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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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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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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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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.
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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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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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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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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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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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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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[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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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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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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Reference Pressure = 1 [atm]
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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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
BOUNDARY CONDITIONS:
WALL INFLUENCE ON FLOW:
Option = Free Slip
END
END
END
FLUID: WaterBOUNDARY CONDITIONS:
WALL INFLUENCE ON FLOW:
Option = No Slip
END
END
END
WALL CONTACT MODEL:
Option = Use Volume Fraction
END
END
BOUNDARY: Sparger
Boundary Type = INLET
Location = Sparger
BOUNDARY CONDITIONS:
FLOW REGIME:
Option = Subsonic
END
MASS AND MOMENTUM:
Option = Fluid Velocity
END
END
FLUID: Air at 25 C
BOUNDARY CONDITIONS:
VELOCITY:
Normal Speed = 0.3 [m s^-1]
Option = Normal Speed
ENDVOLUME FRACTION:
Option = Value
Volume Fraction = 0.25
END
END
END
FLUID: Water
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BOUNDARY CONDITIONS:
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
BOUNDARY CONDITIONS:
FLOW REGIME:Option = Subsonic
END
MASS AND MOMENTUM:
Option = Degassing Condition
END
END
END
BOUNDARY: DraftTube
Boundary Type = WALL
Location = DraftTube
BOUNDARY CONDITIONS:
WALL ROUGHNESS:
Option = Smooth Wall
END
END
FLUID: Air at 25 C
BOUNDARY CONDITIONS:
WALL INFLUENCE ON FLOW:
Option = Free Slip
END
END
END
FLUID: Water
BOUNDARY CONDITIONS:
WALL INFLUENCE ON FLOW:
Option = No Slip
ENDEND
END
WALL CONTACT MODEL:
Option = Use Volume Fraction
END
END
BOUNDARY: DraftTube Other Side
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Boundary Type = WALL
Location = F10.B1.P3
BOUNDARY CONDITIONS:
WALL ROUGHNESS:
Option = Smooth Wall
END
END
FLUID: Air at 25 C
BOUNDARY CONDITIONS:
WALL INFLUENCE ON FLOW:
Option = Free Slip
END
END
END
FLUID: Water
BOUNDARY CONDITIONS:
WALL INFLUENCE ON FLOW:
Option = No Slip
END
ENDEND
WALL CONTACT MODEL:
Option = Use Volume Fraction
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:
Reference Pressure = 0 [atm]
END
END
FLUID: Air at 25 C
FLUID MODELS:
FLUID BUOYANCY MODEL:
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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:
