Modeling Multiphase Flows

1
Modeling Multiphase Flows
2
Outline

• Definitions Examples of flow regimes
• Description of multiphase models in FLUENT 5 and
  FLUENT 4.5
• How to choose the correct model for your
  application
• Summary and guidelines

3
Definitions

• Multiphase flow is simultaneous flow of
• Matters with different phases( i.e. gas, liquid
  or solid).
• Matters with different chemical substances but
  with the same phase (i.e. liquid-liquid like
  oil-water).
• Primary and secondary phases
• One of the phases is considered continuous
  (primary) and others (secondary) are considered
  to be dispersed within the continuous phase.
• A diameter has to be assigned for each secondary
  phase to calculate its interaction (drag) with
  the primary phase (except for VOF model).
• Dilute phase vs. Dense phase
• Refers to the volume fraction of secondary
  phase(s)
• Volume fraction of a phase

Volume of the phase in a cell/domain Volume of
the cell/domain
4
Flow Regimes

• Multiphase flow can be classified by the
  following regimes
• Bubbly flow Discrete gaseous or fluid bubbles in
  a continuous fluid
• Droplet flow Discrete fluid droplets in a
  continuous gas
• Particle-laden flow Discrete solid particles in
  a continuous fluid
• Slug flow Large bubbles (nearly filling
  cross-section) in a continuous fluid
• Annular flow Continuous fluid along walls, gas
  in center
• Stratified/free-surface flow Immiscible fluids
  separated by a clearly-defined interface

free-surface flow
5
Flow Regimes

• User must know a priori what the flow field looks
  like
• Flow regime,
• bubbly flow , slug flow, etc.
• Model one flow regime at a time.
• Multiple flow regime can be predicted if they are
  predicted by one model e.g. slug flow and annular
  flow may coexist since both are predicted by VOF
  model.
• turbulent or laminar,
• dilute or dense,
• bubble or particle diameter (mainly for drag
  considerations).

6
Multiphase Models

• Four models for multiphase flows currently
  available in structured FLUENT 4.5
• Lagrangian dispersed phase model (DPM)
• Eulerian Eulerian model
• Eulerian Granular model
• Volume of fluid (VOF) model
• Unstructured FLUENT 5
• Lagrangian dispersed phase model (DPM)
• Volume of fluid model (VOF)
• Algebraic Slip Mixture Model (ASMM)
• Cavitation Model

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Dispersed Phase Model
8
Dispersed Phase Model

• Appropriate for modeling particles, droplets, or
  bubbles dispersed (at low volume fraction less
  than 10) in continuous fluid phase
• Spray dryers
• Coal and liquid fuel combustion
• Some particle-laden flows
• Computes trajectories of particle (or droplet or
  bubble) streams in continuous phase.
• Computes heat, mass, and momentum transfer
  between dispersed and continuous phases.
• Neglects particle-particle interaction.
• Particles loading can be as high as fluid loading
• Computes steady and unsteady (FLUENT 5) particle
  tracks.

Particle trajectories in a spray dryer
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Particle Trajectory Calculations

• Particle trajectories computed by solving
  equations of motion of the particle in Lagrangian
  reference frame
• where represents additional forces due to
• virtual mass and pressure gradients
• rotating reference frames
• temperature gradients
• Brownian motion (FLUENT 5)
• Saffman lift (FLUENT 5)
• user defined

10
Coupling Between Phases

• One-Way Coupling
• Fluid phase influences particulate phase via drag
  and turbulence transfer.
• Particulate phase have no influence on the gas
  phase.
• Two-Way Coupling
• Fluid phase influences particulate phase via drag
  and turbulence transfer.
• Particulate phase influences fluid phase via
  source terms of mass, momentum, and energy.
• Examples include
• Inert particle heating and cooling
• Droplet evaporation
• Droplet boiling
• Devolatilization
• Surface combustion

11
DPM Calculation Procedure

• To determine impact of dispersed phase on
  continuous phase flow field, coupled calculation
  procedure is used
• Procedure is repeated until both flow fields are
  unchanged.

continuous phase flow field calculation
interphase heat, mass, and momentum exchange
particle trajectory calculation
12
Turbulent Dispersion of Particles

• Dispersion of particle due to turbulent
  fluctuations in the flow can be modeled using
  either
• Discrete Random Walk Tracking (stochastic
  approach)
• Particle Cloud Tracking

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User Defined Function Access in DPM

