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Eprints ID : 17869
To link to this article : DOI:10.1016/j.cherd.2017.01.028
URL :
http://dx.doi.org/10.1016/j.cherd.2017.01.028
To cite this version : Bennani, Lokman and Neau, Hervé and Baudry,
Cyril and Laviéville, Jérôme and Fede, Pascal and Simonin, Olivier
Numerical simulation of unsteady dense granular flows with rotating
geometries. (2017) Chemical Engineering Research and Design, vol.
120. pp. 333-347. ISSN 0263-8762
Any correspondence concerning this service should be sent to the repository
Numerical
simulation
of
unsteady
dense
granular
flows
with
rotating
geometries
L.
Bennani
a,∗,
H.
Neau
a,
C.
Baudry
b,
J.
Laviéville
b,
P.
Fede
a,
O.
Simonin
aaInstitutdeMécaniquedesFluidesdeToulouse(IMFT),UniversitédeToulouse,CNRS-INPT-UPS,France bEDF–R&D,DépartementMFEE,6quaiWatier,FR-78401Chatou,France
Keywords: Granularflow Numericalsimulation Unsteady Rotatingmesh Stirredreactor Frictionalstress
a
b
s
t
r
a
c
t
Inchemicalengineeringapplications,itisnotuncommontoencounterreactorsfeaturing rotatingparts.Astheserotatingpartsarepresentinordertoenhanceprocessessuchas chemicalreactionsand/orensurehomogeneity,itisessentialtotakethemintoaccountto performpredictivenumericalsimulations.Thisaspectcanbeparticularlychallenging,even moresowhencomplexindustrialgeometriesaretobetreated.
Inthispaperanumericalmethodologyforsimulatingunsteadygranularflowinrotating geometriesispresented.Themethodisbasedonsplittingthedomainintostaticandrotating parts.Theinformationbetweenrotatingandstaticpartsispassedbyanon-conformalmesh matchingtechnique.Thepresentedmethodologyisvalidatednumericallybycomparingits resultswithotherconventionalmethods.Themethodisthenappliedtoanindustrialscale problem.Theapplicabilityofthemethodandthewayitmaybeusedtoinvestigatecomplex flowisdemonstrated.
Thereforethisapproachenablestoconsiderthefullgeometryofcomplexreactors.It opensthedoortofurtherinvestigation,optimizationanddesignofindustrialscalechemical processes.
1.
Introduction
Flowofgranularmaterialsinrotatinggeometriesis encoun-teredinawiderangeofchemicalengineeringapplications, suchasthemixingprocessinaxiallystirredchemical reac-torsorgraindryinginrotarykilns(Boateng,2011;Soaresand McKenna,2013).Theserotatingpartsareoftenpresentsoasto enhancethechemicalprocess.Thegaininperformancecanbe investigatedexperimentally.However,experimental investi-gationoflargescalechemicalreactorsinvolvingsuchamotion isoftencomplicated.Hencethereisaneedforefficient numer-icalsimulationstrategieswhichareabletocapturetheflowof granularmaterialsinrotatinggeometries.
A possible numerical approach is the Discrete Element Method(DEM).Ineffect,thismethodhasbeenusedto
inves-∗ Correspondingauthor.
E-mailaddresses:[email protected](L.Bennani),[email protected](H.Neau),[email protected](C.Baudry), [email protected](J.Laviéville),[email protected](P.Fede),[email protected](O.Simonin).
tigateflatbladedstirrersorrotatingdrums(Zhouetal.,2003; Remyetal.,2009;Alizadehetal.,2014).Thismethodyields accurateresultsandisabletotreatawidevarietyofregimes (seeforexamplethecomparisonsofDEMresultsto experi-mentaldataperformedbyAlizadehetal.,2014).However,DEM islimitedbythenumberofparticlesthathavetobetracked. Indeed,afullscaleindustrialreactormaytypicallycontain severaltonnesofparticles(SoaresandMcKenna,2013).Asan illustration,supposeanindustrialreactorcontains25tonnes ofpolyethyleneparticles.Thedensityoftheseparticlesisof the order of1000kg/m3 and atypical diameter is 0.001m. Thiscorrespondstomorethan1010particles inthereactor,
whichisatthepresenttimeoutofreachforDEMcomputation (CapecelatroandDesjardins,2013).
