Discrete Element Methods.pptx

Discrete Element Methods

1. 9/13/2022 Discrete Element Method (DEM) DISCRETE ELEMENT METHOD

2. INTRODUCTION What is DEM? The computation of the motion and effect of a large number of small particles A simulation of discreet elements https://www.youtube.com/watch?v=-j1lCCznSrU 1 9/13/2022 Discrete Element Method (DEM)

3. a way of simulating discrete matter a numerical model capable of describing the mechanical behavior of assemblies of discs and spheres a particle-scale numerical method for modeling the bulk behavior of granular materials and many geomaterials (coal, ores, soil, rocks, aggregates) capture dual nature of materials CONT’D…. 2 DEM 9/13/2022 Discrete Element Method (DEM)

4. 9/13/2022 Discrete Element Method (DEM) CONT’D…. 3 Discretization of Space! Lagrangian (ex: DEM) Discontinuous Classical mechanics interaction (general) Resolution at particle level Computationally expensive Eulerian (ex: FEM) Continuous Related stresses/stains via constitutive EQs Resolution filled throughout grid Computationally cheaper Track position and velocity of moving particle Track velocity (or flux) at fixed grid locations http://15462.courses.cs.cmu.edu/fall2018/lecture/pdes/

5. 9/13/2022 Discrete Element Method (DEM) 4 CONT’D…. What is unique about DEM? Each particle has its own rotational, positional, radial, and momentum vectors that can be calculated using simple Newtonian physics (Kong, 2019) Simulation consists of three parts Small timesteps must be used, as solution is only conditionally stable (O’Sullivan & Bray, 2004) Ideal for modeling separate, discrete particle situations, like, Colloids, granular mater. Powder, bulk materials in storage, progressive fracture and failure

6. 9/13/2022 Discrete Element Method (DEM) Industrial applications of DEM? CONT’D…. Chemicals Pharmaceuticals Ceramics Metals Food Agriculture 5

7. 9/13/2022 Discrete Element Method (DEM) Modeling movement of individual particles Micromechanical level of analysis Coupled with FEM, CFD Complex particle geometries and arrangements Complicated validation process Computationally expensive Advantages and Disadvantages of DEM CONT’D…. ADVANTAGES DISADVANTAGES 6

8. 9/13/2022 Discrete Element Method (DEM) 7 HISTORY OF DEM 1971 1974 1978 1985 1992 Cundall develops DEM to assist with modeling rock mechanics (Cundall 1971) Cundall translates the method into an RBM code (Cundall 1974) Cundall translates the method into a FORTRAN code (Cundall et. al 1978) Williams and Mustoe generalize the method, comparing it to FEM (Williams & Mustoe 1985) Cundall & Hart develop codes to perform the DEM in 3 dimensions (Cundall & Hart 1985) Shi develops Discontinuous Deformation Analysis (Shi 1992) DEM

9. 9/13/2022 Discrete Element Method (DEM) General Principles Newton’s Second Law of Motion Conservation of momentum Particle motion Force Displacement Law Stiffness Friction 8 Displacement / velocity boundary condition Force boundary conditions Force Displacement Law (e.g. stiffness, friction) Newton’s Second Law of Motion DEM

10. 9/13/2022 Discrete Element Method (DEM) CONT’D…. Soft Sphere and Hard Sphere Rigid particles but small overlap is allowed Evaluates forces accurately Simultaneous contacts possible Impulsive forces Exchange of momentum One collision at a time Soft Sphere Hard Sphere 9

11. 9/13/2022 Discrete Element Method (DEM) CONT’D…. Advantage of Soft Sphere             Pre-contact Contact (Instant contact and no overlap) Post-contact             Pre-contact Contact (Long lasting contact) Post-contact                 Overlap 10