Homogeneous Model = False
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:
Homogeneous Model = False
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:
Option = Automatic with Value
U = 0 [m s^-1]
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V = 0 [m s^-1]
W = 0 [m s^-1]
END
EPSILON:
Option = Automatic
END
K:
Option = Automatic
END
VOLUME FRACTION:
Option = Automatic with Value
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
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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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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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| 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|
+----------------------+------+---------+---------+------------------+
======================================================================
OUTER LOOP ITERATION = 2 CPU SECONDS = 6.930E+00
----------------------------------------------------------------------
| Equation | Rate | RMS Res | Max Res | Linear Solution |
+----------------------+------+---------+---------+------------------+
| U-Mom-Air at 25 C | 0.16 | 5.7E-06 | 1.3E-04 | 2.2E-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|
+----------------------+------+---------+---------+------------------+
| K-TurbKE-Water | 0.42 | 1.2E-02 | 8.7E-02 | 9.7 4.6E-02 OK|
| E-Diss.K-Water | 0.90 | 5.5E-02 | 1.0E+00 | 8.4 5.4E-02 OK|
+----------------------+------+---------+---------+------------------+
======================================================================
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OUTER LOOP ITERATION = 3 CPU SECONDS = 1.337E+01
----------------------------------------------------------------------
| Equation | Rate | RMS Res | Max Res | Linear Solution |
+----------------------+------+---------+---------+------------------+
| U-Mom-Air at 25 C | 2.28 | 1.3E-05 | 1.5E-04 | 9.7E-03 OK|
| V-Mom-Air at 25 C | 0.02 | 3.2E-04 | 2.1E-03 | 1.0E-03 OK|
| W-Mom-Air at 25 C | 2.29 | 5.2E-06 | 7.7E-05 | 2.1E-02 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-Air at 25 C | 0.68 | 7.7E-03 | 1.4E-01 | 9.9 2.9E-02 OK|
| Mass-Water | 0.51 | 1.7E-02 | 3.5E-01 | 9.7 2.0E-02 OK|
+----------------------+------+---------+---------+------------------+
| K-TurbKE-Water | 0.71 | 8.2E-03 | 6.5E-02 | 9.7 3.9E-02 OK|
| E-Diss.K-Water | 1.63 | 8.9E-02 | 1.0E+00 | 8.4 3.0E-02 OK|
+----------------------+------+---------+---------+------------------+
======================================================================OUTER LOOP ITERATION = 4 CPU SECONDS = 1.898E+01
----------------------------------------------------------------------
| Equation | Rate | RMS Res | Max Res | Linear Solution |
+----------------------+------+---------+---------+------------------+
| U-Mom-Air at 25 C | 6.01 | 7.8E-05 | 5.8E-04 | 1.8E-02 OK|
| V-Mom-Air at 25 C | 0.94 | 3.0E-04 | 1.8E-03 | 1.3E-02 OK|
| W-Mom-Air at 25 C | 4.30 | 2.2E-05 | 3.3E-04 | 4.2E-02 OK|
| 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-Air at 25 C | 0.64 | 5.0E-03 | 9.0E-02 | 9.9 2.9E-02 OK|
| Mass-Water | 0.51 | 8.6E-03 | 1.5E-01 | 9.9 2.3E-02 OK|
+----------------------+------+---------+---------+------------------+
| K-TurbKE-Water | 9.55 | 7.8E-02 | 5.3E-01 | 5.3 1.1E-02 OK|
| E-Diss.K-Water | 0.97 | 8.7E-02 | 5.9E-01 | 8.3 1.8E-03 OK|
