CFD Modeling of Multiphase Flow. Focus on Liquid-Solid Flow

FACE/IFE Multiphase Flow Workshop
13-14 November, 2013
Lecturer: Luis R. Rojas-Solórzano, Ph.D.
Luis Rojas-Solórzano, Ph.D.
Associate Professor, School of Engineering
Nazarbayev University
Astana, Rep. of Kazakhstan
E-mail: luis.rojas@nu.edu.kz
Kjeller, Norway
November 13rd , 2013
The Multiphase Flow Assurance Centre
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OUTLINE
1. Introduction to Flow Assurance and Motivation.
2. Multiphase flows. Approaches to solution. Generalities.
3. Slurry flows. Antecedents.
4. Modeling Slurry Flows using CFD. Theory of some models
available in CFD tools (specifically with emphasis in ANSYS).
5. Case Study: Liquid-Particles Flow in Horizontal Pipe, using
ANSYS-Fluent.
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1. Flow Assurance
 Many large slurry pipelines are currently operating
around the world (mining, oil, energy, etc.).
 In the oil industry the transport of particles is a
common outcome of its operation (cementing,
drilling and oil production along with hydrates
dispersions, rock and sand particles).
 On this regard, flow assurance is a term used in
the O&G industry referring to ensure the
hydrocarbon flow from reservoir to delivery in a
safety, environmentally friendly and economical
way.
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Matoušek V., (2005), Hydraulic Transport of Settling Slurries in Pipelines – Practical Application and Optimization of Pipeline Operation.

Introduction and motivation
 For flow assurance, O&G pipeline
designers need accurate information
regarding pressure drop, and velocity-
concentration profiles in slurry flows.
 The flow assurance guarantees a better
selection of pumps, optimization of power
consumption and reduction of secondary
effects, such as wall abrasion.
 Today Lecture: Modeling Slurry Flows using
CFD (Special focus on ANSYS tools).
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Lahiri S.K and Ghanta K.C , Prediction of pressure drop and concentration profile by semi empirical correlations-modification of
Wasp model, under review, Chemical engineering progress(2009)