• User defined functions (UDFs) are provided for
  access to the discrete phase model. Functions
  are provided for user defined
• drag
• external force
• laws for reacting particles and droplets
• customized switching between laws
• output for sample planes
• erosion/accretion rates
• access to particle definition at injection time
• scalars associated with each particle and access
  at each particle time step (possible to integrate
  scalar variables over life of particle)

FLUENT 5
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Eulerian-Eulerian Multiphase ModelFLUENT 4.5
10s
70s
120s

Becker et al. 1992
water
air
Locally Aerated Bubble Column
15
Eulerian Multiphase Model

• Appropriate for modeling gas-liquid or
  liquid-liquid flows (droplets or bubbles of
  secondary phase(s) dispersed in continuous fluid
  phase (primary phase)) where
• Phases mix or separate
• Bubble/droplet volume fractions from 0 to 100
• Evaporation
• Boiling
• Separators
• Aeration
• Inappropriate for modeling stratified or
  free-surface flows.

Volume fraction of water
Stream function contours for water
Boiling water in a container
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Eulerian Multiphase Model

• Solves momentum, enthalpy, continuity, and
  species equations for each phase and tracks
  volume fractions.
• Uses a single pressure field for all phases.
• Interaction between mean flow field of phases is
  expressed in terms of a drag, virtual and lift
  forces.
• Several formulations for drag is provided.
• Alternative drag laws can be formulated via UDS.
• Other forces can be applied through UDS.

Gas sparger in a mixing tank contours of volume
fraction with velocity vectors
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Eulerian Multiphase Model

• Can solve for multiple species and homogeneous
  reactions in each phase.
• Heterogeneous reactions can be done through UDS.
• Allows for heat and mass transfer between phases.
• Turbulence models for dilute and dense phase
  regimes.

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Mass Transfer

• Evaporation/Condensation.
• For liquid temperatures ? saturation temperature,
  evaporation rate
• For vapor temperatures ? saturation temperature,
  condensation rate
• User specifies saturation temperature and, if
  desired, time relaxation parameters rl and rv .
  (Wen Ho Lee (1979))
• Unidirectional mass transfer, is constant
  
• User Defined Subroutine for mass transfer

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Eulerian Multiphase Model Turbulence

• Time averaging is needed to obtain smoothed
  quantities from the space averaged instantaneous
  equations.
• Two methods available for modeling turbulence in
  multiphase flows within context of standard k-e
  model
• Dispersed turbulence model (default) appropriate
  when both of these conditions are met
• Number of phases is limited to two
• Continuous (primary) phase
• Dispersed (secondary) phase
• Secondary phase must be dilute.
• Secondary turbulence model appropriate for
  turbulent multiphase flows involving more than
  two phases or a non-dilute secondary phase.
• Choice of model depends on importance of
  secondary-phase turbulence in your application.

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Eulerian Granular Multiphase Model FLUENT 4.5
Volume fraction of air 2D fluidized bed with a
central jet
21
Eulerian Granular Multiphase Model

• Extension of Eulerian-Eulerian model for flow of
  granular particles (secondary phases) in a fluid
  (primary)phase
• Appropriate for modeling
• Fluidized beds
• Risers
• Pneumatic lines
• Hoppers, standpipes
• Particle-laden flows in which
• Phases mix or separate
• Granular volume fractions can vary from 0 to
  packing limit

Solid velocity profiles
Contours of solid volume fraction
Circulating fluidized bed, Tsuo and Gidaspow
(1990).
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Eulerian Granular Multiphase Model Overview

• The fluid phase must be assigned as the primary
  phase.
• Multiple solid phase can be used to represent
  size distribution.
• Can calculate granular temperature (solids
  fluctuating energy) for each solid phase.
• Calculates a solids pressure field for each solid
  phase.
• All phases share fluid pressure field.
• Solids pressure controls the solids packing limit
• Solids pressure, granular temperature
  conductivity, shear and bulk viscosity can be
  derived based on several kinetic theory
  formulations.
• Gidaspow -good for dense fluidized bed
  applications
• Syamlal -good for a wide range of applications
• Sinclair -good for dilute and dense pneumatic
  transport lines and risers

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Eulerian Granular Multiphase Model

• Frictional viscosity pushes the limit into the
  plastic regime.
• Hoppers, standpipes
• Several choice of drag laws
• Drag laws can be modified using UDS.
• Heat transfer between phases is the same as in
  Eulerian/Eulerian multiphase model.
• Only unidirectional mass transfer model is
  available.
• Rate of mass transfer can be modified using UDS.
• Homogeneous reaction can be modeled.
• Heterogeneous reaction can be modeled using UDS.
• Can solve for enthalpy and multiple species for
  each phase.
• Physically based models for solid momentum and
  granular temperature boundary conditions at the
  wall.
• Turbulence treatment is the same as in
  Eulerian-Eulerian model
• Sinclair model provides additional turbulence
  model for solid phase

