous multiphase flow is a

limiting case of Eulerian-Eulerian multiphase flow where all fluids share the same velocity


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fields and other relevant fields such as temperature, turbulence, etc. The pressure field is

also shared by all fluids.

Lagrangian Particle Tracking. The Particle Transport model is capable of modeling

dispersed phases which are discretely distributed in a continuous phase. The modeling

involves the separate calculation of each phase with source terms generated to account for

the effects of the particles on the continuous phase. Water droplets dispersed in air from the

liquid sprays of a cooling tower and solid particles dispersed in air from the pneumatic

transport of solids or transport of airborne particulates are such practical occurrences.

The implementation of particle transport modeling in Ansys CFX can be thought of as a

multiphase flow in which the particles are a dispersed phase. Instead of using an Eulerian

transport model for the dispersed phase, a Lagrangian transport model is used. All

continuous phases must use the Eulerian model. With the dispersed phase, each particle

interacts with the fluid and other particles discretely. Therefore, another method is required

to calculate the particle behavior. The most widely applied method available to determine

the behavior of the dispersed phase is to track several individual particles through the flow

field. Each particle represents a sample of particles that follow an identical path. The

behavior of the tracked particles is used to describe the average behavior of the dispersed

phase. This method is called separated flow analysis. The term particle is used to describe

an individual discrete element of the dispersed phase. The particle could represent a solid,

droplet or bubble. In cases where the flow contains small particles, there may be millions of

particles per second being injected. To avoid the cost of modeling each individual particle,

a smaller number of particles are injected as a representative sample of the actual number


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of particles. Several different forces affect the motion of a particle in a fluid. In Ansys CFX

the forces which have been included in the particle equation of motion are: viscous drag,

buoyancy force, virtual mass and pressure gradient forces, and the centripetal and Coriolis

forces in rotating reference frames [1-5].

5. BACKGROUND

Blazej et al. [2004] simulated the two-phase flow of an experimental airlift reactor using

Fluent software. Data from the simulation was compared with the experimental data

obtained by tracking of a magnetic particle and analysis of the possible pressure drop to

determine the gas holdup. Comparison between vertical velocity and gas holdup were made

for a serious of experiments where the superficial gas velocity in the riser was adjusted

between 0.01 and 0.075 m/s. In the case of gas phase holdup and liquid phase velocities in

the riser appropriate trends are followed and values are modeled to good accuracy, but the

downcomer flow characterization is poor due to effects caused by the choice of the bubble

size, volume fraction equation and mesh resolution used. Therefore to accurately model the

motion of gas and liquid phases in airlift reactors, the use of complex multiple gas/discrete

phase model equation must be implemented, where each discrete phase presents a single

bubble size for the same gas phase composition.

Dhotre and Joshi [2004] developed a CFD model for the prediction of flow pattern

and eddy diffusivity profiles in bubble columns. A low Reynolds number k-e model has

been incorporated for the near wall region. Simulations have been carried out for a wide


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range of superficial gas velocity, column diameter, column height and the nature of the gas-

liquid system. The model predictions have been compared with all the experimental data

available in the literature. Complete energy balance has been established in all the cases.

The validated CFD model has been extended for the prediction of pressure drop and heat

transfer for the two-phase gas-liquid flows in bubble columns. The predicted values of the

two-phase friction multiplier and heat transfer coefficient were in good agreement with the

experimental data.

Van Baten et al. [2003] studied the hydrodynamics of two configurations of internal

airlift reactors, both with a riser diameter of 0.1 m, operating with an air-water system. The

experimental results were compared with a model using computational fluid dynamics with

Eulerian descriptions of the gas and liquid phases. Interactions between the bubbles and the

liquid are taken into account by means of a momentum exchange, or drag coefficient based

on a literature correlation. The turbulence in the liquid phase is described using the k-

model. The CFD model shows excellent agreements with the measured data on gas holdup,

liquid velocity in the downcomer and the riser. The developed CFD model has the potential

of being applied as a tool for scaling-up.

Mudd and Van Den Akker [2001] presented 2D and 3D simulations of an airlift

reactor under steady state conditions at low gas flow rates. The simulations are based on a

two-fluid model with a k- model for the turbulence and as little as possible ad hoc closure

terms. The result are compared with an one-dimensional mechanical energy balance and are

found to be in good agreement. The 2D results show sensitivity to the gas inlet geometry:


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whether or not the gas is partially sparged into the liquid directly next to a wall affects the

liquid velocity distribution and the gas disengagement at the top of the airlift. The three-

dimensional calculations make a more realistic geometry possible. The friction in the

system is found to be about a factor of two larger in the 3D case at the same gas inlet

conditions. For a given gas flow rate, the mean gas fraction in the riser is the same for the

2D and 3D simulations, the liquid circulation rate is about 30% higher in the 2D case than

the 3D one. A comparison is made with experimental data obtained in an airlift of the same

dimensions. The simulated overall gas fraction is in agreement with the experimental

findings. The simulated superficial gas velocity in the riser is compared to LDA data. For

the lowest superficial velocities the LDA data coincide with the results from the 2D

simulations, for higher gas flow rates the LDA results switch over towards the 3D results.

Ekambara and Joshi [2003] measured the liquid phase mixing time in 0.2 and 0.4 m

i.d. columns over a wide range of superficial gas velocity (0.07-0.295 m/s) and height-to-

diameter ratio (1-10). A CFD model was developed for the prediction of flow pattern in

terms of mean velocity and eddy diffusivity profiles. The predications agree favorably with

all experimental data published in the literature. Complete energy balance was established

in all cases. The validated CFD model was extended for the simulation of the macro-mixing

process by incorporating the effects of both the bulk motion and the eddy diffusion.

Excellent agreement was observed between the CFD predicted and experimental values of

the mixing time over the entire range of D, VG, and HD/D covered. The model was further

extended for the predication of residence time distribution and hence the axial dispersion


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coefficient DL. The predicted values of DL were found to agree very well with all

experimental data published in the literature.

Van Baten and Krishna [2003] used computational fluid dynamics to compare the

hydrodynamics and mass transfer of an internal airlift reactor with that of a bubble column

reactor, operating with an air/water system in the homogenous bubble flow regime. The

liquid circulation velocities are significantly higher in the airlift configuration than in the

bubble columns, leading to significantly lower gas holdup. Within the riser of the airlift, the

gas and liquid phases are virtually in plug flow, whereas in bubble column the gas and

liquid phases follow parabolic velocity distributions. When compared at the same

superficial gas velocity, the volumetric mass transfer coefficient kLa, for an airlift is

significantly lower than that for a bubble column. However, when the results are compared

at the same values of gas holdup, the values of kLa are practically identical.

