QUANTIFYING THE UNCERTAINTY OF KINETIC-THEORY PREDICTIONS OF CLUSTERING
May 31st, 2012 University Coal Research Conference
Pittsburgh, PA
Peter P. Mitrano, Sofiane Benyahia, Steven R. Dahl, John R. Zenk, Andrew M. Hilger, Christopher J. Ewasko,
Christine M. Hrenya
University of Colorado at Boulder Chemical and Biological Engineering
Motivation: Granular instabilities
Gasifier
Oxygen
• Coal • Biomass • Petroleum
coke • Municipal
solid waste
Syngas (CO, H2) Feedstock
Fluid Analogy: Continuous vs. Discrete
Continuum perspective
Molecular perspective
Navier Stokes eqns Newton’s laws
System of Interest: Granular Flow
• The Homogeneous Cooling System (HCS) – No external forces – Periodic boundaries – No gradients in the hydrodynamic variables
• Particle properties – Constant coefficient of restitution (e) – Monodisperse particles – No enduring contacts
Background
Velocity field Particle locations
Molecular dynamics (MD) simulations of the HCS
• Dissipative collisions • Sufficiently large
system domain
Goldhirsch, Tan, Zanetti, J. Sci. Comput. (1993)
Vortices Clusters
Velocity field
Background
Kinetic-Theory-based stability analysis: Garzó, 2005 Mitrano et al., Phys. Fluids (2011)
Solids Fraction (ϕ )
MD
MD
Vort
ex
Objectives
Quantitatively assess Kinetic-theory-based predictions of instabilities via MD simulations
• Clustering instabilities
– MD vs. CFD theory solution
• Effect of friction on instabilities – MD vs. linear stability analysis (LSA) of theory
Molecular Dynamics
• Input – System length scale (L/d) – Restitution coefficient (e) – Volume fraction (ϕ)
• 3-dimensional domain • Hard sphere collision model
– Binary, instantaneous collisions • Relevant Output
– Particle positions & velocities
MD: Fourier Analysis
Goldhirsch, Tan, Zanetti, J. Sci. Comput. (1993)
“Mass Mode” vs. wavenumber Particle positions (2D MD simulation)
At 400 collisions per particle (cpp)
MD: Fourier Analysis
0
5
10
15
20
0 2 4 6 8 10
Mas
s Mod
e k/π
2 cpp
40 cpp
800 cpp
Mitrano et al., PRE (2012)
e=0.6, ϕ=0.2 N=2000
collisions/particle
MD: Fourier Analysis
0
5
10
15
20
0 2 4 6 8 10
Mas
s Mod
e k/π
2 cpp
40 cpp
800 cpp
Mitrano et al., PRE (2012)
e=0.6, ϕ=0.2 N=2000
MD: Fourier Analysis
0
5
10
15
20
0 2 4 6 8 10
Mas
s Mod
e k/π
2 cpp
40 cpp
800 cpp
Mitrano et al., PRE (2012)
e=0.6, ϕ=0.2 N=2000
MD: Fourier Analysis
0
5
10
15
20
0 2 4 6 8 10
Mas
s Mod
e k/π
2 cpp
40 cpp
800 cpp
Mitrano et al., PRE (2012)
e=0.6, ϕ=0.2 N=2000
MD: Fourier Analysis
0
5
10
15
20
0 2 4 6 8 10
Mas
s Mod
e k/π
2 cpp
40 cpp
800 cpp
Mitrano et al., PRE (2012)
e=0.6, ϕ=0.2 N=2000
CFD: Cluster Detection
time
CFD: Cluster Detection
(%) e = 0.8 ϕ = 0.1
0
10
20
30
40
50
60
70
0.05 0.1 0.15 0.2 0.25 0.3
L clu
ster
/d
φ
e = 0.8 CFD
LSA
MD
Clustering Onset: CFD-MD-LSA
0
10
20
30
40
50
60
70
0.05 0.1 0.15 0.2 0.25 0.3
L clu
ster
/d
φ
e = 0.8 CFD
LSA
MD
Theory does well even though velocity gradients are present
Clustering Onset: CFD-MD-LSA
0
10
20
30
40
50
60
70
0.05 0.1 0.15 0.2 0.25 0.3
L clu
ster
/d
φ
e = 0.8 CFD
LSA
MD Nonlinear contributions to clustering are important
Clustering Onset: CFD-MD-LSA
Clustering Onset: CFD-MD-LSA
0
10
20
30
40
50
60
70
0.05 0.1 0.15 0.2 0.25 0.3
L clus
ter /
d
e = 0.6
0
10
20
30
40
50
60
70
0.05 0.1 0.15 0.2 0.25 0.3
L clus
ter /
d
ϕ
e = 0.7
0.05 0.1 0.15 0.2 0.25 0.3
e = 0.8 CFD
LSA
MD
0.05 0.1 0.15 0.2 0.25 0.3 φ
e = 0.9
Types of Dissipation
• Normal dissipation – Constant normal restitution coefficient 0 ≤ e ≤ 1
• Tangential dissipation – Constant tangential restitution coefficient -1 ≤ β ≤ 1
N
T
Types of Dissipation