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Algebraic Slip Mixture ModelFLUENT 5
Courtesy of Fuller Company
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Algebraic Slip Mixture Model

• Can substitute for Eulerian/Eulerian,
  Eulerian/Granular and Dispersed phase models
  Efficiently for Two phase flow problems
• Fluid/fluid separation or mixing
• Sedimentation of uniform size particles in
  liquid.
• Flow of single size particles in a Cyclone.
• Applicable to relatively small particles
  (lt50 microns) and low volume fraction (lt10) when
  primary phase density is much smaller than the
  secondary phase density.

Air-water separation in a Tee junction Water
volume fraction

• If possible, always choose the fluid with higher
  density as the primary phase.

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ASMM

• Solves for the momentum and the continuity
  equations of the mixture.
• Solves for the transport of volume fraction of
  secondary phase.
• Uses an algebraic relation to calculate the slip
  velocity between phases.
• It can be used for steady and unsteady
  flow. is the drag function

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Oil-Water Separation
Fluent 5 Results with ASMM
Fluent v4.5 Eulerian Multiphase
Courtesy of Arco Exploration Production
Technology Dr. Martin de Tezanos Pinto
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Cavitation Model ( Fluent 5)

• Predicts cavitation inception and approximate
  extension of cavity bubble.
• Solves for the momentum equation of the mixture
• Solves for the continuity equation of the
  mixture
• Assumes no slip velocity between the phases
• Solves for the transport of volume fraction of
  vapor phase.
• Approximates the growth of the cavitation bubble
  using Rayleigh equation
• Needs improvement
• ability to predict collapse of cavity bubbles
• Needs to solve for enthalpy equation and
  thermodynamic properties
• Solve for change of bubble size

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Cavitation model
30
VOF Model
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Volume of Fluid Model

• Appropriate for flow where Immiscible fluids have
  a clearly defined interface.
• Shape of the interface is of interest
• Typical problems
• Jet breakup
• Motion of large bubbles in a liquid
• Motion of liquid after a dam break (shown at
  right)
• Steady or transient tracking of any liquid-gas
  interface
• Inappropriate for
• Flows involving small (compared to a control
  volume) bubbles
• Bubble columns

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Volume Fraction

• Assumes that each control volume contains just
  one phase (or the interface between phases).
• For volume fraction of kth fluid, three
  conditions are possible
• ?k 0 if cell is empty (of the kth fluid)
• ?k 1 if cell is full (of the kth fluid)
• 0 lt ?k lt 1 if cell contains the interface
  between the fluids
• Tracking of interface(s) between phases is
  accomplished by solution of a volume fraction
  continuity equation for each phase
• Mass transfer between phases can be modeled by
  using a user-defined subroutine to specify a
  nonzero value for S?k .
• Multiple interfaces can be simulated
• Can not resolve details of the interface smaller
  than the mesh size

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VOF

• Solves one set of momentum equations for all
  fluids.
• Surface tension and wall adhesion modeled with an
  additional source term in momentum eqn.
• For turbulent flows, single set of turbulence
  transport equations solved.
• Solves for species conservation equations for
  primary phase .

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Formulations of VOF Model

• Time-dependent with a explicit schemes
• geometric linear slope reconstruction (default in
  FLUENT 5)
• Donor-acceptor (default in FLUENT 4.5)
• Best scheme for highly skewed hex mesh.
• Euler explicit
• Use for highly skewed hex cells in hybrid meshes
  if default scheme fails.
• Use higher order discretization scheme for more
  accuracy.
• Example jet breakup
• Time-dependent with implicit scheme
• Used to compute steady-state solution when
  intermediate solution is not important.
• More accurate with higher discretization scheme.
• Final steady-state solution is dependent on
  initial flow conditions
• There is not a distinct inflow boundary for each
  phase
• Example shape of liquid interface in centrifuge
• Steady-state with implicit scheme
• Used to compute steady-state solution using
  steady-state method.
• More accurate with higher order discretization
  scheme.
• Must have distinct inflow boundary for each phase
• Example flow around ships hull

Decreasing Accuracy
35

Comparison of Different Front Tracking Algorithms
2nd order upwind
Donor - Acceptor
Geometric reconstruction with tri mesh
Geometric reconstruction
36
Surface Tension

• Cylinder of water (5 x 1 cm) is surrounded by air
  in no gravity
• Surface is initially perturbed so that the
  diameter is 5 larger on ends
• The disturbance at the surface grows because of
  surface tension

37
Wall Adhesion

• Wall adhesion is modeled by specification of
  contact angle that fluid makes with wall.
• Large contact angle (gt 90) is applied to water
  at bottom of container in zero-gravity field.
• An obtuse angle, as measured in water, will form
  at walls.
• As water tries to satisfy contact angle
  condition, it detaches from bottom and moves
  slowly upward, forming a bubble.