Krishna et al [2001] measured the gas holdup with air-water system in bubble

columns of 0.1, 0.15 and 0.38 m diameter, equipped with identical distribution devices. For

operation with superficial gas velocity in the range 0-0.04 m/s, the total gas holdup was

found to decrease with increasing column diameter. Of all the literature correlations for the

gas holdup, only Zhener correlation anticipated the decrease in the gas holdup with increase

in column diameter. The reason for this scale dependence is because the strength of the

liquid circulations increases with increasing scale. The reason for this scale dependence is

because the strength of the liquid circulations increases with increasing scale. Such

circulations accelerate the bubbles traveling upwards in the central core. Computational


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fluid dynamics simulations were carried out using the Eulerian description for both the gas

and the liquid phases in order to verify the scale dependence of the hydrodynamics.

Interactions between the bubbles and the liquid are taken into account in terms of a

momentum exchange, or drag, coefficient. The drag coefficient is determined from the

Mendelson correlation for bubble rise velocity. The turbulence in the liquid phase is

described using k-e model. The simulation results verify the trends predicted by the Zhener

(1989) correlation. It was concluded that Eulerian simulations are tools for scaling up

bubble columns.

6. SIMULATIONS AND RESULTS

6.1 PREPARATION

The bubble column will be analyzed using the Eulerian-Eulerian multiphase model using

CFX code. The bubble column has internal draft tube (which promote axial mixing) used to

direct the recirculation of the flow. The model is based on dispersion of air bubbles in

water. The flow profiles in the bubble column determine important reactor performance

parameters such as conversion and selectivity. The gas is supplied through a sparger at the

bottom of the draft tube. The bubble column four zones are:

- the riser zone where the gas is injected and flows upward with liquid

- the head zone where most of the gas is separated from the liquid

- the downcomer zone where the liquid flows back to the bottom of the apparatus

- the bottom zone where the liquid re-enters the riser.


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Fig. 3 shows schematic diagram of the bubble column reactor. The geometry of the bubble

column consists of a cylindrical vessel of 700 liter. A sparger, which has 35 holes of 1 mm

diameter, is used to distribute the gas in the riser zone (draft tube region). To avoid

excessive mesh refinement in the sparger region due to different length scales of the sparger

holes in relation to the overall bubble column length scale, an inlet boundary condition is

applied over the top half of the sparger to model gas entering the column. To reduce CPU

time the bubble column reactor is modeled as a thirty-degree segment as axisymmetric flow

is assumed.

Gas in

Gas out

Fig. 3: Schematic diagram of the experimental setup.


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Domain: Steady state and transient, stationary frame of reference, reference pressure of 1.0

bar, buoyant.

Fluid properties: The fluids used are air (dispersed phase at STP) and water (continuous

phase at RTP). The bubble diameter is set at 5 mm.

Fluid models: Isothermal, k-e (liquid), dispersed phase zero equation (gas), Grace

correlation used for interphase momentum transfer (with an exponent of 4 for dense bubble

effects) and turbulent dispersion (with a turbulent dispersion coefficient of 0.1).

Boundary conditions:

Inlet: a region corresponding to the sparger dimensions is cut out of the mesh. The

top surface of this region has an inlet boundary condition applied which is set to

conserve the volume flux of gas entering the bubble column through the sparger.

Outlet: degassing condition applied, which acts as an outlet to the dispersed phase

and a free slip wall to the continuous phase.

Other boundaries: free-slip walls for the gas phase, no-slip walls to the liquid phase, and

scalable wall functions used. Symmetry planes are used at segment sides.

Solver parameters: Time step: 5 s (0.1 s for transient runs), Residual target: 1x10-4 (RMS).


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Mesh: 22947 tetrahedral 3-D mesh used, with mesh inflation applied at external and

internal walls. number of nodes 5501, number of elements 22947.

Fig. 4 shows the thirty-degree segment axisymmetric mesh. Fig. 5 and 6 show enlargement

of the mesh at the top and bottom, respectively.

Fig. 4: Thirty-degree segment axisymmetric mesh.


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Fig. 5: Enlargement of the thirty-degree segment axisymmetric mesh at the top.

Fig. 6: Enlargement of the thirty-degree segment axisymmetric mesh at the bottom.


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6.2 CONVERGENCE

The solution was deemed converged when the residuals were below 1x10-4

level and the

gas holdup in the riser reached a steady value. Convergence was generally achieved

between 60 and 100 iterations (see Appendix A), which was dependent on the superficial

gas velocity. Fig. 7 shows the RMS for momentum and mass, Fig. 8 shows RMS for

turbulence quantities, and Fig. 9 shows RMS for volume fraction.

Fig. 7: RMS for momentum and mass.


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Fig. 8: RMS for turbulence quantities.


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Fig. 9: RMS for volume fraction.

6.3 RESULTS

At low gas velocities the problem can be solved as steady state since the riser suppresses

any transient effects. The modified grace Grace correlation has allowed good

correspondence between the calculated holdup in the riser and the experimental data. Fig.

10 is the front view of the bubble column. Fig. 11 shows the riser air volume fraction

contour plot in the bubble column. It can be seen that the gas plume moves towards the

draft tube in the riser and moves towards the centre of the column and degasses when


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leaving the draft tube. Note that some of the gas is entrained with the recirculating liquid

phase into the top of the downcomer. Fig. 12 shows a water velocity vector plot where the

recirculation patterns at the top and bottom of the column can be clearly seen in Figs. 13

and 14, respectively.

The hydrodynamics of a bubble column are inherently transient. However, in a bubble

column steady-state solutions are possible as lateral oscillations are suppressed by the draft

tube at low superficial gas velocities. As the superficial gas velocity is increased, transient

vortices begin to form around the top of the draft tube and a steady-state solution is not

possible.

Fig. 15 shows the air velocity vector plot at the top of the column and Fig. 16 shows the

vector plot for air velocity at the bottom. It is clear that the air velocity is at its maximum at

the sparger location and reduces with the axial length of the column. Air volume fraction at

the downcomer is shown in Fig. 17. It is clear from the enlargement graph (Fig. 18) that

there is only slight gas present at the edge of the draft tube, while liquid predominantly

present in the downcomer. This is in agreement with the experimental results. By turning

the 30-degree symmetrical segment planes, we can see the volume fraction of the air is at

its maximum value at the top of the column.


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34

Fig. 10: Front view for the bubble column before simulation at symmetry wall.

Fig. 11: Graphical representation of volume fraction in the riser and in the downcomer.


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Fig. 12: Vector presentation of water velocity.

Fig. 13: Vector presentation of water velocity at the top.


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Fig. 14: Vector presentation of water velocity at the bottom.

Fig. 15: Vector presentation of air velocity at the top.


