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

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

Flowing Granular Material Flowing Granular Material System Particles (catalyst pellets, Gas powders, gravel, rice grains, cement, coal, ores, sand, glass beads, ball Dense flow bearings, …) Medium (usually air) Solid Interparticle interactions dominate – no effect of medium IIT Bombay

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 IIT Bombay

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 Blast furnace: steel manufacture IIT Bombay

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) IIT Bombay

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 IIT Bombay

Single Particle Motion Single Particle Motion Lower density tracer particles in sheared annulus = ρ − ρ F ( ) Vg Force p pt Song et al. , PNAS , 102 , 2299-2304 (2005) IIT Bombay

Effective Temperature Effective Temperature (Not granular temperature) 2 〉= 2 Dt 〈Δ z Stokes-Einstein D = T E ξ Parisi, PRL , 1997 Berthier and Barrat, JCP , 2002 t (s) 〈Δ z 〉 〈Δ z 〉= Ft /ξ 2 〉= 2 T E 〈Δ z F IIT Bombay

System Definition System Definition Sedimentation of a higher mass particle in a flowing layer of otherwise identical particles with diameter d ● DEM simulations in 3d – soft particles ● Viscoelastic force model (L3 model of Silbert et al. 2001) IIT Bombay

Theory: Single Particle Motion Theory: Single Particle Motion Effective medium approach Net force on heavy particle V g y v H F H = m H g y −ρ V E g y y weight buoyancy x Density: ρ=( m L / V )ϕ V E g y v H V E = V /ϕ Effective volume: y F H =( m H − m L ) g y x m : mass, V : volume of particles Drag force: Modified Stokes Law H : Heavy; L : Light F d = c π η v H d IIT Bombay

Drag Force Drag Force Terminal velocity ( F H = F d ) ( m H − m L ) g y = c π η v H d Dimensionless form ( ̄ m H − 1 ) cos θ= c π ̄ η̄ v H Low Re (0.01-0.18) Tripathi and Khakhar, PRL , 2011 IIT Bombay

Details Details Disturbance velocity and Number density Tripathi and Khakhar, PRL , 2011 IIT Bombay

Theory: Mixtures Theory: Mixtures Effective medium approach g y g y Net force on heavy particle v H F H = m H g y −〈ρ〉 V E g y y x weight buoyancy g y Density: 〈ρ〉=[ m H f + m L ( 1 − f )]ϕ/ V v H V E = V /ϕ Effective volume: y x F H =( 1 − f )( m H − m L ) g y m : mass, V : volume of particles H : Heavy; L : Light f : number fraction of H Segregation velocity v H = F H / c π η d IIT Bombay

Theory Theory Segregation flux: nD T E = D [ c π η d ] = = − − − s J nfv ( m m ) g f ( 1 f ) H H H L y T E Diffusion flux: Equilibrium: df = − J H nD s + = J J 0 dy H H Self-diffusivity = Binary diffusivity for equal size particles Equilibrium profile dy =−( m H − m L ) g y 1 df f ( 1 − f ) T E Sarkar and Khakhar, EPL , 2008 IIT Bombay

Mixture Profiles Mixture Profiles IIT Bombay

Results Results Diffusivity ( D yy ) Mean square displacement Granular temperature Effective temperature IIT Bombay

Results Results IIT Bombay

Experimental Study Experimental Study Equal size particles – different density 2 mm steel balls + 2 mm glass beads High speed video (500 frames/s), Image analysis Sarkar and Khakhar, EPL , 2008 IIT Bombay

Profiles Profiles IIT Bombay

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

Comparison to Theory Comparison to Theory Good agreement between theory and experiment – no fitted parameters; two compositions Sarkar and Khakhar, EPL , 2008 IIT Bombay

Rheology of Dense Flows Rheology of Dense Flows P γ ˙ Macroscopic time scale 1 / ˙ γ Microscopic time scale 1 / 2 d (ρ p / P ) d , ρ p P : pressure γ : shear rate ˙ ρ p ,d : particle density, diameter Inertial number γ d ˙ Dense flows: Low I I = 1 / 2 ( P /ρ p ) IIT Bombay

Rheology of Dense Flows Rheology of Dense Flows Friction coefficient Viscosity μ= τ xy η= τ xy γ =μ P P =μ( I ) γ ˙ ˙ Solid volume fraction empirical functions ϕ=ϕ( I ) Pouliquen et al. 2004 Da Cruz et al. 2005 Lois et al. 2005 IIT Bombay

Shear Flow Results Shear Flow Results Bagnold Frictional, inelastic particles IIT Bombay Tripathi and Khakhar, Phys. Fluids , 2011

(I), φ (I) µ (I), φ (I) µ IIT Bombay Tripathi and Khakhar, Phys. Fluids, 2011

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

Extension to Mixtures Extension to Mixtures Inertial number γ d mix ˙ I = √ P /ρ p,mix d mix = d 1 ϕ 1 + d 2 ϕ 2 ϕ 1 +ϕ 2 ρ p,mix =ρ p , 1 ϕ 1 +ρ p, 2 ϕ 2 ϕ 1 +ϕ 2 Tripathi and Khakhar, Phys Fluids, 2011 IIT Bombay

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

Full Model Predictions Full Model Predictions IIT Bombay

Theory: Single Particle Motion Theory: Single Particle Motion Different size particles Net force on big particle g y v B F B = m B g y −ρ V E g y y weight buoyancy x Density: ρ=( m S / V S )ϕ g y v B V E = ? Effective volume: y x Drag force: Modified Stokes Law F d = c π η v B d B m : mass, V : volume of particles B : Big, S : Small m B g y =ρ V E g y − c πη v B d B IIT Bombay

Results Results ∗ =ρ V E m B m B =ρ V E − c π η v B d B / g y IIT Bombay

Results Results IIT Bombay

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 IIT Bombay