2. Multiphase flow
 Refers to flows where there are more than one
immiscible fluid or at least one fluid and one solid
phase involved.
 The term “phase” has a broader mean than usual in
thermodynamics. “Phase” in this context refers to
different chemical species; e.g., air, oil, water, sand,
etc.
 It may or may not contain multi-components
capable to mix at a molecular level. These
components diffusion may be modeled using Fick’s
law.
 Multiphase flows tend to mix at macroscopic levels;
e.g., bubbles in water or droplets in gas. Each
phase has its own velocity and temperature field.
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Based on Alan Burns (CFX, Europe-Harwell) notes on: Computational Fluid Dynamics Modeling of Multiphase Flows
GLCC Experiments vs. CFD
CEMFA-USB, 2004
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Approaches to modeling Multiphase flows
 Semi-analytical: complexity difficults analytical close solution. Therefore,
mechanistic closure models are proposed in semi-empirical one-
dimensional solutions to liquid/liquid and liquid-gas flows.
 Mechanistic models as opposed to phenomenological-statistical models*, come
from lumped conservation equations which are used to fit experimental
observations.
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(*) Based on finding the best fit between data
Source: Sarica C., et al., 2011. “Sensitivity of Slug Flow Mechanistic Models on Slug Length”, J. Energy Resour. Tech., 133(4)
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 Semi-analytical
 Mechanistic models begin by assuming a
particular flow regime or pattern FP
(validated by experimental observations).
Then stability analysis of the flow pattern is
assessed.
 If FP is stable, the procedure ends.
 If FP is not stable under given conditions,
a new FP is assumed and procedure
repeats until a new stable is found.
 Procedure repeated over range of Qg/Ql
to construct FP map.
 Ultimate objective: Pressure drop and
Liquid hold-up.
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A. Barns course .mp
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 Semi-analytical
 Mechanistic models. Transitions:
 Large velocity at interface causes wave that may become
unstable. Eventually, liquid bridges across pipe section.
 Causes transition to Slug Flow or Annular flow, according to
stratified liquid hold-up (hl/D < 0.5  Annular Flow;
hl/D > 0.5  Slug Flow).
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A. Barns course .mp
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 Semi-analytical
 Mechanistic models. Transitions:
Intermittent-to-Disperse Bubble Transition
 Governed by breakup of large slug bubbles by turbulence.
 Occurs when turbulent fluctuations overcome buoyancy that
keeps large plug bubbles on top of pipe. Taitel [132]
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(*) Based on finding the best fit between data
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 Semi-analytical. Experimental data supporting Mechanistic
models
 Similar transitions maps have been developed for inclined and
vertical pipelines (e.g. oil-water):
 Transitions governed by Helmholtz
instability  breaking waves.
 In general, experimentalists aim to
universal flow regime maps. But,
not single pair of non-dimensional
parameters suffices (Taitel et al. [132], [133])
 Taitel and Dukler [133] found 5 non-dimensional parameters to
build a generalize flow regime map, but still not very accurate.
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 Semi-analytical. Experimental data supporting
Mechanistic models
 Empirical Transitions map for horizontal pipes:
 Maps depend on fluid properties and pide diameter
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Valid only for Air-Water at STP
2.5 cm pipe diameter
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 Numerical: two approaches according to Reference Framework
 Eulerian-Lagrangian Statistical framework (EL-FW).
 Very intuitive. Tracks the motion of individual particles, droplets or
bubbles. Allows different velocities for separate particles at same cell
location (as opposed to EE).
 Continuous phase is treated via E-FW and disperse phase is treated
via L-FW.
 Interphase momentum, mass and heat transfer may occur between
particles and continuous phase. One-way or two-way coupling
between particles and surrounding fluid.
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 Eulerian-Lagrangian Statistical framework (EL-FW). (cont´d)
 Particle collisions are treated statistically.
 Requires a particle-sensitivity assessment, that many times may lead
to 10k´s or 100k´s particles (in very dense particles distribution it´s
unpractical), wich require large time of computation and analysis.
 Better than EE in unsteady-transient analysis, but accuracy level is
comparable to EE in steady analysis, despite its (EL) higher cost(**).
 EL is not very well suited for parallel computing since it requires the
coupling of two different solvers.
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(**) Zhang, Z., Chen, Q., 2007. Comparison of the Eulerian and Lagrangian methods for predicting
particle transport in enclosed spaces. Atmospheric Environment 41(25), 5236-5248.
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 Eulerian-Direct Numerical Lagrangian or Discrete Element Method
(DEM). Has become a very accepted technique to model the motion
of large number of particles. DEM theory is close to molecular
dynamics.
 Considers complex particle geometries and rotational degrees-of-
freedom. Accounts for frictional, gravity and inelastic-contact particle-
particle and particle-wall forces (main difference with Lagrangian
Multiphase), as well as other attractive potential and molecular forces
(e.g., droplet-droplet bridging).
 Adds all forces on each particle and enforces Newton´s 2nd law. The
position and velocity of each particle is updated every time-step using
a suitable integration algorithm.
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 Eulerian-Direct Numerical Lagrangian or Discrete Element Method
(DEM). (cont´d)
 Time-stepping uses closest-neighbor-sorting algorithm to reduce
possible contact pairs and thus, the computational needs.
 Widely accepted for granular and powder flows.
 Is very computational intensive, as typical real granular flows may
involve 1M´s of particles. It may limit length of simulaton or number of
particles to be considered.
 Allows a detailed study of particle micro-dynamics.
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(cont´d)
 Eulerian-Homogeneous Eulerian framework (EHE-FW)
 Homogeneous phases.
 Phases are either completely mixed or are totally separated, except at the
interphase.
 May be treated as one-phase and interphase tracking (truly VOF, i.e.,
Fluent, Star-CD and Flow-3D) or multiphase with interphase compressive
algorithm (i.e., CFX). Surface tension may be included as a body force
acting on the interface.
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(cont´d)
 Eulerian-Homogeneous Eulerian framework (EHE-FW) (cont´d)
 Homogeneous phases.
 In multiphase model, a single velocity, pressure and temperature fields are
assumed. However, the volume fractions ¨r¨ are either 1 or 0 everywhere,
except on cells at interface. No interphase momentum transfer is
considered as both phases are considered always at same velocity at
similar cells.
 Only 1 momentum equation is used. In multiphase model, the properties
are weighted by the volume fraction of each phase (i.e., Density = (rα.ρα+
rβ.ρβ)).
 Ideal for free-surface, wavy-stratified regimes.
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(cont´d)
 Eulerian-Disperse Eulerian framework (EDE-FW)
 Continuous and disperse phases are treated as a inter-penetrating
continua.
 Phases are mixed on length-scales larger than molecular, but smaller than
grid resolution; hence two phases may co-exist in a given control volume,
as result of an averaging procedure.
 However, reduced vision of 3D phenomena in complex geometries or
under non-equilibrium conditions.
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(cont´d)
 Eulerian-Disperse Eulerian framework (EDE-FW) (cont´d)
 1 momentum equation/phase and interphase momentum transfer is used
to account for interaction between disperse phase(s) and continuous
phase. Therefore, most of current models don’t account for particle
interaction.
 A velocity and temperature field is calculated for each phase, but common
pressure field.
 Inter-phase transfer models are provided as empirical input (i.e., drag for
sphere in a fluid). Models are highly dependent on problem. Careful
analyst´s judge is required.
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(cont´d)
 Eulerian-1D vs Eulerian-Eulerian/Lagrangian Multi-dimensional models
(cont´d)
 Transient 1D models (1DM) are currently succesfully implemented in
pipelines-network analysis (e.g., OLGA). Inexpensive calculations per cell,
but may become large simply because of accounts for pipeline length and
networking.
http://www.software.slb.com/products/foundation/Pages/olga.aspx
 1DM may include all the physics through empirical correlations for wall
resistance, interphase momentum-energy transfer, etc. and are very useful
in Flow Assurance analyses. However, cannot predict details of cross-
section (2D-3D) features, useful in single equipment´s optimization (e.g.,
G-L, L-L, L-L-G separators).
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(cont´d)
 Eulerian-1D vs Eulerian-Eulerian/Lagrangian Multi-dimensional
models (cont´d)
 Multidimensional models (MM) may predict details of phase distribution in
sections of equipments or pipes.
 MM are too computationally expensive to be used for pipeline networking.
Therefore, is not suitable to determine pressure drops in lenghty pipeline
systems (+100m, e.g., Star-CCM+ has a 100m riser in details).
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Source: Star-CD, Flow Assurance Booklet

