Extended discrete element method: Difference between revisions

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{{Short description|Granular material interaction simulation technique}}

[[File:Internal temperature distribution in a particle.png|thumb|An internal temperature distribution for a spherical particle versus radius and time under a time-varying [[heat flux]].]]

The '''extended discrete element method''' (XDEM) is a numerical technique that extends the dynamics of granular material or particles as described through the classical [[discrete element method]] (DEM) (Cundall<ref>{{cite journal

| first1=P. A.

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| volume=29

| pages=47–65

| doi=10.1680/geot.1979.29.1.47

}}</ref> and Allen<ref>{{cite book

| first1=M. P.

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| authorlink2=D. J. Tildesley

| title=Computer Simulation of Liquids

| publisher=ClaredonClarendon Press Oxford

| year=1990}}</ref>) by additional properties such as the [[thermodynamic]] state, [[Stress (mechanics)|stress]]/[[Deformation (mechanics)|strain]] or [[electro-magnetic]] field for each particle. Contrary to a [[continuum mechanics]] concept, the XDEM aims at resolving the particulate phase with its various processes attached to the particles. While the discrete element method predicts position and orientation in space and time for each particle, the extended discrete element method additionally estimates properties such as internal [[temperature]] and/or [[species]] distribution or mechanical impact with structures.

[[File:Internal temperature distribution in a particle.png|thumb|An internal temperature distribution for a spherical particle versus radius and time under a time-varying [[heat flux]].]]

==History==

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| year=1959

| volume=31

| issue=2

| pages=459}}</ref> and early 1960s by Rahman<ref>{{cite journal

| pages=459–466

| doi=10.1063/1.1730376

| bibcode=1959JChPh..31..459A}}</ref> and early 1960s by Rahman<ref>{{cite journal

| first1=A.

| last1=Rahman

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| year=1964

| volume=136

| issue=2A

}}</ref> may be regarded as a first step toward the extended discrete element method, although the forces due to collisions between particles were replaced by energy potentials e.g. [[Lennard-Jones]] potentials of [[molecules]] and [[atoms]] as long range forces to determine interaction.

| doi=10.1103/physrev.136.a405

| pages=A405–A411

|bibcode = 1964PhRv..136..405R }}</ref> may be regarded as a first step toward the extended discrete element method, although the forces due to collisions between particles were replaced by energy potentials e.g. [[Lennard-Jones]] potentials of [[molecules]] and [[atoms]] as long range forces to determine interaction.

Similarly, the fluid dynamic interaction of particles suspended in a flow were investigated. The [[drag (physics)|drag]] forces exerted on the particles by the relative velocity by them and the flow were treated as additional forces acting on the particles. Therefore, these [[multiphase flow]] phenomena including a solid e.g.~particulate and a gaseous or fluid phase resolve the particulate phase by discrete methods, while gas or liquid flow is described by continuous methods, and therefore, is labelled the combined continuum and discrete model (CCDM) as applied by Kawaguchi et al.,<ref>{{cite journal

| first1=T.

| last1=Kawaguchi

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| year=1993

| volume=77

| doi=10.1016/0032-5910(93)85010-7

| pages=79–87

}}</ref> Hoomans,<ref>{{cite journal

| first1=B. P. B.

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| year=1996

| volume=51

| doi=10.1016/0009-2509(95)00271-5

| pages=99–118

| citeseerx=10.1.1.470.6532

| s2cid=17460834

}}</ref> Xu 1997<ref>{{cite journal

| first1=B. H.

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| year=1997

| volume=52

| pagesissue=278516

| pages=2785–2809

| doi=10.1016/s0009-2509(97)00081-x

}}</ref> and Xu 1998.<ref>{{cite journal

| first1=B. H.