Nomenclature
Subscript
k k=g:gasphase,k=p:particulatephase
Superscript g granular f frictional Latinsymbols Cd dragcoefficient[–] dp particlediameter[m] D rotatingdrumdiameter[m]
ec particle–particlenormalrestitutioncoefficient
[–]
F constantinequationforpc[Pa] g0 radialdistributionfunction[–]
gi ithcomponentofthegravitationalacceleration
[m/s2]
Kp granulardiffusivity[m2/s]
Kcol
p collisionalgranulardiffusivity[m2/s]
Kkin
p kineticgranulardiffusivity[m2/s] L half-lengthofthefreesurface[m]
np particlenumberdensity[m−3] mp particleaveragemass[kg] Pg gaspressure[Pa]
kg turbulencekineticenergyofthegas[m2/s2] pc criticalstatepressure[Pa]
q2
p randomparticlekineticenergy[m2/s2] qpg gas-particlecovariance[m2/s2] Rep particleReynoldsnumber[–] r exponentinequationforpc[–] s exponentinequationforpc[–] Sp,ij particlestrainratetensor[s−1]
Uk,i ith component of the mean velocity of the
phasek[m/s]
Ufs freestreamvelocity[m/s] Greeksymbols
˛p solidvolumefraction(˛p=npmp/p)[–]
˛p,max maximumsolidpacking[–]
˛p,min minimum solid volume fraction for the
fric-tionalviscosity[–] ıij Kronecker’sdelta[–]
t timestep[s]
p bulkgranularviscosity[kg/m/s]
g dynamicalgasviscosity[kg/m/s]
p granularviscosity[kg/m/s]
p kineticviscosityofthephasek[m2/s]
col
p collisionalgranularviscosity[m2/s]
kin
p kineticgranularviscosity[m2/s]
angleofinternalfriction[–] g gasdensity[kg/m3]
p particledensity[kg/m3]
k,ij stresstensorofthephasek[kg/m/s2]
fp,ij frictionalstresstensor[kg/m/s2]
gp,ij granular stress tensor (gp,ij=kin p,ij+collp,ij)
[kg/m/s2]
kin
p,ij kineticstresstensor[kg/m/s 2]
collp,ij collisionalstresstensor[kg/m/s2]
F
gp particleresponsetime[s]
ω rotationspeed[rads−1]
Anothermethod,whichistheoneadoptedinthiswork, is to solve the flow equations in an Eulerian framework. Althoughitismoredifficulttoaccountforcomplex phenom-enawiththisapproach,itdoesnotrequirethetrackingofevery particle.Hencemoresuitedforthesimulationoflargescale reactors(Zerenetal.,2012).Nevertheless,takingrotatingparts intoaccountisacomplextask.Indeed,ifthegeometryof inter-est featuresrotationalsymmetriesapossiblesolutionisto solvethegoverningequationsinarotatingframeofreference or byusingsliding wallboundary conditions(welladapted to rotatingdrums forinstance). For example, Huanget al. (2013)andSantosetal.(2013)haveinvestigatedthegranular flowinarotatingdrumusinganEulerianframeworkandby imposingano-slipboundaryconditionattherotatingwall.In addition,Ahmadzadehetal.(2008)studiedarotatingfluidized bedreactorusingatwodimensionalrotatingframeapproach. However,inmostproblemssuchsimplificationsarenot possi-ble.Thiscanbedue,forexample,tothepresenceofchimneys orextractionpipes.Hence,thesimulationofindustrial reac-torscontainingrotatingpartsremainsagreatchallengeanda moreelaboratenumericalstrategyisneeded.
In this paper, a rotating mesh method for simulating unsteady granularflowsin complexlarge scale geometries ispresented.Itisafirststeptowardssolvingthe aformen-tionnedproblems.Inaddition,theunsteadinessisanaspect of prime importance for practical applications, in particu-larwhenchemicalreactionsoccur.Themainideaistosplit thedomainintostaticandrotatingpartsandtousea non-conformalmeshmatchingtechniquetoconnectthedomains (EDF,2015).