(cont´d)
 Eulerian-1D vs Eulerian-Eulerian/Lagrangian Multi-dimensional
models (cont´d)
 Interphase interactions in MM are modeled based on local physics.
 MM have led to significant results in bubbly and stratified-wavy flows.
 Interfacial and turbulence modeling are very important in MM.
 Current link between 1D and 3D models is possible either by conjugate
user interface (e.g., StarCCM-OLGA) or by interconnecting 1D-3D input-
output manually (CFX/Fluent-OLGA). Potential advantage to get the most
of complex oil-transport systems (e.g., applications to Flow Assurance).
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3. Slurry flows. Antecedents.
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• Prediction of Particle-fluid flow is a very complex phenomenon of
interest in many practical fields.
• Particle diameter, concentration, fluid properties and flow velocity →
free-particles macro-motion:
Creeping, Saltation and Suspension
Creep, saltation, and suspension of particles by wind.
From Wind Erosion Research Unit, Wind Erosion Simulation Models. (source:
http://oceanworld.tamu.edu/resources/environment-book/aeoliantransport.html)
Homogenous mixture with all
the solids in the suspension
Heterogeneous mixture with
all the solids in the suspension
Flow with a base/I milk
mobile/movable and saltation
(with or without suspension)
Flow with a base/I milk
immovable
Homogeneous mixture w/all
particles in suspension
Heterogeneous mixture w/all
particles in suspension
Fluidized mobile bed w/ or
w/o particles in suspension
Non-mobile particle bed w/
or w/o part. in suspension
Newitt et al., 1955
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Antecedents on slurry flows
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The study of slurry flows has been addressed via:
1) Experimental approach.
2) Theoretical approach.
3) Multiphase flow modeling approach.
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Source: Baha Abulnaga, (2002), Slurry Systems Handbook, Mc. Graw Hill.
Volumetric concentration
Pressure drop
Antecedents. Experimental-Theoretical approach
Fig. (a)
Fig. (b)
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• Early works aimed at giving a practical classification via empirical
observations:
Slurry Classification Particle Diameter
Homogenous suspensions < 40μm
Maintained suspensions 40μm a 0.15 mm
Suspension with saltation 0.15 mm y 1.5 mm
Saltation >1.5mm
According to Duran and Condolios, 1952
Newitt et al., 1955
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• Then, researchers observed that it was a multi-variable problem and
gave a more general qualitative/quantitative description:
Source: Baha Abulnaga, (2002), Slurry Systems Handbook, Mc. Graw Hill.
Shen, 1970
Newitt, et al. (1955), Thomas (1964), Shen (1970)
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Source: Chemloul et al. (2009), Simultaneous Measurements of the Solid Particles Velocity and Concentration Profiles in Two Phase Flow by Pulsed Ultrasonic Doppler Velocimetry, J. of the Braz.
Soc. of Mech. Sci. & Eng. , October-December 2009, Vol. XXXI, No. 4 / 333
Antecedents. Experimental approach
However, saltation regimes have proven to depict a complex behavior close to
bottom wall, as shown in latest works by Chemloul et al. (2009), using Pulsed
Ultrasonic Doppler Velocimetry (PUDV) in a 20mm pipe diameter and glass
bead particles (for Cv ≤ 2%):
Effect of the mean flow velocity (heterogeneous-saltation regimes)
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Source: Chemloul et al. (2009), Simultaneous Measurements of the Solid Particles Velocity and Concentration Profiles in Two Phase Flow by Pulsed Ultrasonic Doppler Velocimetry, J. of the Braz.
Soc. of Mech. Sci. & Eng. , October-December 2009, Vol. XXXI, No. 4 / 333
Antecedents. Experimental approach
Chemloul et al. (2009), cont´d:
Effect of particle volumetric concentration
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• In summary, past 50 years research on horizontal slurry pipeline flow:
• Most investigations were conducted using small pipeline loops (D≤ 55 mm) to
determine pressure gradients and critical velocities.
• Many studies only considered moderate particle volume concentration (≤
26%). In creeping-saltation regime complex distribution of particles close to
bottom wall.
• Many of the models are 1D or 2D semi-empirical models, therefore are very
limited to estimate particles local velocity/concentration.
• Previous models are limited to straight pipelines with circular diameter;
therefore CFD models may, if correctly tune-up, offer a good predictive tool.
Source: Ekambara K., Sanders R., Nandakumar K., Masliyah J. H., “Hydrodynamic Simulation of Horizontal Slurry Pipeline Flow Using ANSYS-CFX”, Ind. Eng. Chem. Res., 48, 8159–8171, 2009.
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Antecedents. Numerical approach
Recent works, based on Eulerian-Eulerian models rely on Granular Kinetic
Theory. Relative success in modeling slurries with Cv = 8-45%; Dpart = 90-
500µm, Vm=1.5-5.5 m/s and Dpipe = 50-500mm (ex., Ekambara et al., 2009;
Hernandez et al., 2008; Lahiri & Ghanta, 2010):
Pressure drop prediction vs. experiments
Ekambara et al., 2009 Hernandez et al., 2008
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Source: Ekambara K., Sanders R., Nandakumar K., Masliyah J. H., “Hydrodynamic Simulation of Horizontal Slurry Pipeline Flow Using ANSYS-CFX”, Ind. Eng. Chem. Res., 48, 8159–8171, 2009.
However, in creeping-saltation regimes, CFD models have failed in predicting
the correct particle concentration close to the bottom wall. (ex., Ekambara et
al., 2009):
Numerical study (Ekambara et al. 2009) vs. experiments (Roco and Shook, 1983;
Schaan et al., 2000; and others)
Good agreement with
experiments. Suspension regime
No agreement between CFD and
experimental data. Saltation regime
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Source: , “Lahiri, S.K., Ghanta, K.C., ¨Slurry Flow Modelling by CFD”, CI & CEQ, 16 (4) 295-308, 2009.
After experimental fitting and optimization, CFD models (e.g. Fluent) have
succeeded in predicting the correct particle concentration in mid-high
concentration regimes (e.g., Lahiri and Ghanta, 2010):
Good agreement with
experiments. Suspension regime
No agreement between CFD and
experimental data. Saltation regime
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 What´s the outcome from Ekambara et al., Hernández et al.
and Lahiri & Ghanta ANSYS models?
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We noticed that all used state-or-the-art slurry modeling, present in ANSYS-CFX-
Fluent, for which:
• GKT proved to be a good model to predict particle concentration and pressure
drop in 3D geometry.
• However, CFX model (current limitation of ANSYS-CFX), lacks one of the viscous
components (frictional term).
• In Marval (2009) we succeeded to model sand-air transport in saltation regime,
using GKT as well, but including all viscous components and the turbulence
dispersion, using ANSYS-Fluent.
• Lahiri and Ghanta (2010) used full GKT in Fluent, but with 1st-order discretization
scheme and no special attention to turbulence dispersion, who may account for a
large deal of suspension regime.
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4. Modeling Slurry Flows using CFD. Theory of some models
available in CFD tools (specifically with emphasis in ANSYS)
 General overview of CFD multiphase modeling
 Liquid-Solid modeling
 Lagrangian approach
 Features of current models (theoretical lecture)
 Advantages
 Disadvantages
 Examples
 Eulerian approach
 Features of current models (theoretical lecture)
 Advantages
 Disadvantages
 Examples
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ANSYS-CFX: Modeling of Slurry Flows
• Eulerian-Eulerian Modeling
– phases are mixed on length scales larger than molecular but smaller
than mesh resolution
Lagrangian Eulerian
– each phase assigned a volume fraction, ra, in each control volume
– each phase has its own field variables: velocity, pressure,
temperature, etc., which are coupled by interphase transfer models
for momentum, heat, and mass
– simplest models assume a shared pressure field
Modelo Lagrangiano Modelo Euleriano
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Lagrangian-Eulerian approach
• Light particles (ρp << ρf) :
– streamlines/streaklines for steady/unsteady respect.
– follow x(t):
– integrate equation using ODE solver
Based on: AEA Technology, CFX European Conference. Strasbourg, 2002
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• Heavy particles O(ρp) ~ O(ρf) :
– Solve Newton’s 2nd Law of motion (vectorial) for each
particle, with mp (particle mass) and F = force on particle:
– F depends on local relative velocity between fluid and
particle. 1-way coupling (fluid  particle) or 2-way
coupling possible (fluid  particle). Second case
requires source/sink in fluid momentum equations.
– Also solves ODE’s for heat and mass transfer between
fluid and particle when necessary.
– Can include fluid turbulence effects on particles by
applying stochastic term, function of k on F equation.
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• Heavy particles O(ρp) ~ O(ρf) (cont’d):
– Momentum transfer including turbulence interaction:
– Solution of fluid flow with turbulence closure leads to , mean
fluid velocity.
– Turbulence interaction on particles is included through a stochastic
treatment of fluid velocity in momentum transfer, via ODE:
–
– u’f are randomly generated such that lead to an isotropic (normal)
distribution with mean 0.0 and variance of (2/3)k.
– More accurate models are also available (e.g., Berlemont et al.,
1990).
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fU
fff uUU '