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| year=1998

| volume=53

| issue=14

| pages=2646–2647

| doi=10.1016/s0009-2509(98)00086-4

}}</ref> Due to a discrete description of the solid phase, [[constitutive]] relations are omitted, and therefore, leads to a better understanding of the fundamentals. This was also concluded by Zhu 2007 et al.<ref>{{cite journal

}}</ref> Due to a discrete description of the solid phase, [[constitutive equation|constitutive]] relations are omitted, and therefore, leads to a better understanding of the fundamentals. This was also concluded by Zhu 2007 et al.<ref>{{cite journal

| first1=H. P.

| last1=Zhu

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| year=2007

| volume=62

| issue=13

| pages=3378-3396

| pages=3378–3396

| doi=10.1016/j.ces.2006.12.089

}}</ref> and Zhu 2008 et al.<ref>{{cite journal

| first1=H. P.

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| year=2008

| volume=63

| issue=23

| pages=5728–5770

| doi=10.1016/j.ces.2008.08.006

}}</ref> during a review on particulate flows modelled with the CCDM approach. It has seen a mayor development in last two decades and describes motion of the solid phase by the [[Discrete Element Method]] (DEM) on an individual particle scale and the remaining phases are treated by the [[Navier-Stokes]] equations. Thus, the method is recognized as an effective tool to investigate into the interaction between a particulate and fluid phase as reviewed by Yu and Xu,<ref>{{cite journal

| first1=B. H.

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| year=2003

| volume=78

| issue=2–3

| pages=111–121

| doi=10.1002/jctb.788

}}</ref> Feng and Yu <ref>{{cite journal

| first1=Y. Q.

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| year=2004

| volume=43

| issue=26

| pages=8378–8390

| doi=10.1021/ie049387v

}}</ref> and Deen et al.<ref>{{cite journal

| first1=N. G.

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| year=2007

| volume=62

| issue=1–2

| pages=28–44

| doi=10.1016/j.ces.2006.08.014

}}</ref> Based on the CCDM methodology the characteristics of spouted and fluidised beds are predicted by Gryczka et al.<ref>{{cite journal

| first1=O.

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| year=2009

| volume=87

| issue=2

| pages=318–328

| doi=10.1002/cjce.20143

}}</ref>.

| citeseerx=10.1.1.335.4108

}}</ref>

The theoretical foundation for the XDEM was developed in 1999 by Peters,<ref>{{cite journal

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| year=1999

| volume=116

| issue=1–2

| pages=297-301

| pages=297–301

| doi=10.1016/s0010-2180(98)00048-0

}}</ref> who described incineration of a wooden moving bed on a forward acting grate.<ref>{{cite journal

| first1=B.

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| year=2002

| volume=131

| issue=1–2

| pages=132–146

| doi=10.1016/s0010-2180(02)00393-0

}}</ref> The concept was later also employed by Sismsek et al.<ref>{{cite journal

| first1=E.

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| year=2009

| volume=193

| issue=3

| pages=266–273

| doi=10.1016/j.powtec.2009.03.011

}}</ref> to predict the furnace process of a grate firing system. Applications to the complex processes of a blast furnace have been attempted by Shungo et al.<ref>{{cite journal

| first1=Shungo

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| year=2010

| volume=50

| issue=2

| pages=207–214

| doi=10.2355/isijinternational.50.207

}}</ref> Numerical simulation of fluid injection into a gaseous environment nowadays is adopted by a large number of CFD-codes codes such as Star-CD of [[CD-adapco]], [[Ansys]] and [[AVL]]-Fire. Droplets of a spray are treated by a zero-dimensional approach to account for heat and mass transfer to the fluid phase.

| doi-access=free

}}</ref> Numerical simulation of fluid injection into a gaseous environment nowadays is adopted by a large number of CFD-codes such as [[Simcenter STAR-CCM+]], [[Ansys]] and AVL-Fire. Droplets of a spray are treated by a zero-dimensional approach to account for heat and mass transfer to the fluid phase.

==Methodology==

[[File:Staggered methodology for software coupling.png|thumb|Staggered methodology for discrete/continuous applications.]]