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Fig. 16: Vector presentation of water velocity at the bottom.

Fig. 17: Air volume fraction in the downcomer.


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Fig. 18: Air volume fraction in the downcomer showing only presence at the edges of the

draft tube.

Fig. 19: Air volume fraction in the riser showing.


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39

The 30-degree segment can be revolved to form the entire bubble column with the draft

tube as seen in the wire frame in Fig. 20. Fig. 21 shows the air-volume fraction in the riser

and Figure 22 shows the volume fraction in the downcomer. Fig. 23 shows the bubble

column volume fraction simulation without the downcomer and Fig. 24 shows the sparger

in the riser section. Gas holdup can be seen for the riser and downcomer as in Figs. 25 and

26.

Fig. 20: Displaying wire-frame of the entire bubble column with 12 rotations.


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40

Fig. 21: Air volume fraction in the riser of the entire bubble column.

Fig. 22: Air volume fraction in the downcomer of the entire bubble column.


41/83

41

Fig. 23: Air volume fraction in the entire bubble column.

Fig. 24: Air volume fraction in the riser showing the sparger.


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42

Fig. 25: Air volume fraction in the bubble column showing the sparger.

Fig. 26: Air volume fraction in the bubble column showing the sparger and the 12

axisymmetrical segments.


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43

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 100000 200000 300000 400000 500000 600000

Number of element in mesh

Gasholdup

Fig. 27: Gas holdup in the riser for a superficial gas velocity of 0.015 m/s for different

mesh sizes.

Fig. 27 shows the effect of number of elements on gas holdup in the reactor riser. The

modified Grace correlation has allowed good correspondence between the calculated

holdup in the riser and experimental data. Fig. 28 shows the variation of gas holdup with

increasing superficial gas velocity. It can be seen that there is an approximate

proportionality between gas holdup and superficial gas velocity, and the predicted holdups

are in good agreement with the experimental data.


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44

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 0.005 0.01 0.015

Superficial gas velocity, m/s

Gasholdupintheriser

Experimental

CFX

Fig. 28: Plot of gas holdup in the riser versus superficial gas velocity.

7. SCALE-UP STRATEGY

In scaling bubble columns usually the experimental methods are based on having several

column sizes for testing, starting form bench scale to pilot-plant scale. After the pilot-plant,

usually the scale-up is based on either fixing one property to be similar in both the pilot-

plant and the commercial plant. The other properties may vary. Some scale-up strategies

are based on rule of thump. However, the use CFD will facilitate the scale-up on the

computer up to the commercial plant without resorting to build costly experimental setups.

The problem is usually relying on the power of the computing of the computers used to

solve millions of differential equations in the system.


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45

8. CONCLUSIONS

The successful simulation of a bubble column reactor has been demonstrated. A mesh

independent study yielded a mesh size that allowed a good compromise between accuracy

and CPU time. For relatively low superficial gas velocities steady-state solutions were

obtained on tetrahedral mesh and compared with experimental data. At higher gas

velocities transient flow features dominate the solution domain.


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46

9. REFERENCES

[1] H., Versteeg, W., Malalasekera, An introduction to computational fluid dynamics.

Prentice Hall, USA, 1995.

[2] P., Wesseling, Principles of computational fluid dynamics, Springler, Berlin,

Germany, 2001.Anderson, J., Computational fluid dynamics the basics with

applications, McGraw Hill, New York, USA, 1995.

[3] B., Andersson, R., Andersson, L., Hakansson, M., Mortensen, Computational fluid

dynamics, 2nd

edition, Sweden, 2005.

[4] Chung, T. J., Computational fluid dynamics, Cambridge University Press,

Cambridge, UK, 2002.

[5] H., Lomax, T., Pulliam, D., Zingg, Fundamentals of computational fluid dynamics,

Springler, Berlin, Germany, 2003.

[6] M., Blazej, G., Gartland Glover, S., Generalis, J., Markos, Gas-liquid simulation of

an airlift bubble column reactor. Chemical Engineering and Processing 43 (2004) 137.

[7] M., Dhotre, J., Joshi, Two-dimensional CFD for the prediction of flow pattern in

bubble column reactors. Chemical Engineering Research and Design 82 (2004) 689.

[8] J., Van Baten, J., Ellenberger, R., Krishna, Hydrodynamics of internal air-lift

reactors: experiments versus CFD simulations. Chemical Engineering and Processing

42 (2003) 733.

[9] R., Mudd, H., van Den Akker, 2D and 3D simulations of internal airlift loop reactor

on the basis of a two-fluid model. Chemical Engineering Science 56 (2001) 6351.


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47

[10] K., Ekambara, J., Joshi, CFD simulation of mixing and dispersion in bubble

columns. Chemical Engineering Research and Design 81 (2003) 987.

[11] M., Van Baten, R., Krishna, Comparison of hydrodynamics and mass transfer in

airlift and bubble column reactors using CFD. Chemical Engineering and Technology

26 (2003) 1074.

[12] R., Krishna, J., van Baten, M. Urseanu, Scale effects on the hydrodynamics of

bubble columns operating in the homogenous flow regime. Chemical Engineering and

Technology 24 (2001) 451.


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48

APPENDIX A

This run of the CFX-10.0 Solver started at 13:25:59 on 12 May 2006 by

user Dr. Waheed Al-Masry on WAHEED (intel_pentium_winnt5.1) using the

command:

"C:\Program Files\ANSYS Inc\CFX\CFX-10.0\bin\perllib\cfx5solve.pl"

-stdout-comms -batch -ccl -

Setting up CFX-5 Solver run ...

+--------------------------------------------------------------------+

| Warning! |

| |

| Your user ID, Dr. Waheed Al-Masry, appears to be longer than 15 |

| characters. This will cause rsh, and therefore the CFX-5 parallel |

| code, to fail on many platforms. Attempting to continue anyway, |

| but if the parallel code fails to start correctly this may be the || cause. |

| |

| If it works, you can suppress this message by setting the variable |

| CFX5_NOCHECK_PPEID, either in your cfx5rc file or the environment. |

+--------------------------------------------------------------------+

+--------------------------------------------------------------------+

| |

| CFX Command Language for Run |

| |

+--------------------------------------------------------------------+

LIBRARY:MATERIAL: Air at 25 C

Material Description = Air at 25 C and 1 atm (dry)

Material Group = Air Data, Constant Property Gases

Option = Pure Substance

Thermodynamic State = Gas

PROPERTIES:

Option = General Material

Thermal Expansivity = 0.003356 [K^-1]

ABSORPTION COEFFICIENT:

Absorption Coefficient = 0.01 [m^-1]