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• Heavy particles O(ρp) ~ O(ρf) (cont’d): Particle-induced turbulence
– 2-way coupling implies the counter-effect from particles onto the
transport fluid.
– Form and friction drag are included in particle-fluid drag
correlation, therefore even for phase-averaged zero shear, drag
may be exerted in both ways.
– Additional source term in k and ɛ-eqns, similar to production
accounts for turbulence reflection from particle onto fluid.
– It’s a field of large space to improve.
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• Heavy particles O(ρp) ~ O(ρf) (cont’d):
– Solution procedure:
– Each particle trajectory represents a large number of identical
particles with given mass flowrate.
– Iterative process starts
– Compute fluid flow field without particles.
– Compute ODE’s to find particle trajectories (1-way type)
– If 2-way coupling, then re-compute flow field according to
source-sink effect from particles on fluid.
– Repeat until flow field and particle trajectories converge.
– Laminar flow may require O(10) iterations, while turbulent
flows may require much more iterations and a sensitivity
analysis based on number of particles in sampling to
guarantee a statistical convergence.
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Eulerian-Eulerian multi-dimensional approach
– Assumes a distinct field of velocities, temperatures, etc, except
pressure.
– Good approximation when phases are either completely separated
or mixed at length scales too small to resolve.
– Phase properties are coupled via inter-phase transfer models.
– Inter-phase transfer models are highly problem dependent and
provided as empirical input by the analyst.
– Pressure field is assumed as shared for all phases. Not appropriate to
all situations.
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– Momentum Transfer
– Drag-Lift Forces
– EE assumes mathematically both phases as continua, but
introduces the concept of pseudo-particles (PP) to account
for intephase transfer between the ¨continuum phase¨ and
each ¨disperse phase¨). While LE uses real particles (RP).
– RP and PP experience a force due to normal and shear
stresses from surrounding fluid: F = D+L
– Drag force (D) in opposite direction to U.
– Lift (F), perpendicular to U.
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Momentum Transfer Modeling for LE and EE multi-dimensional

44
– Drag is correlated in terms of drag coefficient as:
Ap : projected area of RP and PP in direction of the flow.
– For incompressible flow, CD depends only on particle Re:
– dp: diameter of particle.
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– Drag coefficient curve:
– Stokes´ Regime: (viscous effects)
– Transitional or Allen´s Regime: (visc + inertial effects)
– Newton´s Regime: (inert. effects)
– Super-critical Regime: (b.layer transition)
 1Re
Re
 p
p
S
D
C
C

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– Drag coefficient for isolated spherical solid particles:
– Valid for low concentration of particles (isolated particle hypothesis)
– When particles are not spherical, equivalent diameter applies:
– Stokes´ Regime: (calculated analytically)
– Newton´s Regime:
– Super-critical Regime:

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– Drag coefficient for isolated spherical solid particles:
– Allen´s or transition Regime (several correlations):
– Schiller-Naumann [120]:
– Ishii-Zuber [61]:
– Ihme et al.[57]:
– Sub-critical Regime may be covered, in general, by:

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– Treatment of Deformable Particles (e.g., bubbles and droplets in EE
approach only):
– Very well approximated by Ishii and Zuber [61], but includes possible
shape distortions:
– Next correlations are only for single particles (very low concentration)
and based on Particle Reynolds in terms of:

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approach only) Cont’d:
– Viscous of Spherical Regime:
– drops and bubbles are spherical at low
Rep.
– CD similar for solid spherical particles.
– However, drag reduction due to internal flow motion (Clift et al. [19])
– Therefore, Stokes flow CD becomes:
– However, in most real situations impurities attenuate secondary flows
and solid particles correlations work OK. Thus, it’s valid to use Ishii-
Zuber [61] for Allen’s regime:

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– Distorted or Ellipsoidal Particle Regime:
– drops and bubbles deforms as Rep
increases and become ellipsoidal.
– Distortion increase for larger particles.
– Particle path turns unstable and follow zig-zag.
– Terminal velocity depends much more on shape than on dp.
– Drag vs rise velocity is given by:
– Harmathy [48] showed that significant non-dimensional number is the
Eotvos number:
continuous
discrete

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– Distorted or Ellipsoidal Particle Regime (cont’d):
– Drag coefficient is correlated by:
– Hence, the terminal rise velocity is then given as:
– Thus, U∞ is only function of fluid properties

52
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– Spherical Cap Regime:
– As the size of bubble/drop increases,
particle becomes spherical cap shape
– CDcap = 8/3 (from potential theory)
– Droplet may become unstable before
reaching cap regime.
– Terminal velocity increase as dp
2
– Equivalent dp may be found by equilibrium between ellipsoidal and
cap regimes in the limit of ellipsoidal:

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– Implementation in ANSYS-CFX:
– Using Ishii-Zuber Viscosity Number:
– According to Rep and Nµ, CFX checks:

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– Implementation in ANSYS-CFX:
– Alternative correlation (Grace) for air bubbles in water in ellipsoidal
regime:
– Terminal velocity UT:
(µref = 0.0009 kg m-1 s-1)
where: and:

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– Transient response of isolated spherical solid particles:
– Particles horizontal time response from rest. Eqn.:
– Viscous time-scale:
– Stokes, Allen and Newton´s Regime, relaxation time:
– bubbles accelerate faster than oil droplets in water.
– smaller particles accelerate faster than large ones.