Themaingoal ofthis workis tovalidatethepresented methodologybycomparingittootherwellknownmethods. Anotherapproachcouldhavebeentousethemethodof man-ufacturedsolutions(BlaisandBertrand,2015;Oberkampfand Roy,2010).Thereadershouldbearinmindthatthisworkis ofanumericalnature.Thespecificmodelsorboundary con-ditionscanbechanged/implementedwithoutany difficulty relatedtotherotatingmeshmethod.
Thefirst partofthe paperisfocusedon presentingthe physicalmodellingapproachandtherotatingmeshmethod.
Thenumericalvalidationofthemethodisthenaddressed byconsideringasimplerotatingdrumtestcase.Indeed,asthis geometrypresentsrotationalsymmetry,asimulationusinga slidingwallboundaryconditionisalsopossible.Afirst assess-ment isperformedbycomparingthe resultsobtainedwith bothmethods.
Thirdly,aflatbladedstirrerconfigurationinvestigatedby Stewartetal.(2001)issimulated.Thisconfigurationismore complexandfeaturesinterestingflowphysics.Thecasealso presentsarotationalsymmetrymakingitpossibletosimulate inarotatingframeofreference(byaddingtheinertialforces). Comparisonoftheobtainedresultsallowstofurthervalidate themethod.
Finally,themethodispushedintotherealmofindustrial problemsandisappliedtoaprototypeofahorizontallystirred reactor.Thisprovesthatthepresentedapproachenablesto consider the full geometry of complex reactors. Hence, it provides a step further in the investigation and design of industrialscalechemicalprocesses.
2.
Physical
model
The physicalmodelling is based on a particulate Eulerian approach.Itisderivedfrom ajointfluid-particlePDF equa-tionallowingtoobtainthetransportequationsforthemass, momentumandfluctuatingkineticenergyofparticlephases inrapid granular flows (Balzer et al., 1995;Simonin, 1996; Balzer, 2000). These equations are standard in two-fluid modelling. Briefly, the set of equations consists of mass, momentumandfluctuatingkineticenergyequationscoupled byinterfacialtransferterms.Moredetailsonthis approach aregiveninAppendixA.Moreover,thenumerical methodol-ogypresentedhereisgeneralenoughtobeappliedtoother approaches.
Thereadershouldbearinmindthatthegoalofthiswork istoassessthenumericalperformanceoftherotatingmesh method.Thatbeingsaid,giventhehighdegreeofcompaction ofthe particle phaseand the importantpresenceof walls inthecaseswhicharepresentedinthis paper,theparticle phasestresstensorandwallboundaryconditionsarebriefly discussed.
2.1. Particlephasestresstensor
Theflowsunderconsideration inthisstudy presentahigh degreeofcompaction.Thismeansthatfrictionalinteraction betweensolidparticlesisanimportantaspectandhastobe takenintoaccount.Thisisdonethroughtheintroductionofa frictionalpartinthestresstensorinthemomentumequation. Theparticlephasestresstensorpismodelledasthesum
ofakinetic-collisionalpartderivedintheframeofthekinetic theoryofrapidgranularflowmodifiedbythedragforce(Boëlle etal.,1995),gpandafrictionalpart,
f p.
Thecollisionalandkineticpartoftheparticlephasestress tensoriswrittenusingaviscosityassumption:
gp,ij=
Pp−p ∂Up,l ∂xl ıij−pSp,ij, (1)wherePpisthegranularpressure,pthebulkgranular
viscos-ityandpthekinetic-collisionalviscosity(seeAppendixA).