Option = Value

END

DYNAMIC VISCOSITY:Dynamic Viscosity = 1.831E-05 [kg m^-1 s^-1]

Option = Value

END

EQUATION OF STATE:

Density = 1.185 [kg m^-3]

Molar Mass = 28.96 [kg kmol^-1]

Option = Value

END


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49

REFRACTIVE INDEX:

Option = Value

Refractive Index = 1.0 [m m^-1]

END

SCATTERING COEFFICIENT:

Option = Value

Scattering Coefficient = 0.0 [m^-1]

END

SPECIFIC HEAT CAPACITY:

Option = Value

Reference Pressure = 1 [atm]

Reference Specific Enthalpy = 0. [J/kg]

Reference Specific Entropy = 0. [J/kg/K]

Reference Temperature = 25 [C]

Specific Heat Capacity = 1.0044E+03 [J kg^-1 K^-1]

Specific Heat Type = Constant Pressure

END

THERMAL CONDUCTIVITY:

Option = Value

Thermal Conductivity = 2.61E-02 [W m^-1 K^-1]END

END

END

MATERIAL: Water

Material Description = Water (liquid)

Material Group = Water Data, Constant Property Liquids

Option = Pure Substance

Thermodynamic State = Liquid

PROPERTIES:

Option = General Material

Thermal Expansivity = 2.57E-04 [K^-1]

ABSORPTION COEFFICIENT:

Absorption Coefficient = 1.0 [m^-1]

Option = Value

END

DYNAMIC VISCOSITY:

Dynamic Viscosity = 8.899E-4 [kg m^-1 s^-1]

Option = Value

END

EQUATION OF STATE:

Density = 997.0 [kg m^-3]

Molar Mass = 18.02 [kg kmol^-1]

Option = Value

END

REFRACTIVE INDEX:

Option = Value

Refractive Index = 1.0 [m m^-1]END

SCATTERING COEFFICIENT:

Option = Value

Scattering Coefficient = 0.0 [m^-1]

END

SPECIFIC HEAT CAPACITY:

Option = Value


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50


Reference Specific Enthalpy = 0.0 [J/kg]

Reference Specific Entropy = 0.0 [J/kg/K]

Reference Temperature = 25 [C]

Specific Heat Capacity = 4181.7 [J kg^-1 K^-1]

Specific Heat Type = Constant Pressure

END

THERMAL CONDUCTIVITY:

Option = Value

Thermal Conductivity = 0.6069 [W m^-1 K^-1]

END

END

END

END

EXECUTION CONTROL:

PARALLEL HOST LIBRARY:

HOST DEFINITION: waheed

Installation Root = C:\Program Files\ANSYS Inc\CFX\CFX-%v

Host Architecture String = intel_pentium_winnt5.1

ENDEND

PARTITIONER STEP CONTROL:

Multidomain Option = Independent Partitioning

Runtime Priority = Standard

MEMORY CONTROL:

Memory Allocation Factor = 1.0

END

PARTITIONING TYPE:

MeTiS Type = k-way

Option = MeTiS

Partition Size Rule = Automatic

END

END

RUN DEFINITION:

Definition File = D:/BubbleColumn/BubbleColumn.def

Interpolate Initial Values = Off

Run Mode = Full

END

SOLVER STEP CONTROL:

Runtime Priority = Standard

EXECUTABLE SELECTION:

Double Precision = Off

END

MEMORY CONTROL:

Memory Allocation Factor = 1.0

END

PARALLEL ENVIRONMENT:Number of Processes = 1

Start Method = Serial

END

END

END

FLOW:

DOMAIN: BubbleColumn


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51

Coord Frame = Coord 0

Domain Type = Fluid

Fluids List = Air at 25 C,Water

Location = Assembly

BOUNDARY: BubbleColumn Default

Boundary Type = WALL

Location = F14.B2.P3,F3.B1.P3,F5.B1.P3,F8.B1.P3,F9.B1.P3

BOUNDARY CONDITIONS:

WALL ROUGHNESS:

Option = Smooth Wall

END

END

FLUID: Air at 25 C


WALL INFLUENCE ON FLOW:

Option = Free Slip

END

END

END

FLUID: WaterBOUNDARY CONDITIONS:


Option = No Slip

END

END

END

WALL CONTACT MODEL:

Option = Use Volume Fraction

END

END

BOUNDARY: Sparger

Boundary Type = INLET

Location = Sparger


FLOW REGIME:

Option = Subsonic

END

MASS AND MOMENTUM:

Option = Fluid Velocity

END

END

FLUID: Air at 25 C


VELOCITY:

Normal Speed = 0.3 [m s^-1]

Option = Normal Speed

ENDVOLUME FRACTION:

Option = Value

Volume Fraction = 0.25

END

END

END

FLUID: Water


52/83

52


TURBULENCE:

Option = Medium Intensity and Eddy Viscosity Ratio

END

VELOCITY:

Normal Speed = 0 [m s^-1]

Option = Normal Speed

END

VOLUME FRACTION:

Option = Value

Volume Fraction = 0.75

END

END

END

END

BOUNDARY: Top

Boundary Type = OUTLET

Location = Top


FLOW REGIME:Option = Subsonic

END

MASS AND MOMENTUM:

Option = Degassing Condition

END

END

END

BOUNDARY: DraftTube


Location = DraftTube


WALL ROUGHNESS:


END

END

FLUID: Air at 25 C



Option = Free Slip

END

END

END

FLUID: Water



Option = No Slip

ENDEND

END

WALL CONTACT MODEL:


END

END

BOUNDARY: DraftTube Other Side


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53


Location = F10.B1.P3


WALL ROUGHNESS:


END

END

FLUID: Air at 25 C



Option = Free Slip

END

END

END

FLUID: Water



Option = No Slip

END

ENDEND

WALL CONTACT MODEL:


END

END

BOUNDARY: SymP1

Boundary Type = SYMMETRY

Location = Symmetry1

END

BOUNDARY: SymP2

Boundary Type = SYMMETRY

Location = Symmetry2

END

DOMAIN MODELS:

BUOYANCY MODEL:

Buoyancy Reference Density = 997 [kg m^-3]

Gravity X Component = 0 [m s^-2]

Gravity Y Component = -9.81 [m s^-2]

Gravity Z Component = 0 [m s^-2]

Option = Buoyant

BUOYANCY REFERENCE LOCATION:

Option = Automatic

END

END

DOMAIN MOTION:

Option = Stationary

ENDREFERENCE PRESSURE:


END

END

FLUID: Air at 25 C

FLUID MODELS:

FLUID BUOYANCY MODEL:


54/83

54

Option = Density Difference

END

MORPHOLOGY:

Mean Diameter = 6 [mm]

Option = Dispersed Fluid

END

TURBULENCE MODEL:

Option = Dispersed Phase Zero Equation

END

END

END

FLUID: Water

FLUID MODELS:

FLUID BUOYANCY MODEL:

Option = Density Difference

END

MORPHOLOGY:

Option = Continuous Fluid

END

TURBULENCE MODEL:Option = k epsilon

BUOYANCY TURBULENCE:

Option = None

END

END

TURBULENT WALL FUNCTIONS:

Option = Scalable

END

END

END

FLUID MODELS:

COMBUSTION MODEL:

Option = None

END

HEAT TRANSFER MODEL:

Homogeneous Model = False

Option = None

END

THERMAL RADIATION MODEL:

Option = None

END

TURBULENCE MODEL:


Option = Fluid Dependent

END

END

FLUID PAIR: Air at 25 C | WaterSurface Tension Coefficient = 0.072 [N m^-1]

INTERPHASE TRANSFER MODEL:

Option = Particle Model

END

MASS TRANSFER:

Option = None

END


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MOMENTUM TRANSFER:

DRAG FORCE:

Option = Grace

Volume Fraction Correction Exponent = 2

END

LIFT FORCE:

Option = None

END

TURBULENT DISPERSION FORCE:

Option = Lopez de Bertodano

Turbulent Dispersion Coefficient = 0.3

END

VIRTUAL MASS FORCE:

Option = None

END

WALL LUBRICATION FORCE:

Option = None

END

END

TURBULENCE TRANSFER:ENHANCED TURBULENCE PRODUCTION MODEL:

Option = Sato Enhanced Eddy Viscosity

END

END

END

MULTIPHASE MODELS:


FREE SURFACE MODEL:

Option = None

END

END

END

INITIALISATION:

Option = Automatic

FLUID: Air at 25 C

INITIAL CONDITIONS:

Velocity Type = Cartesian

CARTESIAN VELOCITY COMPONENTS:

Option = Automatic with Value

U = 0 [m s^-1]

V = 0.3 [m s^-1]

W = 0 [m s^-1]

END

VOLUME FRACTION:

Option = Automatic

END

ENDEND

FLUID: Water

INITIAL CONDITIONS:

Velocity Type = Cartesian

CARTESIAN VELOCITY COMPONENTS:


U = 0 [m s^-1]


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56

V = 0 [m s^-1]

W = 0 [m s^-1]

END

EPSILON:

Option = Automatic

END

K:

Option = Automatic

END

VOLUME FRACTION:


Volume Fraction = 1

END

END

END

INITIAL CONDITIONS:

STATIC PRESSURE:

Option = Automatic

END

ENDEND

OUTPUT CONTROL:

RESULTS:

File Compression Level = Default

Option = Standard

END

END

SIMULATION TYPE:

Option = Steady State

END

SOLUTION UNITS:

Angle Units = [rad]

Length Units = [m]

Mass Units = [kg]

Solid Angle Units = [sr]

Temperature Units = [K]

Time Units = [s]

END

SOLVER CONTROL:

ADVECTION SCHEME:

Option = High Resolution

END

CONVERGENCE CONTROL:

Maximum Number of Iterations = 100

Physical Timescale = 1 [s]

Timescale Control = Physical Timescale

ENDCONVERGENCE CRITERIA:

Residual Target = 1.E-4

Residual Type = RMS

END

DYNAMIC MODEL CONTROL:

Global Dynamic Model Control = On

END


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END

END

COMMAND FILE:

Version = 10.0

Results Version = 10.0

END

+--------------------------------------------------------------------+

| |

| Solver |

| |

+--------------------------------------------------------------------+

+--------------------------------------------------------------------+

| |

| ANSYS CFX Solver 10.0 || |

| Version 2005.07.11-10.24 Mon Jul 11 10:26:04 GMTDT 2005 |

| |

| Executable Attributes |

| |

| single-32bit-optimised-supfort-noprof-nospag-lcomp |

| |

| Copyright 1996-2005 ANSYS Europe Ltd. |

+--------------------------------------------------------------------+

+--------------------------------------------------------------------+

| Job Information |

+--------------------------------------------------------------------+

Run mode: serial run

Host computer: WAHEED

Job started: Fri May 12 13:26:23 2006

+--------------------------------------------------------------------+

| Memory Allocated for Run (Actual usage may be less) |

+--------------------------------------------------------------------+

Data Type Kwords Words/Node Words/Elem Kbytes Bytes/Node

Real 7802.6 1430.10 340.03 30479.0 5720.40

Integer 1371.7 251.41 59.78 5358.1 1005.62

Character 2400.4 439.95 104.60 2344.1 439.95

Logical 40.0 7.33 1.74 156.2 29.33


58/83

58

Double 162.2 29.73 7.07 1267.0 237.80

+--------------------------------------------------------------------+

| Total Number of Nodes, Elements, and Faces |

+--------------------------------------------------------------------+

Domain Name : BubbleColumn

Total Number of Nodes = 5456

Total Number of Elements = 22947

Total Number of Tetrahedrons = 22947

Total Number of Faces = 5752

+--------------------------------------------------------------------+

| Reference Pressure Information |

+--------------------------------------------------------------------+

Domain Group: BubbleColumn

Pressure has not been set at any boundary conditions.

The pressure will be set to 0.00000E+00 at the following location:

Domain : BubbleColumn

Node : 1 (equation 1)

Coordinates : ( 4.76314E-02, 7.35000E-01, 2.75000E-02).

Domain Group: BubbleColumn

Buoyancy has been activated. The absolute pressure will include

hydrostatic pressure contribution, using the following reference

coordinates: ( 4.76314E-02, 7.35000E-01, 2.75000E-02).

+--------------------------------------------------------------------+

| Average Scale Information |

+--------------------------------------------------------------------+

Domain Name : BubbleColumn

Global Length = 1.0873E-01

Minimum Extent = 4.0000E-02

Maximum Extent = 7.7000E-01

Air at 25 C.Density = 1.1850E+00

Air at 25 C.Dynamic Viscosity = 1.8310E-05

Air at 25 C.Velocity = 3.0000E-01Air at 25 C.Advection Time = 3.6242E-01

Air at 25 C.Reynolds Number = 2.1110E+03

Air at 25 C.Mass (Conservative) = 3.8077E-10

Air at 25 C.Mass (Normalised) = 3.8077E-10

Air at 25 C.Volume = 3.2132E-10

Air at 25 C.Volume Fraction = 2.5000E-07

Water.Density = 9.9700E+02


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59

Water.Dynamic Viscosity = 8.8990E-04

Water.Velocity = 0.0000E+00

Water.Mass (Conservative) = 1.2814E+00

Water.Mass (Normalised) = 1.2814E+00

Water.Volume = 1.2853E-03

Water.Volume Fraction = 1.0000E+00

Air at 25 C | Water.Slip Reynolds Number = 2.0166E+03

+--------------------------------------------------------------------+

| Checking for Isolated Fluid Regions |

+--------------------------------------------------------------------+

No isolated fluid regions were found.

