Modeling of Granular Flows

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Modeling of Granular Flows

• Harald Laux
• SINTEF Materials Technology
• Multiphase Flow Modeling Lecture Series DIO 1013

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Overview

• Multiphase flow models and examples
• What is a granular flow?
• Granular flow examples
• Modeling of granular flows
• Macroscopic equations
• Important concepts
• Collision dynamics
• Rheology
• Boundary conditions
• Large scale structures

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Multiphase flow models (in CFD)
Percentage of projects
Free surface flow models (e.g. VOF)
Mixture model
0
Standard multi-fluid model
Granular multi-fluid model
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DEM
39
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4
0
Lagrangian particle tracking
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Example Free surface model
VOF model of plunging steel jet
Laux et al (2000).
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Example Lagrangian particle tracking
Berg et al. (1999).
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Example Granular multi-fluid model
Sedimentation of spherical mono-sized polymer
particles in water
Laux H. (1998), Ph.D. Thesis NTH 199871.
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Example Standard multi-fluid model
Transparent model alloy NH4CL (70wt) H2O
solution
Beckermann and Wang (1996).
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What is a granular flow?
Simple It is the flow of a granular material!
But then What is a granular material?
Granular material Mixture made up of discrete
solid particles which are dispersed in a fluid
phase.
During flow, the particles come into contact or
near contact with each other and an essential
feature of the granular flow is the interaction
between the solid particles.
Strictly In a granular flow the particle
concentration is high.
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Granular flows I (examples)

• Flow of
• Sand
• Ore
• Metal powders
• Alloy particles
• Free-floating dendrites
• Food particles
• Catalytic particles
• Granular snow
• Pack ice (2D)
• Dynamics of unidirectional freeway traffic
• a.o.

• Flow in form of
• Dry grains
• Powders
• Suspensions
• Slurries
• Pastes

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Typical granular flows II (examples)

• Cold gas-fluidized bubbling bed (dense B
  powder)

Laux H. (1998), Ph.D. Thesis NTH 199871.
Evolution of solids volume fraction over 5 s real
time
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Typical granular flows III (examples)

• Flow of dry sand in an hourglass (dense
  interparticle friction)

Granular multi-fluid model
Laux H. (1998), Ph.D. Thesis NTH 199871.
Discrete Element Method
Curtesy of Leif Rune Hellevik
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Typical granular flows IV (examples)

• Ozone decomposition in a 0.25 m I.D. riser
  (dilute - reactive gas)

Samuelsberg A.E. (1994), Ph.D. Thesis NTH 199424.
Solids volume fraction

• Pyrolysis of biomass in bubbling fluidized bed
  reactor (dense - reactive particles)

gt 0.25
Lathouwers D. and J. Bellan (2001), Int. J.
Multiphase Flow, 27, 2155-2187.
lt 0.15
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Modeling of granular flows I - Volume
averaging

• Traditional method to derive the multi-fluid
  model
• Suitable for simulation of entire range of
  granular flows
• Closure completely empirical
• For granular multi-fluid model
• Derive only the governing equations for the fluid
  phase

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Modeling of granular flows II - Granular
kinetic theory

• Relatively new theory (since 1980) to derive the
  governing equations for dispersed phases in a
  granular flow
• Closure considerably less empirical
• Granular multi-fluid model consists of
• Granular phase(s) conservation equations
• Fluid phase conservation equations
• Granular theory
• Most work done on dilute and dense gas-solid
  particle flows

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Macroscopic equationsfor isothermal flow of
mono-sized particles
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Mass conservation
No difference in equations for standard and
granular multi-fluid model
Volume fraction
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Standard multi-fluid model
Momentum conservation
Momentum transfer with suspending fluid phase
Stress tensor
Dynamic viscosity
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Granular multi-fluid model
Momentum conservation
Momentum transfer with suspending fluid phase
Particle pressure
Stress tensor
Dynamic viscosity
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Granular multi-fluid model Granular
temperature
Dissipation of ? due to suspending fluid phase
Production of ?
Conduction
Dissipation of ?
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Granular transport coefficients

• Functions of volume fraction, granular
  temperature, particle diameter, particle density,
  maximum packing fraction and restitution
  coefficient
• Dynamic viscosity
• Bulk viscosity
• Conductivity
• All can be rigorously derived from kinetic
  theory.

