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Dissertation zur Erlangung des Doktorgrades
der Fakult¨at fu¨r Mathematik und Physik
der Albert-Ludwigs-Universit¨at Freiburg im Breisgau
Numerical Simulations
of Granular Flow and Filling
Claas Sven Bierwisch
27. April 2009Dekan: Prof. Dr. Kay K¨onigsmann
Erstgutachter: Prof. Dr. Michael Moseler
Zweitgutachter: Prof. Dr. Christian Els¨asser
Tag der Disputation: 27. April 2009Peer-reviewed publications concerning the contents of this thesis:
• C. Bierwisch, T. Kraft, H. Riedel, and M. Moseler. Three-
dimensional discrete element models for the granular statics and dynamics
of powders in cavity filling. Journal of the Mechanics and Physics of Solids,
57:10–31, 2009.
• C. Bierwisch, T. Kraft, H. Riedel, and M. Moseler. Die filling op-
timization via three-dimensional discrete element modeling. Submitted to
Powder Technology.
• C.Bierwisch andM.Moseler. Compressible kinematic modeling of gran-
ular flow. Manuscript in preparation.
Patents concerning the contents of this thesis:
• C. Bierwisch, T. Kraft, H. Riedel, and M. Moseler. Verfahren
zur Homogenisierung einer Pulverschu¨ttung bei der Herstellung von Pulver-
presslingen. Pending patent application.
Conference proceeding papers concerning the contents of this thesis:
• C. Bierwisch, B. Henrich, T. Kraft, M. Moseler, and H. Riedel.
3D-Modelling of die filling. In Proceedings of the EURO PM 2005, Volume 3,
Prague, Czech Republic, pages 331–337, Shrewsbury, UK, 2005. European
Powder Metallurgy Association.
• C. Bierwisch,T. Kraft,H. Riedel, andM. Moseler. Predicting den-
sity distributions in die filling. In Proceedings of the EURO PM 2007, Vol-
ume 3, Toulouse, France, pages 305–310, Shrewsbury, UK, 2007. European
Powder Metallurgy Association.
3• H. Riedel, A. Wonisch, C. Bierwisch, T. Kraft, and M. Moseler.
Simulationen von Prozessfolgen mit der Diskrete-Elemente-Methode. In
H.Kolaska, editor. Pulvermetallurgie—KompetenzundPerspektive, pages
153–175, Du¨sseldorf, Germany, 2007. VDI-Gesellschaft Werkstofftechnik.
• C. Bierwisch, T. Kraft, M. Moseler, and H. Riedel. Simulation
of die filling in 3D with special emphasis on vibration supported filling. In
Advances in powder metallurgy & particulate materials 2008, Part 1, pages
130–138, Princeton, USA, 2008. Metal Powder Industries Federation.
¨• C. Bierwisch, B. Weber, R. Kubler, M. Moseler, and G. Kleer.
Contact regimes in the wire sawing process: Explicit 3D modeling of PEG/SiC
slurry. In Proceedings of 23rd European Photovoltaic Solar Energy Con-
ference, Valencia, Spain, pages 1104–1108, Munich, Germany, 2008. WIP-
Renewable Energies.
• T. Kraft, C. Bierwisch, and H. Riedel. New advances in modeling
¨powder metallurgical processing steps. InL. S. Sigl,P. Rodhammer, and
H.Wildner, editors. Proceedingsofthe 17thPlanseeSeminar2009, Reutte,
Austria, 2009. Plansee Group.
Conference talks concerning the contents of this thesis:
• C. Bierwisch, B. Henrich, and M. Moseler. Particle methods in pow-
der technology. The second Russian-German Advanced Research Workshop
on Computational Scienceand High PerformanceComputing, Stuttgart, Ger-
many, March 14–16, 2005.
• C. Bierwisch,T. Kraft,M.Moseler, andH. Riedel. 3D-Modeling of
Powder Flow and Application to Gravity Die Filling. DPG Spring Meeting
of the Condensed Matter Division, Dresden, Germany, March 26–31, 2006.
• C. Bierwisch,T. Kraft,M. Moseler, andH. Riedel. Static-dynamic
transitions: Role of grain shape and coarse graining. DPG Spring Meeting of
the Condensed Matter Division, Regensburg, Germany, March 26–30, 2007.
• C. Bierwisch andM. Moseler. Grain coarsening effects in granular stat-
ics and dynamics. DPG Spring Meeting of the Condensed Matter Division,
Berlin, Germany, February 25–29, 2008.
