NUMERICAL ANALYSIS of PROCESSES

NAP12

DEM (Discrete Element Method) or the method of discrete particles

SPH (Smooth Particle Hydrodynamics)

Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010

NAP12 DEMLagrange method describing continuous gas or liquid phase by partial differential equations while trajectories of particles or droplets by using ordinary differential equations of particle motion was described in the previous lecture NAP11. The most important assumption was the neglect of mutual interactions (collisions of specific particles), which means restriction to systems with a relatively low concentration of particulate phase (for example, a fluidized bed of high density particles had to be modeled based on the idea that the two phases of the continuum, i.e. using a mixture or using Euler-Euler).

Method DEM / DPM (Discrete Element / Particle Method often combined DEM + CFD) is apparently identical with the above Lagrangian methods, but with the crucial difference that the interaction of particles being considered. DEM is the basis for solving previously intractable problems in which the key role played by the contact deformation and mutual force interaction of particles up to a microscopic level (examples are problems with the fragmentation, migration and segregation of particles in a mill, hoppers, separators, etc.). The first application of DEM date back to the early seventies of the last century and are associated with rock mechanics (1971 Peter Cundall completed his doctorate at Imperial College London works: The Distinct Element Method for modeling jointed rock and granular material).

First paper on DEM is Cundall, P.A., Strack, O.D.L., 1979. A discrete numerical model for granular assemblies. Geotechnique 29 (1), 47–65.

NAP12 DEM – history at CTUDevelopment of DEM methods began in the late seventies and early eighties at the Process Engineering Department of CTU equipped with a computer TESLA and a programmable calculator HP -85 with a capacity of about 200 of floating point operations per second. The concept was based upon the following way of reasoning: Nature is discrete (molecules). For a description of this reality continuum mechanics operating with differential equations has been proposed. These equations are solved naturally by analog computers, except that the era of the sixties has subsided and analogs were replaced by more universal digital computers. For them, however, it was necessary to discretize the analog differential equations by finite differences. So why not to skip the continuum mechanics, why not take advantage of parallel processing on multiprocessor computers (each particle has its own processor) and to simulate the interaction directly. A key problem was the force interactions. Interaction of Lennard -Jones type were tested with poor results, even though today the Lennard Jones potential is used quite successfully in solving problems of molecular dynamics, for example, Aoki , Akiyama , Phys.Review 1995. Better simulation results were obtained for the empirical power type interactions

dt

dlF

lElF

n

n

ddynamicmmdm

static

1

12 )( ),

1()(

where F is the force acting between a pair of particles whose distance is ( l is seen as the equilibrium distance corresponding to zero static force). Ed coefficients (effective modulus) and d (viscosity) were determined on the basis of the equilibrium configuration of an infinite number of particles, macroscopic equation of state and viscosity corresponding to the shear flow. Integration of the equations of motion has been implemented in the program PARMOSI (Particle Motion Simulation) for HP85 calculator and used for modeling the dynamics of viscoelastic beam and fluid flow in the cavity. The aim was to demonstrate that it is possible to unify the solution of problems of fluid mechanics and solids, but the HP calculator allowed simultaneously simulate only a maximum of 5-9 particles. Well, it did not work out too well, the final report R.Žitný: Numerical algorithms for modeling of transport processes, CTU FS Prague, 1983, presented at the hearing of the Academy of Sciences was criticized and the DEM project ended.After all, it was just a youthful indiscretion , and in fact only because the results printed on thermal paper fades quickly , I will not even trace a lot of mistakes , nonsense and controversy that the whole project was.

NAP12 DEM

Bierstadt

D.O. Potyondy, P.A. Cundall / International Journal of Rock Mechanics & Mining Sciences 41 (2004) 1329–1364

NAP12 DEM

For each particle three equations of translational motion are solved

viscous forces gravityacceleration distant interactionscontact force

particle-particle

c nc f gii ij ik i i

j k

dum F F F F

dt

inertiaangularmoment

moments generated accelerationby j-th particle

ii ij

j

dI M

dt

together with 3 equations of particle rotation

Note: Typically, considering spherical particles or agglomerates of spheres (and then to calculate the contact forces using relations for spheres)

iv

jv

i

j

Ri

Ri

cijF

ijx

The key assumption DEM is that the simulation time step can be chosen so small that the interaction force applied only immediately adjacent particles i and j