+--------------------------------------------------------------------+

| The Equations Solved in This Calculation |

+--------------------------------------------------------------------+

Subsystem : Momentum and Mass

U-Mom-Air at 25 C

V-Mom-Air at 25 C

W-Mom-Air at 25 C

U-Mom-Water

V-Mom-Water

W-Mom-Water

P-Vol

Subsystem : Volume Fractions

Mass-Air at 25 C

Mass-Water

Subsystem : TurbKE and Diss.K

K-TurbKE-Water

E-Diss.K-Water

CFD Solver started: Fri May 12 13:26:34 2006

+--------------------------------------------------------------------+

| Convergence History |

+--------------------------------------------------------------------+

======================================================================

| Timescale Information |

----------------------------------------------------------------------

| Equation | Type | Timescale |

+----------------------+------------------------+--------------------+

| U-Mom-Air at 25 C | Physical Timescale | 1.00000E+00 |


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60

| V-Mom-Air at 25 C | Physical Timescale | 1.00000E+00 |

| W-Mom-Air at 25 C | Physical Timescale | 1.00000E+00 |

| U-Mom-Water | Physical Timescale | 1.00000E+00 |

| V-Mom-Water | Physical Timescale | 1.00000E+00 |

| W-Mom-Water | Physical Timescale | 1.00000E+00 |

+----------------------+------------------------+--------------------+

| Mass-Air at 25 C | Physical Timescale | 1.00000E+00 |

| Mass-Water | Physical Timescale | 1.00000E+00 |

+----------------------+------------------------+--------------------+

| K-TurbKE-Water | Physical Timescale | 1.00000E+00 |

| E-Diss.K-Water | Physical Timescale | 1.00000E+00 |

+----------------------+------------------------+--------------------+

======================================================================

OUTER LOOP ITERATION = 1 CPU SECONDS = 1.692E+00

----------------------------------------------------------------------

| Equation | Rate | RMS Res | Max Res | Linear Solution |

+----------------------+------+---------+---------+------------------+

| U-Mom-Air at 25 C | 0.00 | 3.5E-05 | 7.5E-04 | 2.9E-03 OK|

| V-Mom-Air at 25 C | 0.00 | 8.2E-06 | 1.7E-04 | 1.0E-02 OK|| W-Mom-Air at 25 C | 0.00 | 1.1E-05 | 2.9E-04 | 5.9E-03 OK|

| U-Mom-Water | 0.00 | 1.6E-12 | 1.2E-11 | 4.2E+07 ok|

| V-Mom-Water | 0.00 | 7.1E-05 | 1.6E-04 | 1.3E+00 ok|

| W-Mom-Water | 0.00 | 1.3E-12 | 9.4E-12 | 3.8E+07 ok|

| P-Vol | 0.00 | 1.7E-02 | 4.3E-01 | 12.6 9.6E-02 OK|

+----------------------+------+---------+---------+------------------+

| Mass-Air at 25 C | 0.00 | 5.2E-02 | 9.6E-01 | 9.9 2.2E-02 OK|

| Mass-Water | 0.00 | 6.5E-02 | 1.0E+00 | 5.4 1.8E-03 OK|

+----------------------+------+---------+---------+------------------+

| K-TurbKE-Water | 0.00 | 2.8E-02 | 2.7E-01 | 5.3 4.4E-02 OK|

| E-Diss.K-Water | 0.00 | 6.1E-02 | 1.1E+00 | 8.4 2.0E-02 OK|

+----------------------+------+---------+---------+------------------+

======================================================================


----------------------------------------------------------------------


+----------------------+------+---------+---------+------------------+


| V-Mom-Air at 25 C |99.99 | 1.4E-02 | 5.2E-02 | 5.3E-05 OK|

| W-Mom-Air at 25 C | 0.21 | 2.3E-06 | 4.9E-05 | 3.3E-02 OK|

| U-Mom-Water |99.99 | 7.4E-04 | 1.1E-02 | 1.6E-02 OK|

| V-Mom-Water |99.99 | 9.7E-02 | 3.3E-01 | 5.0E-04 OK|

| W-Mom-Water |99.99 | 2.8E-04 | 5.6E-03 | 1.6E-02 OK|

| P-Vol | 0.25 | 4.5E-03 | 7.8E-02 | 43.6 1.4E-01 ok|

+----------------------+------+---------+---------+------------------+

| Mass-Air at 25 C | 0.22 | 1.1E-02 | 3.4E-01 | 10.0 2.1E-02 OK|| Mass-Water | 0.50 | 3.3E-02 | 5.7E-01 | 5.4 5.5E-02 OK|

+----------------------+------+---------+---------+------------------+



+----------------------+------+---------+---------+------------------+

======================================================================


61/83

61


----------------------------------------------------------------------


+----------------------+------+---------+---------+------------------+


| V-Mom-Air at 25 C | 0.02 | 3.2E-04 | 2.1E-03 | 1.0E-03 OK|


| U-Mom-Water | 2.07 | 1.5E-03 | 1.3E-02 | 1.3E-02 OK|

| V-Mom-Water | 0.17 | 1.6E-02 | 5.9E-02 | 2.6E-03 OK|

| W-Mom-Water | 2.46 | 6.9E-04 | 7.2E-03 | 8.7E-03 OK|

| P-Vol | 0.69 | 3.1E-03 | 8.0E-02 | 24.2 8.6E-02 OK|

+----------------------+------+---------+---------+------------------+


| Mass-Water | 0.51 | 1.7E-02 | 3.5E-01 | 9.7 2.0E-02 OK|

+----------------------+------+---------+---------+------------------+



+----------------------+------+---------+---------+------------------+

======================================================================OUTER LOOP ITERATION = 4 CPU SECONDS = 1.898E+01

----------------------------------------------------------------------


+----------------------+------+---------+---------+------------------+




| U-Mom-Water | 2.94 | 4.5E-03 | 3.9E-02 | 3.7E-02 OK|

| V-Mom-Water | 0.60 | 9.7E-03 | 1.0E-01 | 2.7E-02 OK|

| W-Mom-Water | 2.30 | 1.6E-03 | 1.7E-02 | 3.4E-02 OK|

| P-Vol | 0.61 | 1.9E-03 | 2.5E-02 | 12.6 5.8E-02 OK|

+----------------------+------+---------+---------+------------------+


| Mass-Water | 0.51 | 8.6E-03 | 1.5E-01 | 9.9 2.3E-02 OK|

+----------------------+------+---------+---------+------------------+


| E-Diss.K-Water | 0.97 | 8.7E-02 | 5.9E-01 | 8.3 1.8E-03 OK|

+----------------------+------+---------+---------+------------------+

======================================================================


----------------------------------------------------------------------


+----------------------+------+---------+---------+------------------+



| W-Mom-Air at 25 C | 0.64 | 1.4E-05 | 2.4E-04 | 4.7E-02 OK|| U-Mom-Water | 0.72 | 3.3E-03 | 2.5E-02 | 3.7E-02 OK|