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– Transient response of isolated solid particles:
– Particle vertical terminal velocity under gravity in stationary fluid.
– Stokes + Newton´s regimes:
– For spherical particles:
>> trelax_Stokes_horizontal
– Newton´s Regime:

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– Additional Forces in Spherical Particles (strictly valid for low Rep):
(Note: for high Rep, equations may have different non-dim coefficients)
– Virtual Mass Force:
– extra-inertia.Only for accelerating flows
– Negligible if ρp >> ρf
– Basset History Force (accounts for b.layer temporal delayed
development -due to relative accelation- and its effect on viscous force
acting on particle):
– Negligible if ρp >> ρf . It may increase relaxation time in small particles.
www.sciencedirect.com

58
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– Lift Forces. Main responsible for particles concentration wall peaking
and coring for flow in a pipe (Mubita et al., 2013).
– Shear Lift Force. Due to rotational flow (different from
aerodynamics):
Uf Up
L
http://web2.clarkson.edu/projects/fluidflow/courses/me637/1_4Lift.pdf

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– Shear Lift Force. Due to rotational flow (different from aerodynamics):
– Lift coefficients may be expressed as:
– S is the shear rate and Ω is the particle angular velocity
Uf Up
L
http://web2.clarkson.edu/projects/fluidflow/courses/me637/1_4Lift.pdf
ReΩ

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– Lift Forces. Main responsible for particles concentration wall peaking
and coring for flow in a pipe (Mubita et al., 2013).
– Saffman Lift Force. Alternative to shear lift and occurs at uniform
mean shear field (S = constant as for laminar Couette flow).
Assuming:
Then:

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– Lift Forces
– Magnus Lift Force (due to particle rotation)
– Magnitude around 10% of shear lift force
– May be used in conjunction with eqn. of particle angular
momentum:
http://commons.wikimedia.org/wiki/File:Magnus_effect.svg

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62
– Additional Forces in Spherical Particles (strictly valid for low Rep)
– Lubrication Force (only implemented in EE within ANSYS-CFX)
– Due to flow field change around a particle close to surfaces.
– May occur between 2 particles in relative motion, but CFX only
implements wall-particle lubrication and NO particle/particle
lubrication.
– Wall lubrication force pushes disperse phase away from the wall.
– Antal et al [4] model it as:
– Clearly will only act in a thin layer close to the wall:
(only a very fine mesh will activate it!!)

63
63
– Heat and Mass Transfer between Particles y Continuous phase
– Heat Transfer
– Particle Temperature (1st Law of Thermodynamics)
– Convective heat transfer

64
64
– Convective heat transfer (cont’d)
– h is correlated in terms of Nusselt number:
– Ranz-Marshall [112] correlation (b. layer theory and recommended):
– Hughmark [55] correlation (recommended for higher Re):
– And therefore, for spherical particles:

65
65
– Heat and Mass Transfer between Particles y Continuos phase
– Heat Transfer associated with Mass Transfer
– Sum over species A undergoing mass transfer across the phase
boundary (corresponds to energy transfer along with mass during
phase change)
– LA = latent heat of vaporisation of species A (e.g., cavitation model)

66
66
– Mass Transfer
– Chemical component A, present in both particles and fluid phase.
– ‘A’ diffuses through the phase boundary
– Analogy to heat transfer:
How do we find m(mass transfer coefficient)?
As: surface area of particle

67
67
– Mass Transfer (cont’d) … How do we find m?
– m is correlated in terms of Sherwood Number (analogous to Nu):
– Ranz-Marshall [112] and Hughmark [55] correlations hold in limited
applications.

68
68
– Multiphase Turbulence Modeling (EE modeling)
– This section addresses how does continuous phase turbulence affect
dispersed phase turbulence?
– Models are based on disperse phase with low (dilute) concentration.
– There is no industrial standard (as k-ɛ or SST for single phase flow).
– Continuous turbulence affects disperse phase turbulence.
– Is one of the fields of more active research today.

69
69
– (For low concentration of particles) disperse phase turbulence may be
assumed proportional to continuous phase turbulence (Drew and Lahey
[27], [28]). It assumes that particles are transported by continuous phase
eddies:
– C = 1 for small tp_relax (e.g.,small particles) & C < 1 for large tp_relax
(e.g.,large solid particles)
– Consequently, disperse phase Re_stresses are proportional to continuous
phase Re_stresses:
Disperse Re_stresses are negligible
when ρd << ρc (e.g. bubbly flow)

70
70
– Dispersed Phase Eddy Viscosity Models
– Zero Equation for Disperse Phase (proportional to continuous phase)
No need to model disperse phase turbulence
σd is the Eddy Viscosity Prandtl Number (user-specified in CFX), as:
σd = 1 for particles with small relaxation time
σd >> 1 for particles with large relaxation time ↔ 1/ C2?? (next)
 tc
c
d
d
td 




1

71
71
– Dispersed Phase Eddy Viscosity Models (cont’d)
– Several models are available for σd =1/C2. Usually as function of
Turbulent Stokes Number:
– If trelax << tturb , e.g. small light particles, particles are driven by
turbulent eddies.
– If trelax >> tturb , e.g. large-heavy particles, particle motion is
independent of turbulent eddies. For example, coarse approx:

72
72
– Particle-Induced Turbulence (EE modeling)
– This section addresses how does dispersed phase turbulence affect
continuous phase turbulence?
– Even for low turbulent Reynolds numbers, phase-averaged equations
already contain Reynolds-stress like terms:
– Hence, Production of Turbulent Kinetic Energy (TKE) in boundary
layers at the continuous-disperse boundaries. This is called “Bubble
Induced Turbulence” or “Burbulence”. It’s dominant contribution to
TKE production and phase distribution mechanism in turbulent
bubbly flows.

73
73
– ¨Burbulence¨ (disperse phase effects on continuous phase
turbulence)
– Traditional eddy viscosity bridge:
– Additional Production term is added in k-equation for continuous
phase. It’s a source term very similar to TKE Production:
(Kataoka and Serizawa [66])
– Similar production term added to ɛ-equation.
– Reasonable agrement with experiments, except in homogenous
turbulence.