NAP12 DEMDynamics of normal impact of two spheres

21

212

2

,mm

mmmF

dt

dm eff

neff

R1

R2

=R1+R2+x1-x2

m1 m2

Fn

x1

x2

2 21 2

1 22 2, n nd x d x

m F m Fdt dt

Initial conditions for time t=0

0(0) 0 (0)d

vdt

t

tn

d/dt

t

tn

v0

vn

0v

ve

nn

The result of previous equations for the specific interaction model are time courses (t), the characteristic time tn (contact time) and the coefficient of restitution en

Kruggel-EmdenH.et al.: Review and extension of normal force models for DEM. Powder Technology 171 (2007), pp.157-173

velocities after and before a contact

compression (overlap) during contact

NAP12 DEM contact forcesTo determine the contact forces models of linear and nonlinear viscoelasticity, viskoplasticity and potential are used. The simplest models are linear viscoelasticity, the parallel combination of linear springs and dampers

2

20n n

eff

d dm k

dt dt

springdamper

n n n dF k

dt

General solution =exp(t) , has to satisfy characteristic equation

212

2 )2

(2

0eff

n

eff

n

eff

nnn

eff mm

ki

mkm

Result (for initial condition (0)=0)1 2

2 2( ) ( ) 2 sin ( )2

n

eff

t n nmt t

eff eff

kt c e e ice t

m m

Time of contact and restitution coefficient

2 2/ ( ) exp( / ( ) )2 2 2

n n n n nn n

eff eff eff eff eff

k kt e

m m m m m

(these parameters depend only on the material and not the speed of collision).

Cundall Strack (1979)

NAP12 DEM contact forcesThe problem of elastic impact balls solved H. Hertz: Über die Berührung elastischer fester Körper, J.

für die Reine und Angewandte Mathematik 92 already in 1882. The dependence of force to displacement is not linear by the Hertz theory, but progressively increasing (with exponent 3/2) and the imaginary spring stiffness increases with the square root of the radius of the sphere. Hertz model is the basis of nonlinear viscoelastic models

4

3eff i jn

i j

E R Rk

R R

3/2

Hertzovskásíla tlumič m=0,1/2 nebo m=1/4

n n n m dF k

dt

The DEM applications used several variants of nonlinear viscoelastic models with different exponent m in the dissipative term (m = 0 Lee Herrmann, exponent m = 0.5 suggest Kuwabara and Kono, am = 0.25 using a model that derived Tsuji).

Plasticity models (hysteresis, different loading/unloading characteristics)

Linear plasticity model was published by Walton, Brown (1986)

0

compression d / 0

( ) unloading d / 0ln

ul

k dtF

k dt

Non-linear model of Hertz force F ~ 3/2 with several stages of plastic deformation development was suggested by Thornton (1997).

0indicates the final plastic deformation (see also the position of the ratchet tooth)

NAP12 DEM noncontact forcesModels of force interactions based upon Lennard Jones potential intermolecular forces (d-particle diameter)

12 6 1[( ) ( ) ]

4ij ij

d d

x x

12 6

13 7

dissipativerepulsive forces attraction forces dumping

12 6[ ]( ) ( )

n n

i j i j

d d dF m

R R R R dt

These interactions were originally used only for simulations in molecular dynamics, but there are also applications for vibration granulation layers (Aoki 1995, Akiyama 1996). Rather than as a contact interaction models are used for modeling the Van der Waalsovských 'remote' forces but using different exponents. Lennard Jones potential, derived only for an isolated pair of molecules, decreases with increasing distance too quickly (to the sixth power). More successful is Hamakerův model (already in 1937), which assumes additivity of interaction of an infinite number of pairs of molecules, and the result is the same type of dependency Van der Waalsovských (dipol) and Coulomb (electrostatic) forces

2

vzdálenost středů

24( )n

i j

AdF

R R

jiij RRxis the distance of centers

these forces decrease with square of distance.

NAP12 DEMPrevious page only applied axial forces. Similar relationships exist for shear interaction. Overview of the relations can be found eg in the review article Zhu et al: "Discrete particle simulation of particulate systems" Theoretical Developments. Chem.Eng.Sci. 62 (2007), pp.3378-3396. The following table of contact and cohesive forces is copied from this paper.