4Contents
1 Introduction 13
2 Numerical methods 17
2.1 Discrete element method . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.1 Force Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.2 Non-rotational spheres . . . . . . . . . . . . . . . . . . . . . 20
2.1.3 Rolling spheres . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.4 Complex grains . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . 24
2.1.6 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . 24
2.1.7 Summary of model parameters . . . . . . . . . . . . . . . . . 24
2.2 Volume fraction computation . . . . . . . . . . . . . . . . . . . . . 26
3 A coarse graining scheme for discrete element modeling 29
3.1 Force scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Coarse graining tests . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.1 Bulk properties . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.2 Slit flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.3 Angle of repose . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 Modeling of real powders in static and dynamic regimes 45
4.1 Model fitting and validation . . . . . . . . . . . . . . . . . . . . . . 46
4.1.1 Fitting of parameters to slit outflow . . . . . . . . . . . . . . 46
4.1.2 Validation by using a larger slit . . . . . . . . . . . . . . . . 50
4.1.3 Fitting of parameters to angle of repose . . . . . . . . . . . . 51
4.1.4 Reciprocal validation of slit outflow and angle of repose . . . 53
4.1.5 Validation with volume fractions . . . . . . . . . . . . . . . 54
4.1.6 Validation with powder filling height in the feeding shoe . . 55
4.1.7 Validation with filling levels in the circular cavity . . . . . . 57
4.1.8 Velocity and volume fraction fields for slit outflow . . . . . . 59
4.2 Influence of further model parameters . . . . . . . . . . . . . . . . . 62
4.2.1 Initial random packing . . . . . . . . . . . . . . . . . . . . . 62
4.2.2 Young’s modulus . . . . . . . . . . . . . . . . . . . . . . . . 62
54.2.3 Dissipative constant . . . . . . . . . . . . . . . . . . . . . . 64
4.2.4 Effective wall friction . . . . . . . . . . . . . . . . . . . . . . 66
4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5 Continuum description and analytic expressions for granular outflow 69
5.1 Continuum description of velocity distributions . . . . . . . . . . . 70
5.1.1 DEM simulations . . . . . . . . . . . . . . . . . . . . . . . . 70
5.1.2 Kinematic modeling . . . . . . . . . . . . . . . . . . . . . . 71
5.1.3 Compressible kinematic modeling . . . . . . . . . . . . . . . 74
5.1.4 Scaling properties . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2 Analytic expressions for mass flow rates . . . . . . . . . . . . . . . . 82
5.2.1 Transient Beverloo equation . . . . . . . . . . . . . . . . . . 82
5.2.2 Mass discharge from moving shoe . . . . . . . . . . . . . . . 85
5.2.3 Experimental validation . . . . . . . . . . . . . . . . . . . . 85
5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6 Homogeneous filling of cavities 89
6.1 Coarse graining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.2 Adjustment of model parameters . . . . . . . . . . . . . . . . . . . 93
6.3 Prediction of density distributions . . . . . . . . . . . . . . . . . . . 94
6.4 Transient filling analyses . . . . . . . . . . . . . . . . . . . . . . . . 96
6.4.1 Density formation during filling . . . . . . . . . . . . . . . . 96
6.4.2 Grain displacement and surface densification . . . . . . . . . 96
6.5 Density (in)homogenization . . . . . . . . . . . . . . . . . . . . . . 100
6.5.1 Influence of shoe velocity . . . . . . . . . . . . . . . . . . . . 100
6.5.2 Cavity and shoe vibrations . . . . . . . . . . . . . . . . . . . 101
6.5.3 Volumetric filling . . . . . . . . . . . . . . . . . . . . . . . . 103
6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7 Summary and outlook 107
A Modeling of wire sawing 111
A.1 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
A.1.1 Modeling of PEG . . . . . . . . . . . . . . . . . . . . . . . . 111
A.1.2 Modeling of SiC grains . . . . . . . . . . . . . . . . . . . . . 113
A.1.3 Modeling of wire and ingot . . . . . . . . . . . . . . . . . . . 113
A.2 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
A.2.1 Dynamic similarity . . . . . . . . . . . . . . . . . . . . . . . 115
A.2.2 Hydrodynamic drag . . . . . . . . . . . . . . . . . . . . . . . 115
A.2.3 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
A.3 Wire sawing simulations . . . . . . . . . . . . . . . . . . . . . . . . 116
A.3.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6A.3.2 Contact regimes . . . . . . . . . . . . . . . . . . . . . . . . . 117
A.3.3 Stress on ingot surface . . . . . . . . . . . . . . . . . . . . . 117
A.3.4 Stress on SiC grains . . . . . . . . . . . . . . . . . . . . . . 119
A.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
B Numerical solution of partial differential equations 123
C Simulation parameters 129
Bibliography 135
7How do we know that the creations of worlds
are not determined by falling grains of sand?
Victor Hugo —Les Miserables
8Abstract
Flow and filling behavior of granular matter was investigated within the framework
of the discrete element method (DEM) in this work. The large number of individual
grains in many systems prevents a one-to-one mapping between experiments and
numerical simulations even on parallel computing clusters. Based on the conser-
vation of local energy density a novel coarse graining scheme was developed which
allows for the representation of a given system using artificially enlarged grains.
Thereby, a dimensional analysis yielded scaling rules for the DEM force laws.
In comparison with experiments, grain models of different mechanical and mor-
phological complexity were assessed with respect to their ability to reproduce and
predict statical properties (angle of repose) as well as dynamical properties (flow
rates, filling behavior) of an iron powder. It was found that an adequate modeling
of the grain shape is of particular importance in addition to the consideration of
inter-granular friction and cohesion.
A reliable grain model was used to analyze kinematics of hopper discharge (com-
pare Fig. 0.1) and density inhomogeneities in cavity filling (compare Fig. 0.2). An
existing continuum model for the velocity field inside a hopper was improved by
taking the influence of the local volume fraction into account. Another contribution
to the theoretical description of granular matter was made by deriving an analytic
expression for the mass discharge through a slit orifice from a moving shoe. Process
variations for cavity filling were developed which yield a considerable homogeniza-
tion of the density distribution.
9Figure 0.1: DEM simulation of granular discharge from a hopper. (a): Color
coded snapshot of velocity magnitude where blue means fast and red
means slow; (b): color coded averaged volume fraction where blue
means dilute and red means dense. The discovery of a relationship
between these two quantities is presented in Chapter 5.
10

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