74
74
– ¨Burbulence¨ (disperse phase effects on continuous phase
turbulence) (cont’d)
– Simplest model (implemented in ANSYS-CFX) implements bubble
diameter dp as length-scale:
– And the particle-induced eddy viscosity is added to continuous
phase eddy viscosity as:
– And the disperse phase eddy viscosity is treated with Zero-Equation
closure to give:

75
75
– Additional Turbulence Phase-Phase Modulation
– Continuous phase TKE (C-TKE) Production may be enhanced by
large particles (“Burbulence”), but also C-TKE Dissipation may be
enhanced.
– Large concentration of small particles attenuate turbulence
(increase C-TKE Dissipation); e.g., Serizawa et al. [122]
– The mechanism depends on particle relaxation time. If tp_relax is
small, then particle are accelerated and withdraw TKE from
Continuous phase due to drag force, according to:
– (Gore and Crowe [39, 40])

76
76
– Multi-Particle Effects (only implemented in ANSYS-CFX for EE models)
– Previous drag correlations apply to low concentration, since are based on
single particle in infinite medium.
– Important modifications are needed due to increase in concentration:
– Spherical - Large spherical
bubbles slow cap bubbles
down accelerate
Momentum Transfer Modeling for EE multi-dimensional

77
77
– Mixture Viscosity Correlation (Ishii-Zuber [61]):
– Effective viscosity of fluid increases due to presence of particles and
therefore, there is an increase in drag.
– Taylor´s linear correlation for mixture viscosity at low concentration is:
– For solid particles, Power law correlation based on max, packing αdm:
αdm ranges from 0.5 to 0.74. However, αdm = 0.62 is OK for most cases

78
78
– Mixture Viscosity Correlation (Ishii-Zuber [61]):
– Generalization for fluid and solid particles:

79
79
– Spherical Particle (Viscous) Regime:
– For small Rep and low particle concentration, fluid and solid can be
treated with similar single-particle drag correlation:
– For large particle concentration, use same correlation, but based on
mixture viscosity:

80
80
– Spherical Particle (Viscous) Regime:
– Power-Law Approximation. Recall:
– Notice that drag increases as αd → 1

81
81
– Newton Regime:
– Ishii-Zuber [61] obtained the ratio between multiple particle and
single particle settling velocities:
which, evaluated at transitional Reynolds (Re∞ ~ 991) gives:
Based on Alan Burns (CFX, Europe-Harwell) notes on: Computational Fluid Dynamics Modeling of Multiphase
Flows

82
82
– Distorted Particle Regime:
– Similar single-multiple particles scaling up as in Newton´s regime:
– Spherical Cap Regime. Reduction of drag due to entrainment of bubbles
in wake of spherical caps.

83
83
– Automatic flow regime selection in CFX.
– If Ishii-Zuber is selected, dense and dilute particles are considered.
The selection of the regime, as before, is based on smooth transition
between regimes:
– Despite Grace correlation was developed for single particles, it can
be generalized by:
– CFX uses p = 0 by default. But, p = -1 fits Ishii-Zuber for small
bubbles; p = 2 fits IZ for large spherical cap; and so on.

84
84
– Solid Particles (only implemented in ANSYS-CFX for EE models)
– Solid Particle Drag Correlations. Dilute Particle Concentrations.
– Richardson and Zaki [113] provided power law correlations for multi-
to-single particle settling velocity. It´s simple to figure CD
– Wen and Yu [140] improved it to give:
– Recommended for particles volume fraction of 0.2 or less.
– Solid Particle Drag Correlations. Dense Particle Concentrations.
– Ergun´s equation based on packed beds is appropriate:

85
85
– Solid Particles (only implemented in ANSYS-CFX for EE models)
– Solid Particle Drag Correlations. Dilute-Dense unification
– Gidaspow [37] uses a combination:
– for αd < 0.2
– for αd > 0.2
Where : and

86
86
– Solid Particle Collision Forces (only implemented for EE models)
– New concept of Solid Pressure “Ps” and Solids Shear Stress “τs” in
momentum equation of solid phase (as fluid):
– Gidaspow Model [37]: asumes pressure force as empirical function of
solid volume fraction:
with:
NO universal constants!!

87
87
– Solid Particle Collision Forces (only implemented for EE models)
– Gidaspow Model [37] is improved by Bouillard et al [13], who gives
constants to:
By:
– Tend to separate solid particles away as they approach maximum packing.
– It’s used along with Gidaspow drag model (Wen Yu and Ergun).
– Later models introduce improvements to model of solid viscosity.

88
Granular Kinetic Theory model
Solid-liquid flow
Heterogeneous-saltation regimes
Granular Kinetic Theory
model basics: kinetic and
collisional contributions are
considered to obtain the bulk
viscosity and pressure of solid
phase
Frictional contribution must
be added to include bottom
interaction
88

89
Solid-liquid flow
Heterogeneous-saltation regimes
Dilute flow, generates kinetic
viscous dissipation
Medium concentration, produces
collissions
High concentration promotes
friction
Granular Kinetic Theory model
89

90
Granular Kinetic Theory model (EE model)
90
– Solid Pressure and Stresses are modelled in terms of:
– Proportional to solid phase mean square velocity fluctuations
– Solid Pressure is linearly proportional to the Granular Temperature:
– No universal form of go(rs). One example is:

91
Granular Kinetic Theory model (EE model)
91
– Solid Pressure and Stresses are modelled in terms of (cont’d):
– Solid stress is modelled using solids bulk and shear viscosities:
– Shear and Bulk viscosities are assumed proportional to square root of
Granular Temperature; e.g., Bulk Viscosity:
– No universal agreement on appropriate form for Shear Viscosity.
– Granular Temperature is solved as another transport equation.