NAP12 DEM

From the same article (Zhu et al.) is taken an overview of models used for interaction with the fluid that flows around the particles

NAP12 DEM - CFD

( ) 0vt

někdy se uvažuje tlakováztráta rozdělená na tekutinu a částiceněkdy jen její působení na tekutinu

( ) ( ) ( )1 p

p

v vv p Rt

For multiphase systems, particle - fluid, dominated by the influence of hydrodynamic forces, is sometimes used detailed description of flow between discrete particles (direct numerical simulation of the Navier-Stokes equations DNS DEM). Less computationally intensive methods are combining description of fluid flow as a continuum by CFD (particulate matter affects only the existence of a source term exchange forces in the momentum balance equation) and the particle describes methods DEM, including the consideration of mutual collisions.CFD equations are de-facto Euler-Euler equations described in NAP11, but simplified for incompressible two-phase (solid and liquid, whose share is characterized by porosity )

NAP12 DEM software

Turner

NAP12 DEM softwareAs an example, freely available software program will bring YADE, developed since the nineties at the University Joseph Fourier, Grenoble, Professor Frederic Donze (description YADE as dissertation 2007 J.Kozicki). The research team development YADE are also students of CTU, Václav Šmilauer (thesis ČVUT) and Jan Stránský (jan.stransky@fsv.cvut.cz department of mechanics at civil engineering CTUJ, see paper Macroscopic elastic properties of particle models).

YADE – documentation, download (and many examples)

DEM program YADE enables connection with CFD software, for example OPENFOAM (some applications, such as 2D flow in the soil, DEM-CFD model column chromatography, as described in the dissertation Chen University of Tennessee)

Commercially available software EDEM , CFDEM

NAP12 DEM application examples

Turner

Powerpoint prezentace CSIRO –Australia 2006

NAP12 DEM aerocycloneK.W. Chu, B.Wang, D.L.Xu, Y.X.Chen, A.B.Yu: CFD–DEM simulation of the gas–solid flow in a cyclone separator. Chemical Engineering Science 66 (2011) 834–847

This article is used to simulate the motion of particles in aerocyklonu nonlinear viscoelastic model of contact (Hertzovské force). Two-phase flow of air turbulence model is described RSM (Reynolds Stress Modelling). Technically, the solution based on the program Fluent in which the code is DEM integrated as a UDF (User Defined Function).

Simulations are carried out for the maximum volume concentration of solids 1% (seemingly few), but the effect of particle-particle interactions and particle-wall far more significant than-air particles (statistical power possible to determine the abrasion and scratch the particles). Simulations have been limited to large spherical monodisperse particles (2 mm) - smaller particles dramatically increases the calculation time.Along with the numerical model was done in a laboratory experiment (the same diameter of cyclone 0.2 meters, the same particles and flow rates), which confirmed the predicted pressure loss (pressure gradient descent with an increasing fraction of solids!) and the spatial distribution of particle concentration.

NAP12 DEM sieveJ. Li et al. / Powder Technology 133 (2003) 190–202Sifting particles (with different diameters 2,3,5,8 mm) on an inclined vibrating sieve. DEM is used with linear viscoelastic particle interactions.

Optimal time step for polyethylene particles 10-7 s.

Similarly, the article looks K. J. Dong et al. / Minerals Engineering 22 (2009) 910-920 of banana screens. It's not about sifting bananas, but the new banana curved vibrating screens and their modeling with DEM.

NAP12 DEM heat transferK.F. Malone, B.H. Xu: Particle-scale simulation of heat transfer in liquid-fluidised beds. Powder Technology 184 (2008) 189–204

The article describes the heat transfer in the fluidized bed (liquid-solid phase) using the CCDM (I think it is the same as the DEM-CFD) and more or less standard models of particle interactions and CFD (two-component model with the equation of energy balance).

Very interesting is modeling of heat transfer by conduction through contact particle

22

Watt

Q d T a

This relationship is not at all trivial and follows from the analytical solution of the stationary temperature field near the contact of two large smooth bodies with the radius of contact a. Result is incredibly simple

thermal conductivity

2Q a T

and was derived at 1967 by M.Yovanovich.

NAP12 DEM spouted bed

Bing Ren, Yingjuan Shao, Wenqi Zhong, Baosheng Jin, Zhulin Yuan, Yong Lu. Investigation of mixing behaviors in a spouted bed with different density particles using discrete element method. Powder Technology 222 (2012) 85–94

Bing Ren, Wenqi Zhong, Baosheng Jin, Yingjuan Shao, Zhulin Yuan: Numerical simulation on mixing behavior of corn-shaped particles in a spouted bed. Powder Technology (2012)

Bing Ren,Wenqi Zhong, Yu Chen,Xi Chen,Baosheng Jin,Zhulin Yuan,Yong Lu.: CFD-DEM simulation of spouting of corn-shaped particles. Particuology xxx (2012) xxx–xxx

Trinity articles by the same authors, the same year, almost the same content ... do not expect anything new, the current method of assessment of science leads to repeated publications of identical procedures and results ...

