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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.
| last1=Cundall
Line 11 ⟶ 12:
| volume=29
| pages=47–65
| doi=10.1680/geot.1979.29.1.47
}}</ref> and Allen<ref>{{cite book
| first1=M. P.
| last1=Allen
Line 19 ⟶ 21:
| authorlink2=D. J. Tildesley
| title=Computer Simulation of Liquids
| publisher=
| 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.
==History==
Molecular
| first1=B. J.
| last1=Alder
Line 40 ⟶ 35:
| year=1959
| volume=31
| issue=2
| 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
Line 47 ⟶ 45:
| year=1964
| volume=136
| issue=2A
| 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
Line 68 ⟶ 61:
| year=1993
| volume=77
| doi=10.1016/0032-5910(93)85010-7
| pages=79–87
}}</ref> Hoomans,<ref>{{cite journal
| first1=B. P. B.
| last1=Hoomans
Line 81 ⟶ 76:
| 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.
| last1=Xu
| first2=A. B.
| last2=Yu
| title=Numerical simulation of the gas-solid flow in a fluidized bed by combining discrete particle method with computational fluid dynamics
| journal=Chemical Engineering Science
| year=1997
| volume=52
|
| pages=2785–2809
| doi=10.1016/s0009-2509(97)00081-x
}}</ref> and Xu 1998.<ref>{{cite journal
| first1=B. H.
| last1=Xu
| first2=A. B.
| last2=Yu
| title=Comments on the paper numerical simulation of the gas-solid flow in a fluidized bed by combining discrete particle method with computational fluid dynamics
| journal=Chemical Engineering Science
| 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 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
Line 118 ⟶ 117:
| year=2007
| volume=62
| issue=13
| pages=3378–3396
| doi=10.1016/j.ces.2006.12.089
}}</ref> and Zhu 2008 et al.<ref>{{cite journal
| first1=H. P.
| last1=Zhu
Line 132 ⟶ 133:
| 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.
| last1=Xu
Line 143 ⟶ 145:
| 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.
| last1=Feng
Line 157 ⟶ 161:
| year=2004
| volume=43
| issue=26
| pages=8378–8390
| doi=10.1021/ie049387v
}}</ref> and Deen et al.<ref>{{cite journal
| first1=N. G.
| last1=Deen
Line 171 ⟶ 177:
| 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.
| last1=Gryczka
Line 189 ⟶ 197:
| year=2009
| volume=87
| issue=2
| pages=318–328
| doi=10.1002/cjce.20143
| citeseerx=10.1.1.335.4108
}}</ref>
The theoretical foundation for the
| first1=B.
| last1=Peters
Line 199 ⟶ 210:
| year=1999
| volume=116
| issue=1–2
| 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.
| last1=Peters
Line 207 ⟶ 220:
| 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.
| last1=Simsek
Line 223 ⟶ 238:
| 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
| last1=Natsui
Line 244 ⟶ 260:
| year=2010
| volume=50
| issue=2
| pages=207–214
| doi=10.2355/isijinternational.50.207
| 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.]]
Many engineering problems exist that include continuous and discrete phases, and those problems cannot be simulated accurately by continuous or discrete approaches. XDEM provides a solution for some of those engineering applications.
Although research and development of numerical methods in each domains of discrete and continuous solvers is still progressing, software tools are available. In order to couple discrete and continuous approaches, two major approaches are available:
*'''Monolithic approach''': The equations describing multi-physics phenomena are solved simultaneously by a single solver producing a complete solution.
*'''Partitioned or staggered approach''': 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 other.
The former approach requires a solver that handles all physical problems involved, therefore it requires a larger implementation effort. However, there exist scenarios for which it is difficult to arrange the coefficients of combined [[differential equations]] in one [[Matrix (mathematics)|matrix]].
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.
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.
{| 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
! [[Equation]]
! [[Variable (mathematics)|Variable]]
|-
| Mass (compressible medium) || Continuity || Pressure/density
Line 315 ⟶ 297:
| Species mass || Species transport || Mass fractions
|-
| Charge, current || [[Maxwell's equations|Maxwell]] || electric, magnetic field, electric displacement field
|}
The solution of these equations in principle defines a three-dimensional and transient field of the relevant variables such as temperature or species. However, the application of these conservation principles to a large number of particles usually restricts the resolution to at most one representative dimension and time due to CPU time consumption. Experimental evidence
at least in reaction engineering supports the assumption of one-dimensionality as pointed out by Man and Byeong,<ref>{{cite journal
| first1=Y. H.
| last1=Man
Line 335 ⟶ 311:
| 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.
| last1=Lee
Line 360 ⟶ 323:
| year=1996
| volume=105
| issue=4
| pages=591–599
| doi=10.1016/0010-2180(96)00221-0
}}</ref>
==Applications==
Line 368 ⟶ 332:
[[File:Particles impacting on a conveyer belt.png|thumb|Deformation of a conveyor belt due to impacting granular material.]]
Problems that involve both a continuous and a discrete phase are important in applications as diverse as pharmaceutical industry e.g.~drug production, agriculture food and processing industry, mining, construction and agricultural machinery, metals manufacturing, energy production and systems biology. Some predominant examples are coffee, corn flakes, nuts, coal, sand, renewable fuels e.g.~biomass for energy production and fertilizer.
Initially, such studies were limited to simple flow configurations as pointed out by Hoomans,<ref>{{cite journal
| first1=B. P. B.
| last1=Hoomans
Line 390 ⟶ 347:
| 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.
| last1=Chu
Line 399 ⟶ 360:
| 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.
| last1=Zhou
Line 415 ⟶ 376:
| 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.
| last1=Chu
Line 435 ⟶ 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).
The CCDM approach has also been applied to fluidised beds as reviewed by Rowe and Nienow<ref>{{cite journal
| first1=P. N.
| last1=Rowe
Line 449 ⟶ 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.
| last1=Feng
Line 463 ⟶ 426:
| 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.
| last1=Feng
Line 473 ⟶ 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.
| last1=Kafuia
Line 487 ⟶ 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 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
| [[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
|}
==References==
{{Reflist|30em}}
{{Numerical PDE}}
[[Category:Numerical differential equations]]
[[Category:
| |||