+----------------------+------+---------+---------+------------------+
======================================================================
OUTER LOOP ITERATION = 5 CPU SECONDS = 2.411E+01
----------------------------------------------------------------------
| Equation | Rate | RMS Res | Max Res | Linear Solution |
+----------------------+------+---------+---------+------------------+
| U-Mom-Air at 25 C | 0.51 | 3.9E-05 | 5.8E-04 | 2.6E-02 OK|
| V-Mom-Air at 25 C | 0.44 | 1.3E-04 | 9.4E-04 | 1.9E-02 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-Air at 25 C | 0.95 | 4.7E-03 | 1.4E-01 | 5.3 7.1E-02 OK|
| Mass-Water | 0.78 | 6.7E-03 | 1.9E-01 | 5.4 5.9E-02 OK|
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+----------------------+------+---------+---------+------------------+
| K-TurbKE-Water | 0.49 | 3.8E-02 | 3.1E-01 | 5.3 9.3E-03 OK|
| E-Diss.K-Water | 0.62 | 5.4E-02 | 5.3E-01 | 8.4 6.9E-04 OK|
+----------------------+------+---------+---------+------------------+
======================================================================
OUTER LOOP ITERATION = 6 CPU SECONDS = 2.898E+01
----------------------------------------------------------------------
| Equation | Rate | RMS Res | Max Res | Linear Solution |
+----------------------+------+---------+---------+------------------+
| U-Mom-Air at 25 C | 1.86 | 7.3E-05 | 1.2E-03 | 1.5E-02 OK|
| V-Mom-Air at 25 C | 0.48 | 6.5E-05 | 7.9E-04 | 1.6E-02 OK|
| W-Mom-Air at 25 C | 1.67 | 2.4E-05 | 5.2E-04 | 2.6E-02 OK|
| 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-Air at 25 C | 1.10 | 5.2E-03 | 1.7E-01 | 5.5 5.3E-02 OK|
| Mass-Water | 1.26 | 8.4E-03 | 4.0E-01 | 5.3 5.7E-02 OK|+----------------------+------+---------+---------+------------------+
| K-TurbKE-Water | 0.39 | 1.5E-02 | 1.0E-01 | 5.3 1.9E-02 OK|
| E-Diss.K-Water | 0.68 | 3.7E-02 | 3.5E-01 | 8.3 6.3E-03 OK|
+----------------------+------+---------+---------+------------------+
======================================================================
OUTER LOOP ITERATION = 7 CPU SECONDS = 3.386E+01
----------------------------------------------------------------------
| Equation | Rate | RMS Res | Max Res | Linear Solution |
+----------------------+------+---------+---------+------------------+
| U-Mom-Air at 25 C | 1.02 | 7.4E-05 | 1.2E-03 | 7.3E-03 OK|
| V-Mom-Air at 25 C | 0.93 | 6.0E-05 | 1.1E-03 | 9.8E-03 OK|
| W-Mom-Air at 25 C | 1.07 | 2.6E-05 | 6.0E-04 | 1.6E-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-Air at 25 C | 0.81 | 4.2E-03 | 2.0E-01 | 5.3 4.5E-02 OK|
| Mass-Water | 1.07 | 9.0E-03 | 4.5E-01 | 5.3 5.2E-02 OK|
+----------------------+------+---------+---------+------------------+
| K-TurbKE-Water | 0.76 | 1.1E-02 | 1.1E-01 | 5.5 3.4E-02 OK|
| E-Diss.K-Water | 1.16 | 4.2E-02 | 5.5E-01 | 8.4 8.8E-03 OK|
+----------------------+------+---------+---------+------------------+
======================================================================
OUTER LOOP ITERATION = 8 CPU SECONDS = 3.867E+01----------------------------------------------------------------------
| Equation | Rate | RMS Res | Max Res | Linear Solution |
+----------------------+------+---------+---------+------------------+
| U-Mom-Air at 25 C | 0.74 | 5.4E-05 | 1.0E-03 | 1.4E-02 OK|
| V-Mom-Air at 25 C | 1.05 | 6.3E-05 | 1.5E-03 | 1.9E-02 OK|
| W-Mom-Air at 25 C | 0.78 | 2.0E-05 | 4.6E-04 | 2.6E-02 OK|
| 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-Air at 25 C | 0.80 | 3.4E-03 | 1.2E-01 | 5.4 2.2E-02 OK|