| V-Mom-Water | 0.57 | 5.6E-03 | 4.9E-02 | 4.2E-02 OK|

| W-Mom-Water | 0.70 | 1.1E-03 | 1.2E-02 | 4.5E-02 OK|

| P-Vol | 1.24 | 2.3E-03 | 3.9E-02 | 8.7 6.6E-02 OK|

+----------------------+------+---------+---------+------------------+


| Mass-Water | 0.78 | 6.7E-03 | 1.9E-01 | 5.4 5.9E-02 OK|


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+----------------------+------+---------+---------+------------------+



+----------------------+------+---------+---------+------------------+

======================================================================


----------------------------------------------------------------------


+----------------------+------+---------+---------+------------------+




| U-Mom-Water | 0.75 | 2.5E-03 | 3.4E-02 | 3.1E-02 OK|

| V-Mom-Water | 0.66 | 3.7E-03 | 3.1E-02 | 3.5E-02 OK|

| W-Mom-Water | 0.74 | 8.2E-04 | 1.5E-02 | 4.1E-02 OK|

| P-Vol | 0.91 | 2.1E-03 | 3.2E-02 | 8.7 4.1E-02 OK|

+----------------------+------+---------+---------+------------------+


| Mass-Water | 1.26 | 8.4E-03 | 4.0E-01 | 5.3 5.7E-02 OK|+----------------------+------+---------+---------+------------------+



+----------------------+------+---------+---------+------------------+

======================================================================


----------------------------------------------------------------------


+----------------------+------+---------+---------+------------------+




| U-Mom-Water | 1.01 | 2.5E-03 | 6.2E-02 | 2.2E-02 OK|

| V-Mom-Water | 0.96 | 3.5E-03 | 5.1E-02 | 2.4E-02 OK|

| W-Mom-Water | 1.07 | 8.7E-04 | 2.9E-02 | 4.0E-02 OK|

| P-Vol | 0.75 | 1.6E-03 | 3.1E-02 | 8.7 3.7E-02 OK|

+----------------------+------+---------+---------+------------------+


| Mass-Water | 1.07 | 9.0E-03 | 4.5E-01 | 5.3 5.2E-02 OK|

+----------------------+------+---------+---------+------------------+



+----------------------+------+---------+---------+------------------+

======================================================================

OUTER LOOP ITERATION = 8 CPU SECONDS = 3.867E+01----------------------------------------------------------------------


+----------------------+------+---------+---------+------------------+




| U-Mom-Water | 0.97 | 2.4E-03 | 5.3E-02 | 4.1E-02 OK|


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| V-Mom-Water | 1.03 | 3.6E-03 | 9.0E-02 | 3.9E-02 OK|

| W-Mom-Water | 0.87 | 7.6E-04 | 1.9E-02 | 5.8E-02 OK|

| P-Vol | 0.67 | 1.1E-03 | 2.3E-02 | 8.7 6.2E-02 OK|

+----------------------+------+---------+---------+------------------+


| Mass-Water | 0.89 | 8.1E-03 | 2.5E-01 | 5.3 4.9E-02 OK|

+----------------------+------+---------+---------+------------------+



+----------------------+------+---------+---------+------------------+

======================================================================


----------------------------------------------------------------------


+----------------------+------+---------+---------+------------------+




| U-Mom-Water | 1.01 | 2.4E-03 | 5.6E-02 | 4.0E-02 OK|| V-Mom-Water | 0.99 | 3.6E-03 | 5.8E-02 | 4.4E-02 OK|

| W-Mom-Water | 1.01 | 7.7E-04 | 1.9E-02 | 5.2E-02 OK|

| P-Vol | 0.73 | 7.9E-04 | 1.5E-02 | 8.7 8.1E-02 OK|

+----------------------+------+---------+---------+------------------+


| Mass-Water | 0.90 | 7.3E-03 | 1.9E-01 | 5.4 4.5E-02 OK|

+----------------------+------+---------+---------+------------------+



+----------------------+------+---------+---------+------------------+

======================================================================


----------------------------------------------------------------------


+----------------------+------+---------+---------+------------------+




| U-Mom-Water | 0.88 | 2.1E-03 | 3.7E-02 | 3.6E-02 OK|

| V-Mom-Water | 0.89 | 3.2E-03 | 6.3E-02 | 4.1E-02 OK|

| W-Mom-Water | 0.90 | 6.9E-04 | 1.7E-02 | 4.4E-02 OK|

| P-Vol | 0.82 | 6.5E-04 | 1.1E-02 | 8.7 7.3E-02 OK|

+----------------------+------+---------+---------+------------------+


| Mass-Water | 1.00 | 7.3E-03 | 2.2E-01 | 5.3 5.1E-02 OK|

+----------------------+------+---------+---------+------------------+| K-TurbKE-Water | 0.81 | 9.3E-03 | 1.0E-01 | 5.3 1.4E-02 OK|


+----------------------+------+---------+---------+------------------+

======================================================================


----------------------------------------------------------------------


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+----------------------+------+---------+---------+------------------+




| U-Mom-Water | 0.78 | 1.7E-03 | 2.4E-02 | 3.0E-02 OK|

| V-Mom-Water | 0.86 | 2.8E-03 | 4.9E-02 | 3.8E-02 OK|

| W-Mom-Water | 0.80 | 5.5E-04 | 1.2E-02 | 3.8E-02 OK|

| P-Vol | 0.99 | 6.4E-04 | 1.8E-02 | 8.7 5.5E-02 OK|

+----------------------+------+---------+---------+------------------+


| Mass-Water | 1.09 | 7.9E-03 | 2.3E-01 | 5.3 3.1E-02 OK|

+----------------------+------+---------+---------+------------------+



+----------------------+------+---------+---------+------------------+

======================================================================


----------------------------------------------------------------------| Equation | Rate | RMS Res | Max Res | Linear Solution |