92
Granular Kinetic Theory (GKT) + 2 fluid model
http://www.granular-volcano-group.org/granular_medium.html
• Two momentum eqns: liquid (fluid) and particles, Ishii (1975) and
Enwald (1995), respectively.
• Fluid turbulent viscosity with k- closure to begin with.
• Particle stress-strain tensor from GKT including kinetic and collisional
contributions (as Boltzmann´s stat); also granular temperature.
• Frictional sub-model to consider particle-bed (from plastic friction).
• Gravity, lift and drag forces included.
• Transient regime.
• Total of 9 equations: 2 momentum, 2 constitutive, 2 turbulence,
granular temperature, continuity and volumetric fraction.
92

𝜕
𝜕𝑡
𝛼 𝑓 𝜌 𝑓 𝑣 𝑓 + ∇ ∙ 𝛼 𝑓 𝜌 𝑓 𝑣 𝑓 𝑣 𝑓
= −𝛼 𝑓 ∇𝑝 𝑓 + ∇ ∙ 𝜏 𝑓 + 𝛼 𝑓 𝜌 𝑓 𝑔 − 𝛽 𝑣 𝑓 − 𝑣 𝑠 + 𝐹𝑙𝑖𝑓𝑡
𝜕
𝜕𝑡
𝛼 𝑠 𝜌 𝑠 𝑣 𝑠 + ∇ ∙ 𝛼 𝑠 𝜌 𝑠 𝑣𝑠 𝑣 𝑠
= −𝛼 𝑠∇𝑝 𝑓 − ∇𝑝 𝑠 + ∇ ∙ 𝜏 𝑠 + 𝛼 𝑠 𝜌 𝑠 𝑔 + 𝛽 𝑣 𝑓 − 𝑣 𝑠 − 𝐹𝑙𝑖𝑓𝑡
93
Model Equations
Fluid
Solid
Fluid stress tensor
Stress tensor of solid phase
How do we get these artificial coefficients?
What are the independent variables?
Juan Pedro Marval, Ph.D. thesis, julio 2009
Eqn.1
Eqn.2
Eqn.3
Eqn.4
93
𝜏𝑓 = 𝛼𝑓μt,f ∇𝑣𝑓 + ∇𝑣𝑓
T
−
2
3
𝛼 𝑓 𝑘𝑓 + 𝜇 𝑓∇ ∙ 𝑣𝑓 𝐼
𝜏 𝑠 = 𝛼 𝑠μs ∇𝑣𝑠 + ∇𝑣𝑠
T
− 𝛼 𝑠 𝜆 𝑠 −
2
3
𝜇 𝑠 ∇ ∙ 𝑣𝑠 𝐼

94
Model: Equations of Granular Kinetic Theory
        






 3:
2
3
SsSsSSSSS vIpv
t
S

Granular Temperature Equation (Lun et al. 1984)
2
3
1
sv

  Ivvv ssss
T
sssss







 
3
2
Solid Stress Tensor
Solid viscosity
Bulk viscosity by Lun
et al. (1984)
friccolkins    



 egdpsss 1
3
4
0
Conductivity
  2/30
2
2
112 



p
ss
s
d
g
e
 dissipation due to
inelastic collisions
 dissipation due to the
suspended phase
friccolkins ppp  /
Solid Pressure
Granular transport coefficients are a function
of volume fraction, diameter and density of
particles, maximum packing and restitution
coefficient.
"All transport coefficients are derived
from the Granular Kinetic Theory"
Eqn.5 Eqn.6
Eqn.7
Eqn.8
94Juan Pedro Marval, Ph.D. thesis, julio 2009

95
GKT viscosity contributions
friccolkins  
[1] D. Gidaspow, R. Bezburuah, and J. Ding. Hydrodynamics of Circulating Fluidized Beds, Kinetic Theory Approach. In Fluidization VII, Proceedings of the 7thEngineering Foundation Conference
on Fluidization, pages 75–82, 1992.
[2] P. C. Johnson and R. Jackson. Frictional-Collisional Constitutive Relations for Granular Materials, with Application to Plane Shearing. J. Fluid Mech., 176:67–93, 1987. 95
95
Collisional viscosity according to Lun et al. (1984):
 



 egdpsscol 1
5
4
0
2
 
   






 0131
5
2
1
36
gee
e
d
s
psS
kin 


Kinetic viscosity according to Syamlal et al. (1993):
µ : viscosity
αs : solid volume fraction
dp : particle diameter
e : restitution coefficient
ρs : solid phase density
go : radial distribution function
Θ : granular temperature
 
2222
4
1
6
1
sin
















































x
v
y
u
x
u
y
v
y
v
x
u
P
ssssss
s
fri
fri



Frictional viscosity by Shaeffer (1987):

96
[1] D. Gidaspow, R. Bezburuah, and J. Ding. Hydrodynamics of Circulating Fluidized Beds, Kinetic Theory Approach. In Fluidization VII, Proceedings of the 7thEngineering Foundation
Conference on Fluidization, pages 75–82, 1992.
[2] P. C. Johnson and R. Jackson. Frictional-Collisional Constitutive Relations for Granular Materials, with Application to Plane Shearing. J. Fluid Mech., 176:67–93, 1987.
Frictional pressure (Johnson, 1990):
96
•ϕ original from soil mechanics and based on Mohr-
Coulomb´s failure criterion (slope of shear strength vs
normal stress). Varies between 27˚-48˚. 30˚ usually taken
for GKT analyses.
• This pressure may harden or soften the fluidized bed.
• n and p are taken from Ocone et al. (1993)
GKT solid pressure contributions
Kinetic/collisional pressure (Lun et al., 1984):
   0/ 121 gep SSScolkin 
µf : frictional viscosity
φ : angle of internal friction

97
[1] D. Gidaspow, R. Bezburuah, and J. Ding. Hydrodynamics of Circulating Fluidized Beds, Kinetic Theory Approach. In Fluidization VII, Proceedings of the 7thEngineering Foundation
Conference on Fluidization, pages 75–82, 1992.
[2] P. C. Johnson and R. Jackson. Frictional-Collisional Constitutive Relations for Granular Materials, with Application to Plane Shearing. J. Fluid Mech., 176:67–93, 1987.
97
GKT granular ¨thermal¨ conductivity
According to Syamlal et al. (1993)
According to Hrenya and Sinclair (1997)
 



































 0
20
2
0 25
512
3341
345121
5
968
1
1
128
25
g
g
g
R
d
s
s
s
mfp
ps
s 






 
    








 00
2
3341
15
16
34
5
12
1
33414
15
gg
d
ss
sss
s 





0 0.1 0.2 0.3
Solid volume fraction
0.001
0.01
0.1
1
10
Dimensionlesssolidconductivity(kd-1
Hrenya & Sinclair 
Hrenya & Sinclair 
Hrenya & Sinclair 
Hrenya & Sinclair 
Syamlal et al.