| Mass-Water | 0.89 | 8.1E-03 | 2.5E-01 | 5.3 4.9E-02 OK|
+----------------------+------+---------+---------+------------------+
| K-TurbKE-Water | 0.93 | 1.1E-02 | 8.5E-02 | 5.5 4.5E-02 OK|
| E-Diss.K-Water | 0.77 | 3.3E-02 | 4.3E-01 | 8.4 8.1E-03 OK|
+----------------------+------+---------+---------+------------------+
======================================================================
OUTER LOOP ITERATION = 9 CPU SECONDS = 4.350E+01
----------------------------------------------------------------------
| Equation | Rate | RMS Res | Max Res | Linear Solution |
+----------------------+------+---------+---------+------------------+
| U-Mom-Air at 25 C | 0.78 | 4.2E-05 | 7.1E-04 | 2.3E-02 OK|
| V-Mom-Air at 25 C | 0.91 | 5.8E-05 | 1.2E-03 | 3.1E-02 OK|
| W-Mom-Air at 25 C | 0.80 | 1.6E-05 | 3.2E-04 | 3.7E-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-Air at 25 C | 0.70 | 2.4E-03 | 5.6E-02 | 5.4 1.1E-02 OK|
| Mass-Water | 0.90 | 7.3E-03 | 1.9E-01 | 5.4 4.5E-02 OK|
+----------------------+------+---------+---------+------------------+
| K-TurbKE-Water | 1.07 | 1.2E-02 | 1.1E-01 | 5.5 2.7E-02 OK|
| E-Diss.K-Water | 0.54 | 1.8E-02 | 2.3E-01 | 8.4 3.9E-03 OK|
+----------------------+------+---------+---------+------------------+
======================================================================
OUTER LOOP ITERATION = 10 CPU SECONDS = 4.835E+01
----------------------------------------------------------------------
| Equation | Rate | RMS Res | Max Res | Linear Solution |
+----------------------+------+---------+---------+------------------+
| U-Mom-Air at 25 C | 0.79 | 3.3E-05 | 6.2E-04 | 2.6E-02 OK|
| V-Mom-Air at 25 C | 0.73 | 4.2E-05 | 8.2E-04 | 4.1E-02 OK|
| W-Mom-Air at 25 C | 0.80 | 1.3E-05 | 3.2E-04 | 3.7E-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-Air at 25 C | 1.00 | 2.4E-03 | 9.1E-02 | 5.5 2.5E-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|
| E-Diss.K-Water | 0.47 | 8.3E-03 | 1.0E-01 | 8.4 2.2E-03 OK|
+----------------------+------+---------+---------+------------------+
======================================================================
OUTER LOOP ITERATION = 11 CPU SECONDS = 5.320E+01
----------------------------------------------------------------------
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| Equation | Rate | RMS Res | Max Res | Linear Solution |
+----------------------+------+---------+---------+------------------+
| U-Mom-Air at 25 C | 0.77 | 2.6E-05 | 4.3E-04 | 1.9E-02 OK|
| V-Mom-Air at 25 C | 0.60 | 2.5E-05 | 5.2E-04 | 4.0E-02 OK|
| W-Mom-Air at 25 C | 0.84 | 1.1E-05 | 3.1E-04 | 2.6E-02 OK|
| 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-Air at 25 C | 1.67 | 3.9E-03 | 1.5E-01 | 5.3 2.9E-02 OK|
| Mass-Water | 1.09 | 7.9E-03 | 2.3E-01 | 5.3 3.1E-02 OK|
+----------------------+------+---------+---------+------------------+
| K-TurbKE-Water | 0.81 | 7.5E-03 | 8.1E-02 | 5.3 1.2E-02 OK|
| E-Diss.K-Water | 0.52 | 4.3E-03 | 4.6E-02 | 8.3 2.1E-03 OK|
+----------------------+------+---------+---------+------------------+
======================================================================
OUTER LOOP ITERATION = 12 CPU SECONDS = 5.808E+01
----------------------------------------------------------------------| Equation | Rate | RMS Res | Max Res | Linear Solution |
+----------------------+------+---------+---------+------------------+
| U-Mom-Air at 25 C | 0.86 | 2.2E-05 | 3.8E-04 | 1.6E-02 OK|