12. 9/13/2022 Discrete Element Method (DEM) CONT’D…. DEM Main micro-parameters of particle system Category Name Intrinsic parameters Poisson’s ratio ( ν) Density ( ρ)/ kg/m3 Shear modulus (G)/Pa Contact parameters between particles Coefficient of restitution Coefficient of static friction Coefficient of rolling friction Contact parameters between particles and geometry Coefficient of restitution Coefficient of static friction Coefficient of rolling friction 11

13. 9/13/2022 Discrete Element Method (DEM) General Principles DEM 12 Particle motion: Governed by Newton’s equation Rotation of particle M = I α M = I   M – torque acting on particle I – moment of inertia α – angular acceleration – angular momentum  

14. 9/13/2022 Discrete Element Method (DEM) DEM Particle motion: Governed by Newton’s equation CONT’D…. Translation of particle F= m α = m   gravitational force (mg) – contact force – not contact force m – mass of the particle – translational velocity   13

15. 9/13/2022 Discrete Element Method (DEM) DEM CONT’D…. 14 Contact forces: Normal force Tangential force   , - damping coefficients , - displacement , - relative velocities  

16. 9/13/2022 Discrete Element Method (DEM) DEM CONT’D…. 15 Non - contact forces: Gravitational force - mass of the particle G – gravitational constant R – distance Molecular forces  

17. 9/13/2022 Discrete Element Method (DEM) DEM 16 CONT’D…. Particle positions: x (t + ∆t) = x(t) + ν (t)∆t Particle velocity: ν (t + ∆t) = ν (t) + a(t)∆t Numerical Integration:

18. 9/13/2022 Discrete Element Method (DEM) DEM 17 Classification of particle interaction force models by contact force and non-contact force. Non-Contact Force Van der Waals force Liquid bridge force Electrostatic force Linear spring model Non – linear spring Hertz- Mindlin Hertz-Mindlin + JKR DMT Model Linear Spring-Dashpot Hysteretic Model Thornton Model Particle interaction Contact Non- Contact Elastic Inelastic Linear model Non- linear model

19. 9/13/2022 Discrete Element Method (DEM) DEM 18 CONTACT MODELS Contact between two particles occurs in the finite area Area consists of the normal and tangential plane Contact force - normal and tangential Overlap ( δ ) = -d Damping forces - friction forces and cohesive forces Determine the acceleration of particles  

20. 9/13/2022 Discrete Element Method (DEM) DEM 19 ELASTIC CONTACT MODELS LINEAR SPRING MODEL Two particles in contact are both normally and tangentially connected by linear spring Energy is not consumed and the contact is considered completely elastic Linear relationship b/w force and displacement Limitation: kinetic energy is dissipated by plastic deformation Normal force Tangential force  

21. 9/13/2022 Discrete Element Method (DEM) DEM 20 HERTZ-MINDLIN MODEL Nonlinear elastic model Contact between two particles in the normal direction – Hertz Contact between two particles in the tangential direction – Mindlin Hertz- Mindlin model – complexity, time-consuming Simplification: no slip – Hertz and Mindlin Accuracy - pharmaceutical industry Eeq , Req and Geq are the equivalent Young’s modulus, equivalent radius and equivalent shear modulus

22. 9/13/2022 Discrete Element Method (DEM) DEM 21 Model the contact of cohesive particles The adhesive theory using a balance between stored elastic energy and loss of surface energy Opposite force owing to the pulling force HERTZ-MINDLIN + JKR MODEL a - contact area γ - surface energy

23. 9/13/2022 Discrete Element Method (DEM) DEM 22 DMT MODEL Cohesion at the contact periphery Hertz- Mindlin + JKR model based on the surface energy Suitable for hard materials Solids with a small tip radius and low surface energy

24. 9/13/2022 Discrete Element Method (DEM) DEM 23 INELASTIC CONTACT MODELS LINEAR SPRING-DASHPOT MODEL Elastic models – accumulation of energy Inelastic models - to model the dissipation of energy Plastic deformation between particles occurs Composed of linear spring and dashpot components Linear spring describes the repulsive forces Dashpot dissipates the relative kinetic energy Normal contact force  

