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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).