| V-Mom-Air at 25 C | 0.69 | 1.7E-05 | 2.9E-04 | 3.4E-02 OK|
| W-Mom-Air at 25 C | 0.95 | 1.0E-05 | 2.8E-04 | 2.6E-02 OK|
| 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-Air at 25 C | 0.92 | 3.6E-03 | 1.5E-01 | 5.5 1.6E-02 OK|
| Mass-Water | 1.03 | 8.1E-03 | 3.6E-01 | 5.3 2.4E-02 OK|
+----------------------+------+---------+---------+------------------+
| K-TurbKE-Water | 0.79 | 6.0E-03 | 5.3E-02 | 5.5 1.4E-02 OK|
| E-Diss.K-Water | 0.81 | 3.5E-03 | 5.2E-02 | 8.4 3.0E-03 OK|
+----------------------+------+---------+---------+------------------+
======================================================================
OUTER LOOP ITERATION = 13 CPU SECONDS = 6.292E+01
----------------------------------------------------------------------
| Equation | Rate | RMS Res | Max Res | Linear Solution |
+----------------------+------+---------+---------+------------------+
| U-Mom-Air at 25 C | 0.98 | 2.2E-05 | 4.0E-04 | 2.1E-02 OK|
| V-Mom-Air at 25 C | 1.05 | 1.8E-05 | 3.3E-04 | 4.5E-02 OK|
| W-Mom-Air at 25 C | 0.97 | 1.0E-05 | 2.3E-04 | 3.6E-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-Air at 25 C | 0.79 | 2.9E-03 | 1.0E-01 | 5.5 1.7E-02 OK|
| Mass-Water | 0.79 | 6.5E-03 | 3.2E-01 | 5.4 4.2E-02 OK|
+----------------------+------+---------+---------+------------------+
| K-TurbKE-Water | 0.72 | 4.3E-03 | 4.3E-02 | 5.4 2.0E-02 OK|
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| E-Diss.K-Water | 0.87 | 3.0E-03 | 6.1E-02 | 8.4 3.1E-03 OK|
+----------------------+------+---------+---------+------------------+
======================================================================
OUTER LOOP ITERATION = 14 CPU SECONDS = 6.782E+01
----------------------------------------------------------------------
| Equation | Rate | RMS Res | Max Res | Linear Solution |
+----------------------+------+---------+---------+------------------+
| U-Mom-Air at 25 C | 0.92 | 2.0E-05 | 4.0E-04 | 1.4E-02 OK|
| V-Mom-Air at 25 C | 1.03 | 1.9E-05 | 3.6E-04 | 3.1E-02 OK|
| W-Mom-Air at 25 C | 0.88 | 8.8E-06 | 2.1E-04 | 2.7E-02 OK|
| 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-Air at 25 C | 0.75 | 2.2E-03 | 7.4E-02 | 5.5 3.4E-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|
+----------------------+------+---------+---------+------------------+
======================================================================
OUTER LOOP ITERATION = 15 CPU SECONDS = 7.280E+01
----------------------------------------------------------------------
| Equation | Rate | RMS Res | Max Res | Linear Solution |
+----------------------+------+---------+---------+------------------+
| U-Mom-Air at 25 C | 0.79 | 1.6E-05 | 3.1E-04 | 1.4E-02 OK|
| V-Mom-Air at 25 C | 0.85 | 1.6E-05 | 3.4E-04 | 3.6E-02 OK|
| W-Mom-Air at 25 C | 0.93 | 8.2E-06 | 2.5E-04 | 2.5E-02 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-Air at 25 C | 0.89 | 1.9E-03 | 4.9E-02 | 5.5 3.6E-02 OK|
| Mass-Water | 0.86 | 3.9E-03 | 1.1E-01 | 5.3 5.7E-02 OK|
+----------------------+------+---------+---------+------------------+
| K-TurbKE-Water | 0.82 | 2.8E-03 | 3.5E-02 | 5.4 2.9E-02 OK|
| E-Diss.K-Water | 0.99 | 3.0E-03 | 5.3E-02 | 8.3 4.7E-03 OK|
+----------------------+------+---------+---------+------------------+
======================================================================
OUTER LOOP ITERATION = 16 CPU SECONDS = 7.783E+01
----------------------------------------------------------------------
