QUANTIFYING THE UNCERTAINTY OF -THEORY P C · QUANTIFYING THE UNCERTAINTY OF KINETIC-THEORY PREDICTIONS OF CLUSTERING May 31 st, 2012 University Coal Research Conference . Pittsburgh,

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