simulation of granular two-phase flow
TRANSCRIPT
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Simulation of Granular Two-phase flow
Contributors to this work:
Hans Kuipers
Renske Beetstra
Christiaan Zeilstra
Sebastian Kriebitzsch
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Important example: gas fluidized beds
Simulation of Granular Two-phase flow
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Euler Lagrange
Euler Lagrange
Euler Euler
Three basic models for two-phase granular flow:
TFM DEM DNS
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Outline
II. DEM models for granular two-phase flow
III. Example: Vibrated granular beds
I. DNS models for granular two-phase flow
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I. DNS models for granular two-phase flow
How model the particles?
Three issues:
How to model the fluid phase?
How to model the fluid-particle interaction?
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2. Models for the fluid phase (CFD, LBM)
3. Fully resolved fluid-particle interaction
1. Models for the solid phase
I. DNS models for granular two-phase flow
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1. Models for the solid phase
- Phase consists of individual particles Lagrangian
- Methods borrowed from classical “molecular dynamics”
- Two different methods A. Soft-sphere model
B. Hard-sphere model
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1A. Solid Phase Models: Soft-Sphere
Interaction force follows from a continuous potential
Time driven scheme
“soft-sphere model”
Newton’s equation of motion:
is integrated numerically:
total force:
Position of particle a:
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• Collision force: Spring-dashpot model
• Electrostatic force
• Cohesive force
spring constant damping coefficient
1A. Solid Phase Models: Soft Sphere
Interaction force:
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soft-sphere model
Coulomb force etc.
1A. Solid Phase Models: Soft-Sphere
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Simplified MD: hard-sphere model
• Collision time between spheres can be calculated analytically:
abR
a
b
Ra
Rb
v ta ab
v tb ab
R Ra b+
• Collision: change of momentum does not follow from forces, but is calculated via:
• Evolution in time: free-flight to nearest collision event followed by instantaneous binary collision (event driven scheme)
1B. Solid Phase Models: Hard-sphere
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1B. Solid Phase Models: Hard-Sphere
Advantages of hard-sphere over soft-sphere
- Much faster for dilute systems
- Soft potential often “too soft” to model e.g. glass spheres
Disadvantages of hard-sphere over soft-sphere
- HS breaks down for dense (close packed) systems
- Update not based on forces: more difficult to include other interactions
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Outline
2. Models for the fluid phase (CFD, LBM)
3. Fully resolved fluid-particle interaction
1. Models for the solid phase
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- Time evolution governed by Navier-Stokes (NS) equation
- Two basic methods for solving the NS equations on a grid
- Continuum description of the phase Eulerian
A. Computational Fluid Dynamics
B. Lattice Boltzmann Method
2. Models for the fluid phase (CFD, LBM)
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2. Numerical Method: Basic CFD 2. Fluid Phase Models: Computational Fluid Dynamics (CFD)
Basic idea: solve the set of differential equations:
by finite difference methods
Closures for and
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2. Numerical Method: Basic CFD
0
2. Fluid Phase Models: Computational Fluid Dynamics (CFD)
finite diff. form
tentative velocity
taking
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Solution procedure to calculate variables at time
Initial guess:
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Divide space up in cells of Define as the pressure at the center of the cell
Finite differences in space: requires discretization of space
Note: velocity is calculated from an equation like:
Then for instance
Requires that velocities are defined at the faces of the cell
Then for instance is again defined at the cell center
2. Fluid Phase Models: Computational Fluid Dynamics (CFD)
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x y z
scalar variables x-velocity component y-velocity component z-velocity component
Define scalar variables at the cell centers, vector variables at the cell faces
Staggered grid
2. Fluid Phase Models: Computational Fluid Dynamics (CFD)
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Stability condition from explicit treatment of
ii) Stress term:
i) Convective term:
Resolution in time and space set by
Any instability will originate from the explicit term in the velocity update
2. Fluid Phase Models: Computational Fluid Dynamics (CFD)
(Courant)
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These 4 variables can be captured by 1 variable:
Time evolution of : the Boltzmann Equation (BE)
2. Fluid Phase Models: Lattice Boltzmann Method (LBM)
Hydrodynamic variables for the gas phase: and
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2. Fluid Phase Models: Lattice Boltzmann Method (LBM)
Discretization of coordinate and velocity space
restricted to lattice sites
discrete velocities , such that is located on a neighboring lattice site
For the 2-D square lattice: 4 velocities:
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Continuous Discrete
2. Fluid Phase Models: Lattice Boltzmann Method (LBM)
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Update in lattice-Boltzmann scheme:
No iterations All calculations are local
output
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2. Fluid Phase Models: Lattice Boltzmann Method (LBM)
Advantages Lattice-Boltzmann: - Easy to program
- Ideally suited for parallelization
- Simple boundary conditions
- Faster than CFD ?