| Equation | Rate | RMS Res | Max Res | Linear Solution |+----------------------+------+---------+---------+------------------+
| U-Mom-Air at 25 C | 1.09 | 1.7E-05 | 3.0E-04 | 2.0E-02 OK|
| V-Mom-Air at 25 C | 0.81 | 1.3E-05 | 3.1E-04 | 6.6E-02 OK|
| W-Mom-Air at 25 C | 1.16 | 9.4E-06 | 3.2E-04 | 3.0E-02 OK|
| 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-Air at 25 C | 1.04 | 2.0E-03 | 4.7E-02 | 5.5 3.1E-02 OK|
| Mass-Water | 0.93 | 3.7E-03 | 8.6E-02 | 5.3 5.2E-02 OK|
+----------------------+------+---------+---------+------------------+
| K-TurbKE-Water | 0.90 | 2.5E-03 | 4.6E-02 | 5.4 2.7E-02 OK|
| E-Diss.K-Water | 0.89 | 2.7E-03 | 5.3E-02 | 8.3 4.3E-03 OK|
+----------------------+------+---------+---------+------------------+
======================================================================
OUTER LOOP ITERATION = 17 CPU SECONDS = 8.260E+01
----------------------------------------------------------------------
| Equation | Rate | RMS Res | Max Res | Linear Solution |
+----------------------+------+---------+---------+------------------+
| U-Mom-Air at 25 C | 1.17 | 2.0E-05 | 4.4E-04 | 1.4E-02 OK|
| V-Mom-Air at 25 C | 0.98 | 1.3E-05 | 2.6E-04 | 6.3E-02 OK|
| W-Mom-Air at 25 C | 1.16 | 1.1E-05 | 4.0E-04 | 2.0E-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-Air at 25 C | 0.92 | 1.8E-03 | 4.3E-02 | 5.5 2.5E-02 OK|
| Mass-Water | 0.92 | 3.3E-03 | 8.1E-02 | 5.4 4.9E-02 OK|
+----------------------+------+---------+---------+------------------+
| K-TurbKE-Water | 1.04 | 2.6E-03 | 5.2E-02 | 5.3 2.7E-02 OK|
| E-Diss.K-Water | 0.97 | 2.6E-03 | 4.7E-02 | 8.3 3.9E-03 OK|
+----------------------+------+---------+---------+------------------+
======================================================================
OUTER LOOP ITERATION = 18 CPU SECONDS = 8.749E+01
----------------------------------------------------------------------
| Equation | Rate | RMS Res | Max Res | Linear Solution |
+----------------------+------+---------+---------+------------------+
| U-Mom-Air at 25 C | 1.09 | 2.2E-05 | 5.6E-04 | 1.1E-02 OK|
| V-Mom-Air at 25 C | 1.15 | 1.5E-05 | 2.7E-04 | 5.3E-02 OK|
| W-Mom-Air at 25 C | 1.07 | 1.2E-05 | 4.3E-04 | 1.5E-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-Air at 25 C | 0.92 | 1.7E-03 | 4.0E-02 | 5.4 2.8E-02 OK|
| Mass-Water | 0.90 | 3.0E-03 | 7.0E-02 | 5.3 4.9E-02 OK|
+----------------------+------+---------+---------+------------------+
| K-TurbKE-Water | 0.99 | 2.6E-03 | 5.0E-02 | 5.3 2.8E-02 OK|
| E-Diss.K-Water | 0.90 | 2.3E-03 | 5.3E-02 | 8.4 3.8E-03 OK|+----------------------+------+---------+---------+------------------+
======================================================================
OUTER LOOP ITERATION = 19 CPU SECONDS = 9.233E+01
----------------------------------------------------------------------
| Equation | Rate | RMS Res | Max Res | Linear Solution |
+----------------------+------+---------+---------+------------------+
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| U-Mom-Air at 25 C | 1.02 | 2.2E-05 | 6.4E-04 | 1.1E-02 OK|
| V-Mom-Air at 25 C | 1.12 | 1.7E-05 | 3.2E-04 | 4.4E-02 OK|
| W-Mom-Air at 25 C | 0.99 | 1.2E-05 | 4.1E-04 | 1.3E-02 OK|
| 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-Air at 25 C | 0.91 | 1.5E-03 | 3.3E-02 | 5.5 3.0E-02 OK|
| Mass-Water | 0.92 | 2.8E-03 | 5.8E-02 | 5.5 4.9E-02 OK|
+----------------------+------+---------+---------+------------------+
| K-TurbKE-Water | 0.96 | 2.5E-03 | 4.3E-02 | 5.3 2.8E-02 OK|
| E-Diss.K-