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Choosing a Multiphase Model Fluid-Fluid Flows
(1)

• Bubbly flow examples
• Absorbers
• Evaporators
• Scrubbers
• Air lift pumps
• Droplet flow examples
• Atomizers
• Gas cooling
• Dryers
• Slug flow examples
• Large bubble motion in pipes or tanks
• Separated flows
• free surface, annular flows, stratified flows,
  liquid films

• Cavitation
• Flotation
• Aeration
• Nuclear reactors

• Combustors
• Scrubbers
• Cryogenic pumping

39
Choosing a Multiphase Model Gas-Liquid Flows (2)
40
Choosing a Multiphase Model Particle-Laden Flow

• Examples
• Cyclones
• Slurry transport
• Flotation
• Circulating bed reactors

• Dust collectors
• Sedimentation
• Suspension
• Fluidized bed reactors

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Solution Guidelines

• All multiphase calculations
• Start with a single-phase calculation to
  establish broad flow patterns.
• Eulerian multiphase calculations
• Use COPY-PHASE-VELOCITIES to copy primary phase
  velocities to secondary phases.
• Patch secondary volume fraction(s) as an initial
  condition.
• For a single outflow, use OUTLET rather than
  PRESSURE-INLET for multiple outflow boundaries,
  must use PRESSURE-INLET for each.
• For circulating fluidized beds, avoid symmetry
  planes. (They promote unphysical cluster
  formation.)
• Set the false time step for underrelaxation to
  0.001
• Set normalizing density equal to physical density
• Compute a transient solution

42
Solution Strategies (VOF)

• For explicit formulations for best and quick
  results
• use geometric reconstruction or donor-acceptor
• use PISO algorithm with under-relaxation factors
  up to 1.0
• reduce time step if convergence problem arises.
• To ensure continuity, reduce termination criteria
  to 0.001 for pressure in multi-grid solver
• solve VOF once per time-step
• For implicit formulations
• always use QUICK or second order upwind
  difference scheme for VOF equation.
• may increase VOF UNDER-RELAXATION from 0.2
  (default ) to 0.5.
• Use proper reference density to prevent round off
  errors.
• Use proper pressure interpolation scheme for
  hydrostatic consideration
• Body force weighted scheme for all types of cells
• PRESTO (only for quads and hexes)

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Summary

• Modeling multiphase flows is very complex, due to
  interdependence of many variables.
• Accuracy of results directly related to
  appropriateness of model you choose
• For most applications with low volume fraction of
  particles, droplets, or bubbles, use ASMM or DPM
  model .
• For particle-laden flows, Eulerian granular
  multiphase model is best.
• For separated gas-liquid flows (stratified,
  free-surface, etc.) VOF model is best.
• For general, complex gas-liquid flows involving
  multiple flow regimes
• Select aspect of flow that is of most interest.
• Choose model that is most appropriate.
• Accuracy of results will not be as good as for
  others, since selected physical model will be
  valid only for some flow regimes.

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Conservation equations

• Conservation of mass
• Conservation of momentum
• Conservation of enthalpy

45
Constitutive Equations

• Frictional Flow
• Particles are in enduring contact and momentum
  transfer is through friction
• Stresses from soil mechanics, Schaeffer (1987)
• Description of frictional viscosity
• is the second invariant of the deviatoric
  stress tensor

46
Interphase Forces (cont.)

• Virtual Mass Effect caused by relative
  acceleration between phases Drew and Lahey
  (1990).
• Virtual mass effect is significant when the
  second phase density is much smaller than the
  primary phase density (i.e., bubble column)
• Lift Force Caused by the shearing effect of the
  fluid onto the particle Drew and Lahey (1990).
• Lift force usually insignificant compared to drag
  force except when the phases separate quickly and
  near boundaries

47
Eulerian Multiphase Model Turbulence

• The transport equations for the model
  are of the form
• Value of the parameters

48
Comparison of Drag Laws
Relative Reynolds number 1 and 1000 Particle
diameter 0.001 mm
Arastoopour
Arastoopour
49
Drag Force Models
Schiller and Naumann
Schuh et al.
Morsi and Alexander
50
Solution Algorithms for Multiphase Flows
Only Eulerian/Eulerian model