NumerousMany challengesengineering in engineeringproblems exist and evolve, that include a continuous and discrete phase simultaneouslyphases, and therefore,those problems cannot be solvedsimulated accurately by continuous or discrete approaches, only. Therefore, XDEM provides a platform, that couples discrete and continuous phasessolution for asome largeof number ofthose engineering applications.

Although research and development of numerical methods in each domains of discrete and continuous solvers is still progressing, respective software tools haveare reached a high degree of maturityavailable. In order to couple discrete and continuous approaches, two major conceptsapproaches are available:

*'''Monolithic conceptapproach''': The equations describing multi-physics phenomena are solved simultaneously by a single solver producing a complete solution.

*'''Partitioned or staggered conceptapproach''': The equations describing multi-physics phenomena are solved sequentially by appropriately tailored and distinct solvers with passing the results of one analysis as a load to the nextother.

The former conceptapproach requires a solver that includes a combination ofhandles all physical problems involved, and therefore, it requires a largelarger implementation effort. However, there exist scenarios for which it is difficult to arrange the coefficients of combined [[differential equations]] in one [[Matrix (mathematics)|matrix]]. A partitioned concept as a coupling between a number of solvers representing individual domains of physics offers distinctive advantages over a monolithic concept.

The latter, partitioned, approach couples a number of solvers representing individual domains of physics offers advantages over a monolithic concept. It encompasses a larger degree of flexibility because it can use many solvers. Furthermore, it allows a more modular software development. However, partitioned simulations require stable and accurate coupling algorithms.

It inherently encompasses a large degree of flexibility by coupling an almost arbitrary number of solvers.

Within the staggered concept of XDEM, continuous fields are described by the solution the respective continuous (conservation) equations. Properties of individual particles such as temperature are also resolved by solving respective conservation equations that yield both a spatial and temporal internal distribution of relevant variables. Major conservation principles with their equations and variables to be solved for and that are employed to an individual particle within XDEM are listed in the following table.

Furthermore, a more modular software development is retained that allows by far more specific solver techniques adequate to the

problems addressed. However, partitioned simulations impose stable and accurate coupling algorithms that convince by their pervasive character.

Within the staggered concept of XDEM, continuous fields are described by the solution the respective continuous (conservation) equations. Properties of individual particles such as temperature are also resolved by solving respective conservation equations that yields both a spatial and temporal internal distribution of relevant variables. Mayor conservation principles with their equations and variables to be solved for and that are employed to an individual particle within XDEM are listed in the

following table.

{| border="2" cellspacing="0" align="right" width="400" cellpadding="3" rules="all" style="border-collapse:collapse; empty-cells:show; margin: 1em 0em 1em 1em; border: solid 1px #aaaaaa;"

|+ Conservation principles of Interfaces

|- class="hintergrundfarbe6"

! [[Conservation Principlelaw (physics)|Conservation law]]

! [[Equation]]

! [[Variable (mathematics)|Variable]]

|-

| Mass (compressible medium) || Continuity || Pressure/density

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| Species mass || Species transport || Mass fractions

|-

| Charge, current || [[Maxwell's equations|Maxwell]] || electric, magnetic field, electric displacement field

|}

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| volume=97

| pages=1–16

| doi=10.1016/0010-2180(94)90112-0

}}</ref> while the importance of a transient behaviour is stressed by Lee et al.<ref>{{cite journal

| first1=J. C.

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| year=1996

| volume=105

| issue=4

| pages=591–599

| doi=10.1016/0010-2180(96)00221-0

}}</ref><ref>{{cite journal

| first1=J. C.

| last1=Lee

| first2=R. A.

| last2=Yetter

| first3=F. L.

| last3=Dryer

| title=Numerical simulation of laser ignition of an isolated carbon particle in quiescent environment

| journal=Combustion and Flame

| year=1996

| volume=105

| pages=591–599

}}</ref>

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| year=1996

| volume=51

| doi=10.1016/0009-2509(95)00271-5

| pages=99–118

| citeseerx=10.1.1.470.6532

| s2cid=17460834

}}</ref> however, Chu and Yu<ref>{{cite journal

| first1=K. W.