• Normal dissipation – Constant normal restitution coefficient 0 ≤ e ≤ 1
• Tangential dissipation – Constant tangential restitution coefficient -1 ≤ β ≤ 1
VT
VT β et
No tangential impulse: “perfectly smooth”
Elastic tang. Impulse: “perfectly rough”
e
VN
Elastic Results
e = 1 ϕ = 0.3
5
7
9
11
13
15
17
19
21
23
25
-1 -0.5 0 0.5 1
Vort
ex C
ritic
al L
engt
h Sc
ale
β (perfectly smooth)
(perfectly rough)
6
8
10
12
14
16
18
20
-1 -0.5 0 0.5 1
Vort
ex C
ritic
al L
engt
h Sc
ale
β
Frictional Results
e = 0.9 ϕ = 0.3
Smooth-particle prediction
Extra note (not in original presentation)
• Very strange behavior for nearly smooth and nearly perfectly (elastically) rough particles can be traced to the energy ratio and more directly the fact that we only allow for “sticking” collisions that depend on the relative tangential overall velocity. Highly rotating particle are caused to separate since the tangential component is so large giving to a large tangential impulse. (vortex motion is dependent on the tangential translation alignment). E_t is a tangential translational restitution coefficient that is well correlated to vortex motion- high et values hinder vortex formation. Next slide shows that the particle rotation is very high on the left side. As particle become more and more rough the tangential impulse is inherently larger. We briefly examine a friction model that allows for either sticking or coulomb-governed sliding collisions a few slides later.
Temperature Ratio (Rotation/Translation)
0.1
1
10
100
1000
10000
-1 -0.5 0 0.5 1
RE/K
E
β
e = 0.9 ϕ = 0.3
Temperature Ratio (DEM-theory comparison)
e = 0.9 ϕ = 0.3
0.1
1
10
100
1000
10000
-1 -0.5 0 0.5 1
RE/K
E
β
DEM
Theory
Theory: Santos, Kremer, Garzó, Prog Theor Phys, Suppl (2010)
6
8
10
12
14
16
18
20
-1 -0.5 0 0.5 1
Criti
cal L
engt
h Sc
ale
β
DEM Theory
Frictional Results
e = 0.9 ϕ = 0.3
Instabilities attenuated
Instabilities enhanced
smooth
Tangential Translational Restitution Coefficient (et)
T
N
e = 1
v
et = -2 et = -1 et = 0 et = 2
Increased rel. tang. velocity: Vortices Suppressed
No change
Decreased rel. tang. velocity: Vortices Enhanced
Onsets normalized to smooth-particle value
0.5
1
1.5
2
2.5
-1 -0.5 0 0.5 1 β
MD Vortex
MD Cluster
e_t
theory vortex
theory cluster
e = 0.9 ϕ = 0.3
Extra note 2
• The et shown is not just averaged
• First take absolute value of et • Take log10 • Average • Raise 10 to the average • This is because we want et=0.1 and 10 to
average to 1 not close to 5
A Coulomb-friction model: Onset of vortices
0
5
10
15
20
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Vort
ex C
ritic
al L
engt
h Sc
ale
β
Vortex Theory
mu=0
mu=inf
mu=0.1
mu=0.5
e = 0.9 ϕ = 0.3
mu=1.0
Concluding Remarks
• MD vs CFD vs LSA – Excellent agreement between kinetic theory and MD
simulations – Small-gradient, molecular chaos assumptions of
theory are not so restrictive – Nonlinear mechanisms are important for clusters
• Frictional dissipation – All dissipation is not created equal – A frictional cooling rate alone does well (other transport coef.’s neglect friction)
Future Work
• Increased system complexity – Polydisperse particles – Non-spherical particles – Fluid phase – Bulk flow – Improved dissipation model – Wall boundaries
QUANTIFYING THE UNCERTAINTY OF KINETIC-THEORY PREDICTIONS OF CLUSTERING
May 31st, 2012 University Coal Research
Pittsburgh, PA
Peter P. Mitrano [email protected]
University of Colorado at Boulder Chemical and Biological Engineering
Sofiane Benyahia (NETL) Steven R. Dahl John R. Zenk Andrew M. Hilger Christopher J. Ewasko Christine M. Hrenya