98
k-e Mixture turbulence model [1]
[1] ANSYS FLUENT 12.0, Theory Guide, Ch 16, Sec. 5, Sub-Sec 11
Turbulent kinetic energy (k)
Turbulence rate of
dissipation (ϵ)
Turbulent Viscosity
Production of (k)
Phase-averaged density
Favré phase-averaged velocity

99
• Spatial discretization: 2nd QUICK (Quadratic Upstream Interpolation for
Convective Kinematics). Second order discretization in time.
• Hexaedral unstructured O-grid mesh with longitudinal symmetry.
• Normalized residuals smaller than 1e-05 for all equations. 0.75% global mass
residual tolerance.
• Computational domain length always: x/d > 50 for development (Ling et al. 2003)
• Each simulation takes about 8 hours in a PC with Intel Core 2 Duo de 2.4 MHz
y 4 GB Ram, Fluid Mechanics Laboratory, USB.
5. Case Study: Liquid-Particles Flow in Horizontal Pipe, using
ANSYS-Fluent v12.
[1] J. Ling et al., Numerical investigations of liquid–solid slurry flows in a fully developed turbulent flow region, Heat and fluid flow 2003,

100
Steps:
1. Mesh verification with diluted case (< 2%). Mesh resolution is evaluated in
order to guarantee developed pressure gradient and concentration. GKT and
Phase-averaged k-ɛ are used.
2. Effect of Turbulence model, Wall Treatment and Lift Force in CFD model.
Validation against Low dilution data from Chemloul et al. (2009).
3. CFD results are firstly tune-up against medium-dense concentrations using
experiments from Kausal (2005). Frictional contribution to solid viscosity and
wall granular temperature are assessed.
4. Numerical simulations are performed to capture the trends of the effects
caused by particle diameter, bulk concentration, flow velocity.
5. Conclusions and ongoing work.
Case Study: Liquid-Particles flow in horizontal pipeline

Step 1: Mesh Verification (Dilute cases)
101
101
Geometry: D = 20 mm and L = 1.5m
< 1%
Convergence in < 3000 iterations

Step 2: Effect of Turbulence Model, Wall Treatment and Lubrication
Force (Dilute cases)
102
102
Base Case
Lift Effect k-e vs RSM

Step 2: Effect of Turbulence Model, Wall Treatment and Lubrication
Force (Dilute cases) (cont’d)
103
103
Base Case
Preliminary Remarks
• The Lift force does not exert important changes on
account for significative changes.
• RSM does not improve prediction of k-epsilon in
diluted conditions. Thus, k-epsilon can be used.
• Scalable wall functions don’t improve std. treatment in
prediction of wall concentration in dilute conditions.
Wall
Function

Step 3: CFD Validations for Medium-Dense Flows
104
Kaushal et al. (2005) “Effect of Particle Size Distribution on Pressure Drop and Concentration in
Pipeline Flow of Highly Concentrated Slurry”, International Journal Multiphase Flow, 31, 809–823
Experimental Dataset
Numerical Set #1
Numerical Set #2 Numerical Set #3
1.0 2.0 3.0

Step 3: CFD Validations for Medium-Dense Flows.
105
Initial Mesh Verification
Chosen
mesh
x/d ≥ 50
Objectives: Effects on Particles Concentration
• Wall Granular Temperature (boundary condition).
• Frictional contribution to solid viscosity.
• Particle diameter (mono-disperse and bidisperse).
• Flow velocity.
• Bulk concentration.

Wall Granular Temperature (monodisperse)
106
Particles Concentration Particles Concentration

Wall Granular Temperature (bidisperse)
107
Particles Concentration

Frictional Contribution to Solid Viscosity (monodisperse)
108

Frictional Contribution to Solid Viscosity (bidisperse)
109

Particle Size mono- and bidisperse
110
Step 3: CFD Trends for Medium-Dense Flows.

Particle Size mono- and bidisperse
111

Particles Bulk Concentration (monodisperse)
112

Particles Bulk Concentration (bidisperse)
113

Mixture velocity (monodisperse)
114

Mixture velocity (bidisperse)
115
Concentration for Dp = 0.125 mm
Concentration for Dp = 0.440 mm

Concluding Remarks
116
116
 A case study for a CFD study using an Eulerian/Eulerian framework has
been presented. Granular Kinetic Theory is used to model the solid
viscosity.
 Predicting capacity of the model is assessed under effects of flow regime,
monodisperse and bidisperse liquid-particle flow, mixture velocity and bulk
concentration.
 Preliminary tune-up of the model is performed based on evaluation of
sensible wall granular temperature and frictional component of solid
viscosity.
 The frictional contribution to solid viscosity improves the computation of
wall shear stresses due to high concentration (above 50%) of particles in
bidisperse mixtures over the wall.

Concluding Remarks (cont’d)
117
117
 The wall granular temperature proved to be a very important element in
the modeling. In fact, tuning its value permits to control the level of particle
agitation near the wall and therefore, affects the prediction of near-wall
concentration.
 The bidisperse mixture depicts a quite interesting behavior as the cross
concentration of particles tend to mantain its heterogeneous distribution
even a flow velocities larger than 2 m/s.
 Mid-size particles show an important response to gravitational force, as
expected, and plays a primary role in the determination of cross-section
concentration profiles.
 Turbulence carrying capacity is appreciated above a threshold value of the
velocity, for which small/mid particles are easily transporte by the fluid.
 The bulk volumetric fraction did not affect consideraby the mixture
transport capacity when the velocity was mantained constant.

Ongoing work
118
118
 Multi-objective Optimization techniques to determine combination of wall
granular temperature and other parameters involved in the prediction of
the solid viscosity (restitution coefficient).
 Improve the model resolution close to the wall by refining the mesh and
using higher order models, with better capabilities to predict the wall shear
within the viscous and buffer sub-layers.

Relevant References (taken from Alan Burns´ Notes on Multiphase Flow)
119
119
• [9] A. Berlemont, P. Desjonqures and G. Gouesbet (1990), Particle Lagrangian simulation in turbulent flows, Int. J.
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Acknowledgements
120
120
• Former Doctoral student: Juan Pedro Marval (Work on Experimental and CFD
approaches using GKT applied to Aeolian Flows)
• Former Master student: Jaime González (Work on horizontal pipelines using
CFD and GKT, reported in this presentation).
• Former Master students: Tania Mubita and Joselin Moreno (Work on CFD
applied to Multiphase Flow liquid-particles in biomedical applications).
• Former Undergraduate student: Cinthia Gutiérrez (Work on optimization of
GKT parameters for horizontal pipelines)

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