Disavantages Lattice-Boltzmann:
- Stability conditions not as clear as in CFD
- Conversion to SI units is less straightforward
- Not straightforward to include heat transer, and/or GLS flow
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Outline
2. Models for the fluid phase (CFD, LBM)
3. Fully resolved fluid-particle interaction
1. Models for the solid phase
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LBM or CFD
Interaction between solid and fluid: no-slip boundary condition
3. Fully-resolved fluid-particle interaction
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Resolved flow with LBM: Bounce Back at boundary nodes
Define boundary node as point halfway an exterior and interior lattice site
3. Fully resolved fluid-particle interaction: LBM
In the propagation step: distribution “bounces back” at boundary nodes, and returns to its original site average flow velocity is zero at boundary site
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boundary nodes(bn) + velocities
Solid phase:
with bounce-back at bn propagated
f p force
Fluid phase: Update with boundary rules
output
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Fluid-particle force for a (infinite) random array (Stokes flow)
3. Fully resolved fluid-particle interaction: LBM
vdH, Beetstra & Kuipers, JFM 2005
Relations have also been obtained for general Re and polydisperse systems
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CFD
Interaction between solid and gas: stick boundary condition at surface
3. Fully resolved fluid-particle interaction: CFD
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Define marker point on the surface, each of which applies a force on the fluid, such that the velocity of the fluid at the marker point is equal to the surface velocity
Immersed Boundary Method (Uhlmann (2005)):
Velocity from update without forcing
2. Numerical Method: Introducing Solid Boundary 3. Fully resolved fluid-particle interaction: CFD
C =
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Update flow field on Eulerian grid:
(note: for simplicity, we assume that the velocity is defined on gridpoints ijk (no staggered grid))
Force at marker point location: Total force density at from all force points in range of
Lagrange Euler
Required: Mapping
Euler Lagrange
3. Fully resolved fluid-particle interaction: CFD
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Euler Lagrange mapping: Volume weighing
S3 S4
S1 S2
Basic idea shown in 2-D (surface weighing)
3. Fully resolved fluid-particle interaction: CFD
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Lagrange Euler mapping: Volume weighing
S3 S4
S1 S2
Basic idea shown in 2-D (surface weighing)
3. Fully resolved fluid-particle interaction: CFD
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Mapping can formally be written as:
Euler Lagrange:
Lagrange Euler:
mapping is of course not restricted to volume-weighing
3. Fully resolved fluid-particle interaction: CFD
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Solution procedure to calculate variables at time including IBM force
Initial guess:
Calculate velocity field without forcing
Map velocity field to marker point locations
Calculate IBM force at marker point locations Map IBM force to Eulerian grid
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Initial guess:
Solid phase:
Output: drag correlation F
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First test: comparison with the exact expression by Hasimoto for a dilute SQ infinite array:
iteration-IBM
3. Fully resolved fluid-particle interaction: CFD
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2. Numerical Method: Hydrodynamic diameter
Use Hasimoto for setting an effective hydrodynamic diameter (calibration)
Idea:
3. Fully resolved fluid-particle interaction: CFD
Note that also in LBM an effective diameter is used!
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With diameter correction Without diameter correction
3. Validation: Drag force for SC arrays 3. Fully resolved fluid-particle interaction: CFD
Validation: drag for a dense square array:
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JFM 2005
3. Validation: Drag force for random arrays 3. Fully resolved fluid-particle interaction: CFD vs LBM
Comparison with LB results for the drag for a dense random array:
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3. Validation: Interaction force between 2 spheres
- Particles fixed at their position
- Particles have equal, but opposite velocities, with Re << 1
- Surface-to-surface distance s varied
- Results compared with exact solution from multipole expansion of the Stokes eq.