25. 9/13/2022 Discrete Element Method (DEM) DEM 24 Uses various spring constant at the loading, Unloading and reloading stages Hysteretic model - linear contact models Normal direction - a partially latched spring force-displacement model Mindlin and Deresiwicz theory - the constant normal force in the tangential direction Limitation: it describes the plastic deformation only in the normal direction where K1 and K2 are the spring constants in the loading and unloading stages HYSTERETIC MODEL

26. 9/13/2022 Discrete Element Method (DEM) DEM 25 Explains plastic deformation Proposed for normal contact between two elastic, perfectly spherical plastic particles Based on the Hertz theory (normal force-displacement relationship during the initial elastic loading ) Plastic deformation occurs if the limiting contact pressure is reached at the center of the contact area THORNTON MODEL where Fny and δ y denote the normal contact force and displacement

27. 9/13/2022 Discrete Element Method (DEM) 26 DEM CONT’D…. DEM Model Workflow Accelerations Velocities Positions Forces Contacts Newton’s Law Contact mechanics

28. 9/13/2022 Discrete Element Method (DEM) DEM 27 Numerical solution software for DEM Open Source Commercial Often written in C, C++, Fortran, python and interacted with within the terminal Many limited to Linux/ Unix, and ran using code- specific commands and functions, or ran from an input file Often more powerful, with convenient GUI’s and built-in system coupling Very expensive for limited licenses ABAQUS: ~ $ 65,000

29. 9/13/2022 Discrete Element Method (DEM) DEM 28 Discrete Element Modeling - DEM Software | Altair EDEM https://www.altair.com › edem

30. 9/13/2022 Discrete Element Method (DEM) DEM 29 1. Rogers, A.; Ierapetritou , M. Challenges and opportunities in modeling pharmaceutical manufacturing processes. Comput . Chem. Eng. 2015, 81, 32–39. 2. U.S. Food and Drug Administration. Guidance for industry: Q8 (R2) pharmaceutical development; Food and Drug Administration: White Oak, MD, USA, 2006. 3. Suresh, P.; Basu , P.K. Improving pharmaceutical product development and manufacturing: Impact on cost of drug development and cost of goods sold of pharmaceuticals. J. Pharm. Innov . 2008, 3, 175–187. 4. Ketterhagen,W.R .; am Ende, M.T.; Hancock, B.C. Process modeling in the pharmaceutical industry using the discrete element method. J. Pharm. Sci. 2009, 98, 442–470. 5. Kremer, D.; Hancock, B. Process simulation in the pharmaceutical industry: A review of some basic physical models. J. Pharm. Sci. 2006, 95, 517–529. References

31. 9/13/2022 Discrete Element Method (DEM) DEM 30 6. Pandey, P.; Bharadwaj, R. Predictive Modeling of Pharmaceutical Unit Operations; Woodhead Publishing: Cambridge, UK, 2016. 7. Björn, I.N.; Jansson, A.; Karlsson, M.; Folestad , S.; Rasmuson , A. Empirical to mechanistic modelling in high shear granulation. Chem. Eng. Sci. 2005, 60, 3795–3803. 8. Reklaitis , G.V.; García-Munoz, S.; Seymour, C. Comprehensive Quality by Design for Pharmaceutical Product Development and Manufacture; JohnWiley & Sons: Hoboken, NJ, USA, 2017. 9. Wassgren , C.; Curtis, J.S. The application of computational modeling to pharmaceutical materials science. MRS Bull. 2006, 31, 900–904. 10. Norton, T.; Sun, D.-W. Computational fluid dynamics ( cfd )—An eective and effcient design and analysis tool for the food industry: A review. Trends Food Sci. Technol. 2006, 17, 600–620. CONT’D….

32. THANK YOU 9/13/2022 Discrete Element Method (DEM)