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| year=2008

| volume=179

| issue=3

| pages=104–114

| doi=10.1016/j.powtec.2007.06.017

}}</ref> demonstrated that the method could be applied to a complex flow configuration consisting of a fluidized bed, conveyor belt and a cyclone. Similarly, Zhou et al.<ref>{{cite journal

| first1=H.

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| year=2011

| volume=90

| issue=4

| pages=1584–1590

| doi=10.1016/j.fuel.2010.10.017

}}</ref> applied the CCDM approach to the complex geometry of fuel-rich/lean burner for pulverised coal combustion in a plant and Chu et al.<ref>{{cite journal

| first1=K. W.

Line 361 ⟶ 396:

| year=2009

| volume=22

| issue=11

| pages=893–909

| doi=10.1016/j.mineng.2009.04.008

}}</ref> modelled the complex flow of air, water, coal and magnetite particles of different sizes in a dense medium [[cyclone]] (DMC).

Line 373 ⟶ 410:

| year=1976

| volume=15

| issue=2

| pages=141–147

| doi=10.1016/0032-5910(76)80042-3

}}</ref> and Feng and Yu<ref>{{cite journal

| first1=Y. Q.

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| year=2004

| volume=43

| issue=26

| pages=8378–8390

| doi=10.1021/ie049387v

}}</ref> and applied by Feng and Yu<ref>{{cite journal

| first1=Y. Q.

Line 397 ⟶ 438:

| year=2008

| volume=6

| issue=6

| pages=549–556

| doi=10.1016/j.partic.2008.07.011

}}</ref> to the chaotic motion of particles of different sizes in a gas fluidized bed. Kafuia et al.<ref>{{cite journal

| first1=K. D.

Line 409 ⟶ 452:

| year=2002

| volume=57

| issue=13

| pages=2395–2410

| doi=10.1016/s0009-2509(02)00140-9

}}</ref> describe discrete particle-continuum fluid modelling of gas-solid fluidised beds. Further applications of XDEM include thermal conversion of biomass on a backward and forward acting grate. Heat transfer in a [[packed bed]] [[reactor]] was also investigated for hot air streaming upward through the packed bed to heat the particles, which dependent on position and size experience different heat transfer rates. The [[deformation]] of a conveyor belt due to impacting [[granular material]] that is discharged over a chute represents an application in the field of [[stress]]/[[strain]] analysis.

}}</ref> describe discrete particle-continuum fluid modelling of gas-solid fluidised beds. Further applications of XDEM include thermal conversion of biomass on a backward and forward acting grate. Heat transfer in thermal/reacting particulate systems was also solved and investigated, as comprehensively reviewed by Peng et al.<ref name="Peng">{{cite journal |last1=Peng |first1=Z. |last2=Doroodchi |first2=E. |last3=Moghtaderi |first3=B. |date=2020 |title=Heat transfer modelling in Discrete Element Method (DEM)-based simulations of thermal processes: Theory and model development |journal=Progress in Energy and Combustion Science |volume=79,100847 |page=100847 |doi=10.1016/j.pecs.2020.100847|s2cid=218967044 }}</ref> The [[deformation (engineering)|deformation]] of a conveyor belt due to impacting [[granular material]] that is discharged over a chute represents an application in the field of [[Stress (mechanics)|stress]]/[[Deformation (mechanics)|strain]] analysis.

{|

| [[File:Temperature distribution on a backward acting grate.png|thumb|Distribution of paticlesparticles' surface temperature on a backward acting grate.]]

| [[File:Char distribution on a forward acting grate.png|thumb|Progress of pyrolysis of straw blades on a forward acting grate, on which straw is converted into charred material.]]

| [[File:Particle temperature in a packed bed reactor.png|thumb|Distribution of porosity inside the packed bed and particle temperature.]]

|}

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[[Category:Numerical differential equations]]

[[Category:PartialComputational differential equationsphysics]]

[[Category:Continuum mechanics]]