Example
Properties which can be measured!
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Granular pressure I
Granular pressure Equation-of-state of
particulate continuum
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Granular pressure II

• Physical significance
• Dispersive force
• Limits compression in a arbitrary volume to
  maximum allowed packing fraction

Measured for example for gas-solid flow by
Campbell and Wang (1990) and for liquid-solid
flow by Zenit et al. (1997)
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Harald Laux Show derivation at borad
Granular temperature

• One of the main characteristics of granular
  flows
• Even if particle cloud moves at a macroscopic
  velocity, the velocity single particles
  fluctuates randomly about this macroscopic
  velocity

Microscopic velocities
Local instantaneous macroscopic velocity
Granular temperature is the kinetic energy
associated with the random velocity fluctuations
due to collisions between particles
Important Do not confuse with turbulence!
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Dissipation of granular temperature

• Dissipation due to inelasticity of
  particle-to-particle collisions (interactions)
• Function of volume fraction, granular
  temperature, particle diameter, particle density,
  maximum packing fraction and restitution
  coefficient

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Collision dynamics I

• During flow, the particles come into contact or
  near contact with each other
• Particles interact

• Kinetic theory of granular flows assumes
• 1) Instantaneous binary collisions
• 2) Energy dissipation due to inelasticity of
  collisions

Restitution coefficient, e
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Collision dynamics II

• Restitution coefficient

Relative velocity after collision
gt Momentum is conserved
gt Energy is dissipated
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Transport mechanisms in granular flows

• (1) Kinetic or streaming mode
• (2) Collisional mode
• (3) Frictional mode
• (4) Radiated mode

Mass transfer (1)
Momentum transfer (1) - (4)
Energy transfer (1), (3?)
Liquid-particle flow
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Granular multi-fluid model
Flow of dry sand in an hourglass
Spontaneous bubbling bed
Sedimentation problem
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Rheology
Harald Laux An der Tafel herleiten Pxy(du/dy)
gamma

• Bagnolds (1954) classical shear cell
  experiments
• Large shear rates
• Small shear rates

Grain inertia regime
It can be shown using granular theory that for
dense flow
Macro-viscous regime
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Harald Laux Constant normal force gt must expand
as granular temperature increases
Rheology simple approach
If particles are interlocked and maintain long
lasting frictional contacts then shear will first
move particles above a critical yield stress.
If frictional contacts are less important, e.g.,
as for perfectly spherical and smooth particles,
the particle bed will expand when sheared and
produce granular temperature.
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Rheology simple approach
Coulomb friction Slip if critical shear stress of
any slip plane is exceeded
Shear stress during slipping
Slip planes
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Rheology simple approach
Dynamic viscosity
   
Mixing length model
Mixing length average particle spacing
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Rheology simple approach
At high shear rates both viscosity and shear
stress increase gt Shear thickening rheology
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Boundary conditions

• Particle velocity at walls
• Particles slip at wall
• Partial-slip boundary condition
• Granular flux at walls
• Dissipative particle-to-wall collisions
• Wall shear produces granular temperature
• Heat flux through wall is balanced by
  difference in dissipation and production at walls

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Large scale structures in granular flows
Stable large scale structures caused by volume
fraction fluctuations (stability analysis of
granular flow equations e.g. by Savage (1992)
Cluster in a dilute riser flow
Spontaneous formation of gas bubbles in a dense
fluidized bed
Dilute wall region due to production of ? at wall
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Standard multi-fluid model versus Granular
multi-fluid model

• Granular multi-fluid model
• Granular pressure gradient
• Dynamic transport coefficients, e.g.
  µf(?d,?,..)
• Compressible granular continuum
• Less empirical input
• Mainly derived for gas-solid flows
• Standard multi-fluid model
• All transport coefficients require empirical
  input
• More flexible easily adopted to new types of
  granular flows if transport coefficients can be
  measured
• Even a force corresponding the granular pressure
  gradient can be introduced (G-modulus)

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Are granular flow models also applicable to
droplet flow and bubbly flow?
E.g. Droplet flow. Is the viscosity of the
droplet material the correct viscosity to use in
the shear stress tensor of the droplet phase?