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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]].]]
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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.
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==
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| year=1959
| volume=31
| issue=2
| pages=459}}</ref> and early 1960s by Rahman<ref>{{cite journal▼
| doi=10.1063/1.1730376
| 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
▲|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
|
| pages=2785–2809
| doi=10.1016/s0009-2509(97)00081-x
}}</ref> and Xu 1998.<ref>{{cite journal
| first1=B. H.
Line 85 ⟶ 101:
| 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
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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.
Line 113 ⟶ 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.
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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
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-
==Methodology==
[[File:Staggered methodology for software coupling.png|thumb|Staggered methodology for discrete/continuous applications.]]
Although research and development of numerical methods in each domains of discrete and continuous solvers is still progressing, respective software tools have reached a high degree of maturity. In order to couple discrete and continuous approaches, two major concepts are available:▼
▲Although research and development of numerical methods in each domains of discrete and continuous solvers is still progressing,
*'''Monolithic concept''': The equations describing multi-physics phenomena are solved simultaneously by a single solver producing a complete solution.▼
*'''Partitioned or staggered concept''': 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 next.▼
▲*'''Monolithic
The former concept requires a solver that includes a combination of all physical problems involved, and therefore, requires a large implementation effort. However, there exist scenarios for which it is difficult to arrange the coefficients of combined [[differential equations]] in one [[matrix]]. A partitioned concept as a coupling between a number of solvers representing individual domains of physics offers distinctive advantages over a monolithic concept.▼
▲*'''Partitioned or staggered
▲The former
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
{| 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 law (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>
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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 349 ⟶ 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).
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| 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.
Line 375 ⟶ 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.
Line 385 ⟶ 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 397 ⟶ 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
{|
| [[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
|}
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[[Category:Numerical differential equations]]
[[Category:
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