Modeling Granular Material Mixing and Segregation Using a Multi-Scale Model Yu Liu, Prof. Marcial Gonzalez, Prof. Carl Wassgren

School of Mechanical Engineering, Purdue University, West Lafayette, IN

MotivationGranular material mixing and segregation• Granular material mixing and segregation plays an important role in many

industries ranging from pharmaceuticals to agrochemicals• Predictive engineering design of industrial powder blenders remains

underdeveloped due to the lack of quantitative modeling tools

ObjectiveDevelop a predictive model of granular material mixing and segregation for industrial equipment • Quantitatively predict the magnitude and rate of powder mixing and segregation• Be capable of modeling industrial-scale equipment• Demonstrate understanding to regulators in particle mixing and segregation

Multi-Scale Model

Diffusion correlations (3-D)• 𝑫 is an anisotropic tensor instead of an isotropic value• Off-diagonal components 𝐷𝑥𝑦 and 𝐷𝑦𝑥 are an order of magnitude smaller than the

diagonal components 𝐷𝑥𝑥 and 𝐷𝑦𝑦

Utter et al. (2004, Phys Rev Lett, Vol. 69); Hsiau et al. (1999, J. Rheol, Vol. 43)

• 𝐷𝑖𝑖 = 𝑘1 𝛾𝑖 𝑑2 + 𝑘2( 𝛾𝑗 + 𝛾𝑘) 𝑑2

𝑘2 = 1.9𝑘1 according to Utter et al. (2004 , Phys Rev Lett, Vol. 69) 𝑘1 can be calibrated from DEM simulations or experiments

Segregation correlations (2-D)• Percolation is one of the most important mechanisms causing segregation• 𝒗𝑝 acts in the direction of gravity

• According to Fan et al. (2014, J. Fluid Mech, Vol. 741): 𝑣𝑝,𝑙 = 𝑆 𝛾 (1 − 𝑐𝑙) & 𝑣𝑝,𝑠 = −𝑆 𝛾 (1 − 𝑐𝑠)

𝑆 can be calibrated from DEM simulations or experiments

FEM ModelModel implementations• The commercial FEM package Abaqus V6.14 is used to perform the simulations• The Coupled Eulerian-Lagrangian (CEL) approach in Abaqus is applied to handle

highly deformable material elements• Within the Eulerian domain, the material stress-strain behavior is modeled using

the Mohr-Coulomb elastoplastic (MCEP) model• Material properties can be measured from independent, standard tests

Bulk internal friction angle 𝜑 and cohesion 𝑐 => Shear test Bulk wall friction angle 𝜙=> Shear test Young’s Modulus 𝐸 and Poisson’s ratio 𝜐 => Uniaxial compression test

FEM simulation results – velocity profile• Rotating drum

• Conical and wedge-shaped hopper

• V blender and Tote blender

3-D Tote blender - mixing• Compared with published experiments of binary mixing of glass beads in an industrial-

scale Tote blender from Sudah et al. (2005, AIChE J., Vol. 51)• All the parameters were calibrated from independent experiments• Predictions of the mixing rate (relative standard deviation, RSD) from the multi-scale

model compare well quantitatively to the published experimental data

2-D rotating drum - segregation• Compared with published DEM simulations of binary segregation in a lab-scale rotating

drum from Schlick et al. (2015, J. Fluid Mech, Vol. 765) • All the parameters were derived directly from the published work• Predictions compare well quantitatively to DEM results

2-D conical hopper - segregation• Compared with published experiments of binary segregation of glass beads in different

conical hoppers from Ketterhagen et al. (2007, Chem Eng Sci, Vol. 62)• All the parameters were calibrated directly from the published work• Predictions from the multi-scale model compare well quantitatively to experiments

Macroscopic scale model• Predicts: advective flow field• Depends on: system geometries material bulk properties boundary conditions

• Method used: FEM

Microscopic scale model• Predicts: local diffusion / segregation rates• Depends on: particle properties local material concentration local shear rate and and solid fraction

• Method used: DEM / Experiments

Advection-diffusion-segregation equation𝜕𝑐𝑖𝜕𝑡

= −𝛻 ∙ 𝒗𝑐𝑖 + 𝛻 ∙ 𝑫𝛻𝑐𝑖 − 𝛻 ∙ 𝒗𝑝𝑐𝑖

• Predicts: global material concentration• Depends on: advective velocity (macro scale) diffusion and segregation rates (micro scale)

Conical Wedge-shaped

FEM simulations

Conical Wedge-shaped

DEM simulations

FEM simulation DEM simulation

V blender

Tote blender

Initial loading conditions

Side-Side Top-Bottom

Results2-D rotating drum - mixing• Compared with DEM simulations of binary mixing in a lab-scale rotating drum • All the parameters were derived from published work by Fan et al. (2015, Phys Rev

Lett, Vol. 115)• Predictions of concentration profiles from the multi-scale model compare well

quantitatively to DEM results

Multi-scale model predictionsFEM simulations

Co

ncen

tration

of red

particles

Co

nce

ntr

atio

n o

f sm

all p

arti

cles

DEM simulation Multi-scale model

Concentration of red particles

Concentration of small particles

Concentration of small particles

B. Utter, R.P. Behringer, Self-diffusion in dense granular shear flows, Phys. Rev. E - Stat. Nonlinear, Soft Matter Phys. 69 (2004) 1–12. O.S. Sudah, P.E. Arratia, A. Alexander, F.J. Muzzio, Simulation and experiments of mixing and segregation in a tote blender, AIChE J. 51 (2005) 836–844.S.-S. Hsiau, Y.-M. Shieh, Fluctuations and self-diffusion of sheared granular material flows, J. Rheol. (N. Y. N. Y). 43 (1999) 1049–1066. C.P. Schlick, Y. Fan, P.B. Umbanhowar, J.M. Ottino, R.M. Lueptow, Granular segregation in circular tumblers: Theoretical model and scaling laws, J. Fluid Mech. 765 (2015) 632–652.Y. Fan, C.P. Schlick, et al., Modelling size segregation of granular materials: The roles of segregation, advection and diffusion, J. Fluid Mech. 741 (2014) 252–279. Y. Liu, M. Gonzalez, C. Wassgren, Modeling Granular Material Blending in a Rotating Drum using a Finite Element Method and Advection-Diffusion Equation Multi-Scale Model, AIChE J. (2018). doi:10.1002/aic.16179.Y. Fan, P.B. Umbanhowar, J.M. Ottino, R.M. Lueptow, Shear-Rate-Independent Diffusion in Granular Flows, Phys. Rev. Lett. 115 (2015) 1–5. Y. Liu, A.T. Cameron, M. Gonzalez, C. Wassgren, Modeling granular material blending in a Tote blender using a finite element method and advection-diffusion equation multi-scale model, Powder Technol. (under review).