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Rheology and Segregation Segregation of of Rheology and Granular - - PowerPoint PPT Presentation

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Rheology and Segregation Segregation of of Rheology and Granular Mixtures in Dense Flows Granular Mixtures in Dense Flows Devang Khakhar Department of Chemical Engineering Indian Institute of Technology Bombay Mumbai, India Unifying

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IIT Bombay

Rheology and Rheology and Segregation Segregation of

  • f

Granular Mixtures in Dense Flows Granular Mixtures in Dense Flows

Devang Khakhar Department of Chemical Engineering Indian Institute of Technology Bombay Mumbai, India

“Unifying Concepts in Materials” J. A. Krumhansl Symposium, 6-8 February, 2012, NCBS, Bangalore

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Flowing Granular Material Flowing Granular Material

Particles (catalyst pellets, powders, gravel, rice grains, cement, coal, ores, sand, glass beads, ball bearings, …) Medium (usually air)

System

Interparticle interactions dominate – no effect of medium

Solid Dense flow Gas

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Flowing Granular Mixtures Flowing Granular Mixtures

  • Spontaneous

segregation due differences in particle properties (e.g., size, density)

  • Rheology depends on

local concentration of different species.

  • Segregation and

rheology are coupled

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Blast furnace: steel manufacture

Blast Furnace Feeding Blast Furnace Feeding

Control of blast furnace: pattern of pouring coke and ore by rotating chute. Bed porosity determines air flow and temperature distribution. Segregation of particles during flow affects bed porosity

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Theoretical Approaches Theoretical Approaches

  • Kinetic theory
  • JT Jenkins and F Mancini, Phys. Fluids A 1, 2050 (1989);

L Trujillo, M Alam,and HJ Herrmann, Europhys. Lett. 64, 190 (2003).

  • Partial stresses
  • JMNT Gray and AR Thornton, Proc. R. Soc. A 461 1447

(2005); Y Fan and KM Hill, New J. Phys. 13, 095009 (2011).

  • Single particle motion
  • DV Khakhar, JJ McCarthy and JM Ottino, Phys. Fluids 9

3600 (1997)

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Outline Outline

  • Single particle motion (buoyancy, drag force)
  • Density segregation of mixtures
  • Rheology of mixtures (size and density)
  • Combined model predictions
  • Conclusions

Acknowledgments

  • Anurag Tripathi
  • Department of Science and Technology
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Single Particle Motion Single Particle Motion

Lower density tracer particles in sheared annulus Vg F

pt p

) ( ρ ρ − = Force Song et al., PNAS, 102, 2299-2304 (2005)

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Effective Temperature Effective Temperature

Stokes-Einstein (Not granular temperature)

t (s)

〈Δ z

2〉=2 Dt

〈Δ z〉=Ft/ξ 〈Δ z

2〉=2T E

〈Δ z〉 F D=T E ξ

Parisi, PRL, 1997 Berthier and Barrat, JCP, 2002

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System Definition System Definition

Sedimentation of a higher mass particle in a flowing layer of

  • therwise identical particles with diameter d
  • DEM simulations in

3d – soft particles

  • Viscoelastic force

model (L3 model of Silbert et al. 2001)

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Effective medium approach

m: mass, V: volume of particles H: Heavy; L: Light

y y x

Theory: Single Particle Motion Theory: Single Particle Motion

x gy vH vH

FH=mH g y−ρV E g y

weight buoyancy

ρ=(mL/V )ϕ V E=V /ϕ FH=(mH−mL)g y Fd=c π ηv H d Net force on heavy particle Density: Effective volume: Drag force: Modified Stokes Law V VE

gy

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Drag Force Drag Force

(mH−mL)g y=c π ηvH d ( ̄ mH−1)cosθ=c π ̄ η̄ vH Terminal velocity (FH=Fd) Dimensionless form

Tripathi and Khakhar, PRL, 2011 Low Re (0.01-0.18)