+----------------------+------+---------+---------+------------------+




| U-Mom-Water | 0.70 | 1.2E-03 | 1.9E-02 | 3.0E-02 OK|

| V-Mom-Water | 0.82 | 2.3E-03 | 3.7E-02 | 4.5E-02 OK|

| W-Mom-Water | 0.71 | 3.9E-04 | 8.2E-03 | 4.5E-02 OK|

| P-Vol | 1.08 | 7.0E-04 | 2.2E-02 | 8.7 6.2E-02 OK|

+----------------------+------+---------+---------+------------------+


| Mass-Water | 1.03 | 8.1E-03 | 3.6E-01 | 5.3 2.4E-02 OK|

+----------------------+------+---------+---------+------------------+



+----------------------+------+---------+---------+------------------+

======================================================================


----------------------------------------------------------------------


+----------------------+------+---------+---------+------------------+




| U-Mom-Water | 0.63 | 7.3E-04 | 1.2E-02 | 5.1E-02 OK|

| V-Mom-Water | 0.81 | 1.8E-03 | 2.6E-02 | 5.3E-02 OK|| W-Mom-Water | 0.64 | 2.5E-04 | 5.1E-03 | 7.5E-02 OK|

| P-Vol | 0.68 | 4.8E-04 | 1.6E-02 | 8.7 9.9E-02 OK|

+----------------------+------+---------+---------+------------------+


| Mass-Water | 0.79 | 6.5E-03 | 3.2E-01 | 5.4 4.2E-02 OK|

+----------------------+------+---------+---------+------------------+



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65


+----------------------+------+---------+---------+------------------+

======================================================================


----------------------------------------------------------------------


+----------------------+------+---------+---------+------------------+




| U-Mom-Water | 0.70 | 5.1E-04 | 6.9E-03 | 3.5E-02 OK|

| V-Mom-Water | 0.86 | 1.6E-03 | 1.9E-02 | 3.0E-02 OK|

| W-Mom-Water | 0.73 | 1.8E-04 | 2.4E-03 | 4.8E-02 OK|

| P-Vol | 0.78 | 3.7E-04 | 1.4E-02 | 12.6 7.5E-02 OK|

+----------------------+------+---------+---------+------------------+


| Mass-Water | 0.71 | 4.6E-03 | 1.7E-01 | 5.3 5.6E-02 OK|

+----------------------+------+---------+---------+------------------+

| K-TurbKE-Water | 0.79 | 3.4E-03 | 4.7E-02 | 5.5 2.8E-02 OK|| E-Diss.K-Water | 1.01 | 3.0E-03 | 6.9E-02 | 8.4 4.6E-03 OK|

+----------------------+------+---------+---------+------------------+

======================================================================


----------------------------------------------------------------------


+----------------------+------+---------+---------+------------------+




| U-Mom-Water | 1.03 | 5.3E-04 | 5.0E-03 | 2.9E-02 OK|

| V-Mom-Water | 0.94 | 1.5E-03 | 1.5E-02 | 2.6E-02 OK|

| W-Mom-Water | 1.01 | 1.9E-04 | 3.1E-03 | 4.0E-02 OK|

| P-Vol | 0.92 | 3.4E-04 | 6.8E-03 | 12.6 7.1E-02 OK|

+----------------------+------+---------+---------+------------------+


| Mass-Water | 0.86 | 3.9E-03 | 1.1E-01 | 5.3 5.7E-02 OK|

+----------------------+------+---------+---------+------------------+



+----------------------+------+---------+---------+------------------+

======================================================================


----------------------------------------------------------------------

| Equation | Rate | RMS Res | Max Res | Linear Solution |+----------------------+------+---------+---------+------------------+




| U-Mom-Water | 0.80 | 4.2E-04 | 4.3E-03 | 5.3E-02 OK|

| V-Mom-Water | 0.83 | 1.2E-03 | 1.3E-02 | 5.1E-02 OK|

| W-Mom-Water | 0.81 | 1.5E-04 | 2.6E-03 | 7.1E-02 OK|


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| P-Vol | 0.96 | 3.3E-04 | 5.9E-03 | 8.7 1.0E-01 OK|

+----------------------+------+---------+---------+------------------+


| Mass-Water | 0.93 | 3.7E-03 | 8.6E-02 | 5.3 5.2E-02 OK|

+----------------------+------+---------+---------+------------------+



+----------------------+------+---------+---------+------------------+

======================================================================


----------------------------------------------------------------------


+----------------------+------+---------+---------+------------------+




| U-Mom-Water | 0.76 | 3.2E-04 | 3.1E-03 | 5.3E-02 OK|

| V-Mom-Water | 0.81 | 1.0E-03 | 1.2E-02 | 5.2E-02 OK|

| W-Mom-Water | 0.79 | 1.2E-04 | 1.5E-03 | 6.5E-02 OK|| P-Vol | 0.83 | 2.7E-04 | 4.2E-03 | 8.7 8.6E-02 OK|

+----------------------+------+---------+---------+------------------+


| Mass-Water | 0.92 | 3.3E-03 | 8.1E-02 | 5.4 4.9E-02 OK|

+----------------------+------+---------+---------+------------------+



+----------------------+------+---------+---------+------------------+

======================================================================


----------------------------------------------------------------------


+----------------------+------+---------+---------+------------------+




| U-Mom-Water | 0.92 | 3.0E-04 | 2.8E-03 | 5.4E-02 OK|

| V-Mom-Water | 0.91 | 9.1E-04 | 9.8E-03 | 4.7E-02 OK|

| W-Mom-Water | 0.91 | 1.1E-04 | 1.1E-03 | 5.9E-02 OK|

| P-Vol | 0.85 | 2.3E-04 | 3.6E-03 | 8.7 7.8E-02 OK|

+----------------------+------+---------+---------+------------------+


| Mass-Water | 0.90 | 3.0E-03 | 7.0E-02 | 5.3 4.9E-02 OK|

+----------------------+------+---------+---------+------------------+


| E-Diss.K-Water | 0.90 | 2.3E-03 | 5.3E-02 | 8.4 3.8E-03 OK|+----------------------+------+---------+---------+------------------+

======================================================================


----------------------------------------------------------------------


+----------------------+------+---------+---------+------------------+


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67




| U-Mom-Water | 0.88 | 2.6E-04 | 2.4E-03 | 6.4E-02 OK|

| V-Mom-Water | 0.93 | 8.5E-04 | 9.4E-03 | 4.4E-02 OK|

| W-Mom-Water | 0.89 | 9.6E-05 | 1.0E-03 | 5.8E-02 OK|

| P-Vol | 0.85 | 2.0E-04 | 2.9E-03 | 8.7 7.3E-02 OK|

+----------------------+------+---------+---------+------------------+


| Mass-Water | 0.92 | 2.8E-03 | 5.8E-02 | 5.5 4.9E-02 OK|

+----------------------+------+---------+---------+------------------+


| E-Diss.K-

bubble analysis with cfd

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