Key words: non-spherical particles such as aggregates balls Hertzovský contact, lift force (Saffman and Magnus), own CFD solver(method SMAC Simplified Marker and Cell, Harlow 1970)

NAP12 DEM mixers blendersM. Lemieux et al. Large-scale numerical investigation of solids mixing in a V-blenderusing the discrete element method. Powder Technology 181 (2008) 205–216V-blender simulated by a classical method of DEM. Hertzovský contact, non-linear viscous damping with exponent m = 0.25. Similarly, the tangential force modeling of nonlinear theory of Mindlin Deresiewicz (now almost a standard). There was no need to consider the gas phase. Atypical but a large number of particles, 225000 and long simulation time 120s (with a time step of less than 10 s that means more than 10 million time steps).

see C. O'Sullivan, J.D. Bray, Selecting a suitable time step for discrete element simulations that use the central difference time integration scheme,Engineering Computations 21 (2004) 278–303.

Such high demands are addressed by periodically updated list of possible conflict of neighbors and dynamic assignment of subdomains to each processor computing cluster according to the load (64 processors AMD). The simulation step was limited by duration of collision pairs of particles, calculated according to

ERt

38 R-effective radius, E-Young modul

and density of particle.

NAP12 DEM mixers blendersH. Mio et al.: Effect of paddle rotational speed on particle mixing behavior in electrophotographic systém by using parallel discrete element method. Advanced Powder Technology 20 (2009) 406–415

DEM applied to mixing of toner particles in the developer tank - in the vane mixer with helical blades in the shape of ellipses. This article is not nothing special in terms of the technology used DEM (linear viscoelastic normal and shear forces in contact, using the method = MPI Message Passing Interface for parallel computing). But there is a comparison with experimental technique PIV (Particle Image Velocimetry)

VIDEO DEM (>106 částic)

NAP12 SPH

WhistlerMonaghan J.J.: Simulating free surface flows with SPH. J.Comp.Phys. 110, p.399-406 (1994)

SPH=Smooth Particle Hydromechanics

is also a method that calculates the trajectory of a particle numerical integration of the equations of motion type mi dui/dt = forces from neighboring particles, but the particles are more or less fictitious (imitating fluid) ai forces are computed little artificial manner designed to meet the need Navier Stokes equations.

0

1

2

0 0.5 1r/s

w

NAP12 SPHSPH method was originally designed and applied for modeling of astrophysical phenomena such as stellar explosions. Lucy LB: A numerical approach to the testing of the fission hypothesis, The Astronomical Journal, 82, 1013-1024, (1997), the first article, which appeared principles SPH - naming SPH However, in this article still does not appear.

Osově symetrická protohvězda v bezrozměrném čase =22

Rozpad na dvě části v bezrozměrném čase =52

The collapse of the star (the star is characterized by a set of finite number of discrete gas particles) is described by differential equations of gas expansion with consideration of its own gravitational forces (momentum equations and the equation of state), plus the equation for entropy.The basis of the numerical solution is an approximation of the spatial distribution of density, pressure, temperature and speed, convolution integrals, which smoothes out "signals" emitted by discrete particles

SPH-convolution

a set of J points in the volume V

1( ) ( ') ( ') ' ( ) ( )h

j jjV

r w r r r dV w r r rJ

Lucy constructs bell weighting function w (kernel convolution integral) as the fourth degree polynomials (s is the characteristic radius of reach of particles)

33

)1)(31(16

105)(

ssssrrr

w

33

)1)(31(16

105)(

ssssrrr

w

0

1

2

0 0.5 1r/s

w

Lucy

Monaghan

NAP12 SPHSPH method was later applied also to the role of gas dynamics, radiation, nuclear reactions and flow with free surface.For example, Monaghan JJ: Simulating free surface flows with SPH. J.Comp.Phys. 110, p.399-406 (1994) analyzes the dynamics of droplets formed by a set of 1884 discrete particles. Drop changing its shape from spherical to ellipsoidal in shape, and the simulations can maintain coherence and is disintegrating. Another problem that was solved in the article is bursting dams (creating waves with free surface after removal of the dam). For both cases the motion of discrete particles of liquids modeled solution to the equations of motion

Historically, the SPH methods are mother of meshless methods to be further discussed in the next lecture.

2 2

viscous gradient of interpolatingforces kernel (function w corresponding terms correponding to distance of particles a-b).to gradient of pressure

( )a a bb ab a ab

b a b

dv p pm w g

dt

Monaghan used as interpolation function kernel convolution integral Gaussian function

))(

exp(1

2

2

2 h

rr

hw ba

ab