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Details Details

Disturbance velocity and Number density

Tripathi and Khakhar, PRL, 2011

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m: mass, V: volume of particles H: Heavy; L: Light f : number fraction of H

y x y x

Theory: Mixtures Theory: Mixtures

Effective medium approach

gy gy gy vH vH

FH=mH g y−〈ρ〉V E g y

weight buoyancy

〈ρ〉=[mH f +mL(1−f )]ϕ/V V E=V /ϕ Net force on heavy particle Density: Effective volume: Segregation velocity v H=F H/c π ηd FH=(1−f )(mH−mL)g y

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Segregation flux: ) 1 ( ) ( f f g m m T nD nfv J

y L H E H s H

− − − = = Diffusion flux: dy df nD J H − = Equilibrium: = +

H s H

J J

Sarkar and Khakhar, EPL, 2008

Theory Theory

T E=D[c π ηd]

1 f (1−f ) df dy=−(mH−mL)g y T E Equilibrium profile

Self-diffusivity = Binary diffusivity for equal size particles

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Mixture Profiles Mixture Profiles

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Results Results

Mean square displacement

Diffusivity (D yy) Effective temperature

Granular temperature

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Results Results

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2 mm steel balls + 2 mm glass beads High speed video (500 frames/s), Image analysis Sarkar and Khakhar, EPL, 2008

Experimental Study Experimental Study

Equal size particles – different density

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Profiles Profiles

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Diffusivity of glass beads and steel balls is nearly the same Diffusivity scales with shear rate

Diffusivity Diffusivity

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Good agreement between theory and experiment – no fitted parameters; two compositions

Sarkar and Khakhar, EPL, 2008

Comparison to Theory Comparison to Theory

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Rheology of Dense Flows Rheology of Dense Flows

Inertial number ˙ γ P Macroscopic time scale 1/ ˙ γ Microscopic time scale d(ρp/P)

1/2

d ,ρp I= ˙ γ d (P/ρp)

1/2

Dense flows: Low I

P : pressure ˙ γ : shear rate ρp,d : particle density, diameter

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Rheology of Dense Flows Rheology of Dense Flows

Viscosity η= τ xy ˙ γ =μ P ˙ γ Friction coefficient μ= τxy P =μ(I) ϕ=ϕ(I) Solid volume fraction empirical functions

Pouliquen et al. 2004 Da Cruz et al. 2005 Lois et al. 2005

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Shear Flow Results Shear Flow Results

Frictional, inelastic particles Tripathi and Khakhar, Phys. Fluids, 2011 Bagnold

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µ µ(I), (I), φ φ(I) (I)

Tripathi and Khakhar, Phys. Fluids, 2011

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Viscosity Viscosity

Symbols: DEM simulation results. Lines: Theory Tripathi and Khakhar, Phys. Fluids, 2011

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Extension to Mixtures Extension to Mixtures

Inertial number

I= ˙ γ dmix

√P/ρp,mix

d mix=d1ϕ1+d2ϕ2 ϕ1+ϕ2 ρp,mix=ρp ,1ϕ1+ρp,2ϕ2 ϕ1+ϕ2

Tripathi and Khakhar, Phys Fluids, 2011

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Results Results

Pure Density Size Size-Density Symbols: DEM simulation results. Lines: Theory

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Full Model Predictions Full Model Predictions

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Different size particles

m: mass, V: volume of particles B: Big, S: Small

y y x

Theory: Single Particle Motion Theory: Single Particle Motion

x gy gy vB vB

FB=mB g y−ρV E g y

weight buoyancy

ρ=(mS/V S)ϕ V E=? Fd=c π ηvB dB Net force on big particle Density: Effective volume: Drag force: Modified Stokes Law mB g y=ρV E g y−c πηvBd B

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Results Results

mB=ρV E−c π ηvBd B/g y mB

∗=ρV E

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Results Results

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Conclusions Conclusions

  • Single particle motion: Buoyancy given by modified

Archimedes principle and drag force by modified Stokes Law.

  • Generalization to density segregation in mixtures:

Role of the effective temperature.

  • Model for rheology of mixtures (size and density).
  • Predictions for combined model for rheology and

density segregation.

  • Size segregation model: Effective volume