• Coupled solver algorithms (more coupling between
  phases)
• Faster turn around and more stable numerics
• High order discretization schemes for all phases.
• More accurate results

51
Heterogeneous Reactions in FLUENT4.5

• Problem Description
• Two liquid e.g. (L1,L2) react and make solids
  e.g. (s1,s2)
• Reactions happen within liquid e.g. (L1--gtL2)
• Reactions happen within solid e.g. (s1---gts2)
• Solution!
• Consider a two phase liquid (primary) and solid
  (secondary)
• liquid has two species L1, L2
• solid has two species s1,s2
• Reactions within each phase i.e. (L1--gtL2) and
  (s1--gts2) can be set up as usual through GUI
  (like in single phase)
• For heterogeneous reaction e.g.
  (L10.5L2--gt0.2s1s2)

52
Heterogeneous Reactions in FLUENT 4.5

• In usrmst.F
• calculate the net mass transfer between phases as
  a result of reactions
• Reactions could be two ways
• Assign this value to suterm
• If the net mass transfer is from primary to
  secondary the value should be negative and vica
  versa.
• The time step and mass transfer rate should be
  such that the net volume fraction change would
  not be more than 5-10.
• In urstrm.F
• Adjust the mass fraction of each species by
  assigning a source or sink value (/-) according
  to mass transfer calculated above.
• Adjust the enthalp of each phase by the net
  amount of heat of reactions and enthalpy transfer
  due to mass transfer. Again this will be in a
  form of a source term.

53
Heterogeneous Reactions in FLUENT 4.5

• Compile your version of the code
• Run Fluent and set up the case
• Enable time dependent, multiphase, temperature
  and species calculations.
• Define phases
• Enable mass transfer and multi-component
  multi-species option.
• Define species, homogeneous reactions within each
  phases
• Define properties
• Enable user defined mass transfer
• GOOD LUCK!!

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Particle size

• Descriptive terms Size range Example
• Coarse solid 5 - 100 mm coal
• Granular solid 0.3 - 5 mm sugar
• Coarse powder 100-300 mm salt, sand
• Fine powder 10-100 mm FCC catalyst
• Super fine powder 1-10 mm face powder
• Ultra fine powder 1 mm paint pigments
• Nano Particles 1e-3 mm molecules

55
Discrete Random Walk Tracking

• Each injection is tracked repeatedly in order to
  generate a statistically meaningful sampling.
• Turbulent fluctuation in the flow field are
  represented by defining an instantaneous fluid
  velocity
• where is derived from the local
  turbulence parameters
• and is a normally distributed random
  number
• Mass flow rates and exchange source terms for
  each injection are divided equally among the
  multiple stochastic tracks.

56
Cloud Tracking

• The particle cloud model uses statistical methods
  to trace the turbulent dispersion of particles
  about a mean trajectory. The mean trajectory is
  calculated from the ensemble average of the
  equations of motion for the particles represented
  in the cloud. The distribution of particles
  inside the cloud is represented by a Gaussian
  probability density function.

57
Stochastic vs. Cloud Tracking

• Stochastic tracking
• Accounts for local variations in flow properties
  such as temperature, velocity, and species
  concentrations.
• Requires a large number of stochastic tries in
  order to achieve a statistically significant
  sampling (function of grid density).
• Insufficient number of stochastic tries results
  in convergence problems and non-smooth particle
  concentrations and coupling source term
  distributions.
• Recommended for use in complex geometry
• Cloud tracking
• Local variations in flow properties (e.g.
  temperature) get averaged away inside the
  particle cloud.
• Smooth distributions of particle concentrations
  and coupling source terms.
• Each diameter size requires its own cloud
  trajectory calculation.

58
Granular Flow Regimes

• Elastic Regime Plastic Regime Viscous Regime
• Stagnant Slow flow Rapid flow
• Stress is strain Strain rate Strain rate
  dependent independent dependent
• Elasticity Soil mechanics Kinetic theory

59
Flow regimes
60
Eulerian Multiphase Model Heat Transfer

• Rate of energy transfer between phases is
  function of temperature difference between
  phases
• Hpq ( Hqp) is heat transfer coefficient between
  pth phase and qth phase.
• Can be modified using UDS.

Boiling water in a container contours of water
temperature
61
Sample Planes and Particle Histograms

• As particles pass through sample planes (lines in
  2-D), their properties (position, velocity, etc.)
  are written to files. These files can then be
  read into the histogram plotting tool to plot
  histograms of residence time and distributions of
  particle properties. The particle property mean
  and standard deviation are also reported.