3. Fully resolved fluid-particle interaction: CFD vs LBM
Interaction force between two particles in relative motion
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3. Validation: Interaction force between 2 spheres 3. Fully resolved fluid-particle interaction: CFD vs LBM
Thesis S.H.L. Kriebitzsch (2011)
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3. Validation: Interaction force between 2 spheres 3. Fully resolved fluid-particle interaction: CFD vs LBM
Thesis S.H.L. Kriebitzsch (2011)
Interaction force between two particles in relative motion
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3. Fully resolved fluid-particle interaction: CFD vs LBM
Thesis S.H.L. Kriebitzsch (2011)
Interaction force between two particles in relative motion
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Euler Lagrange
Euler Lagrange
Euler Euler
Three basic models for two-phase granular flow:
TFM DEM DNS
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DNS DEM
follows from stick boundary conditions
estimated from relations based on the local φ
II. DEM models for granular two-phase flow
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Unresolved flow: implementation similar to resolved flow
DEM models for granular two-phase flow
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Unresolved flow: implementation similar to resolved flow
Note that is the same for all particles that occupy the same cell.
To evaluate , again a Euler-Lagrange mapping is required.
DEM models for granular two-phase flow
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Initial guess:
Solid phase:
Output: drag correlation F
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Initial guess:
Solid phase:
Note: there are other implementations where velocity can be treated fully implicit
Input: drag correlation F
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4. Gas-fluidized systems: unresolved vs fully resolved
Deviation ∆F of true force from DP force
True force Force as calculated in a DEM model from drag correlations
DEM models for granular two-phase flow
Comparison DEM/DNS
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1.DEM drag is 33% lower than the average true drag force 2. For 1 out of every 3 of the particles the individual force for deviates more than 25% with the DEM force.
DEM models for granular two-phase flow
Comparison DEM/DNS
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2. DEM Gas-solid drag force (static systems)
Idea: use individual volume fractions
DEM models for granular two-phase flow
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Discrete element models are useful for obtaining insight, but should not be used for qualitative results.
Drag correlations derived for static system are not applicable to moving particles
Natural spreading in the fluid-particle drag is too large to capture with (advanced) drag models in DEM
DEM models for granular two-phase flow
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III. Example: Effect of air on vibrated granular beds
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Experiments by Burtally, King and Swift (Science 2002)
Equal-sized bronze and glass spheres (100 µm)
Simulations:
• Particles: “molecular dynamics” with soft-sphere model
• Gas phase: computational fluid dynamics model
• Gas-Particle interactions: unresolved, empirical drag force
• System size: Np = 25 000, W x H x D = 8 x 6 x 0.6 mm3 • Parameters: f = 55 Hz, A = 1 mm
No air Air
Vibrated glass-bronze beds
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Burtally, King, Swift & Leaper, Gran.Mat. 2003
f = 55 Hz
A = 1.0 mm
f = 130 Hz
A = 0.07 mm
Vibrated glass-bronze beds
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Experimental phase diagram (Burtally et al)
Vibrated glass-bronze beds
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Why do the light particles sink to the bottom?
Vibrated glass-bronze beds
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Sandwich formation:
- Convection plays an important role
- Sensitive of the particle-particle and partice wall friction
Vibrated glass-bronze beds
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Concluding remarks
Almost all DNS simulations suffer from (large) grid resolution effects. The use of an effective diameter is essential for getting accurate results
DEM simulations are a “cheap” way to incorporate the effect of air, however, they should only be used for getting qualitative insight
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Key features of the model:
• Particles: soft-sphere model, 0.5 mm diameter
• Gas phase: computational fluid dynamics model
• Gas-Particle interactions: unresolved, empirical drag drag
• System size: Np = 14 000, W x H x D = 100 x 50 x 2.1 mm3 • Parameters: f = 6.25 Hz, A = 10 mm
No air Air
First documented by Da Vinci (1500) and Faraday (1831)
With: Devaraj van der Meer Ko van der Weele Gabriel Caballero
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Mechanism for steady state heap