Moreover,Spisdefinedinthefollowingway:
Sp,ij= ∂Up,i ∂xj + ∂Up,j ∂xi − 2 3 ∂Up,l ∂xl ıij. (2)
Concerning the frictional part of the stress tensor, the approachproposed by Srivastavaand Sundaresan(2003) is retained.Itisbasedonrheologicallawsusedinsoilmechanics whichassumethat,inthehighcompactionregime,the gran-ularmaterialfollowsarigid-plasticbehaviour.Inthiscontext, SrivastavaandSundaresanproposethefollowingmodelfor thefrictionalstress:
fp=pcI− √ 2sin()pc Sp
Sp:Sp+(8q2p/3d2p) , (3)Table1–Frictionalmodelparameters.
˛p,min ˛p,max F[Pa] r s
0.5 0.64 0.05 2 5
wheredp istheparticlediameter,istheangleofinternal friction,q2
pistheparticlekineticagitationand8q2p/3d2pisan
estimate forstrain rate fluctuations.pc isthe critical state pressuredefinedby:
pc(˛p)=
⎧
⎪
⎨
⎪
⎩
F˛p−˛p,min r (˛p,max−˛p)s if˛p>˛p,min 0 if˛p≤˛p,min (4)
where F,r, sare constants.The differentparameters used heretodefinethefrictionalmodelaresummedupinTable1. It shouldbenoted that the valueswere taken asgivenby SrivastavaandSundaresan(2003)exceptfor˛p,max.
ThemodellingoffrictionalstressesinanEulerian frame-workisstillanactiveareaofresearch(Schneiderbaueretal., 2012; Chialvo et al., 2012). However, note that the model retainedforthisworksufficestoreflectthebasicphysicsof frictionalregimes.Theimplementationofmoresophisticated frictionalmodelsispartofcurrentworkandshouldpresent nodifficultyassociatedtotherotatingmeshmethod.
2.2. Particlephasewallboundaryconditions
Throughoutthiswork,theparticlephasewallboundary condi-tionthatisusedisano-slipcondition.Therefore,azerofluxis imposedforscalarvariables,inparticularforthesolidrandom kineticenergy(Fedeetal.,2016).Asforvelocities,ano-slip conditionisapplied.
Notethistypeofboundaryconditioncorrespondstothe maximumoffrictionatthewall.Otherwallboundary con-ditions exist and have been investigated (Soleimani et al., 2015;Fedeetal.,2016).However,theirimplementationshould presentnodifficultyassociatedtotherotatingmeshmethod.
3.
The
rotating
mesh
numerical
method
The numerical simulations presented in this paper have beencarriedoutusingaEuleriann-fluidmodelingapproach for gas-solid turbulent polydisperse flows developed and implementedbyIMFT(InstitutdeMécaniquedesFluidesde Toulouse)inNEPTUNECFDversion.NEPTUNECFDisa mul-tiphase flow software developed in the framework of the NEPTUNEproject,financiallysupportedbyCEA(Commissariat l’EnergieAtomique),EDF(ElectricitédeFrance),IRSN(Institut deRadioprotectionetdeSûretéNucléaire)andAREVA-NP.The numericalsolver hasbeendevelopedforHighPerformance Computing(Neauetal.,2010,2013).
3.1. Methodology
Therotatingmeshmethodconsideredhereoperatesbytaking asinputsseveralmeshes,someofthemwillbefixed(stator), otherswillberotating(rotor)(Audebert,2009).Theequations ofmotionaresolvedineachmeshintheabsoluteframeof reference.
For therotatingmesh,thegoverning equationsmustbe formulatedsoastotakeintoaccountitsmotion.Todoso,an ALE-like(ArbitraryLagrangian-Eulerian)approachisused.As
Table2–Particlephasematerialproperties.
Particledensityp Diameterdp Elasticcoefficientec Internalfrictionangle
2500kgm−3 3×10−3m 0.9 25◦
Fig.1–Referenceandtransformedmeshesandtheir associatedcoordinates.
Fig.2–Definitionofthedifferentgeometricalentitiesused inthenumericalscheme.
(adaptedfrom(EDF,2015)).
theALEmethodiswellestablished(Hirtetal.,1974;Hughes etal.,1981;Doneaetal.,1982,2004),onlythebasicgeneral principlesoftheapproacharepresented.
AsshowninFig.1,letxbetheEuleriancoordinatesandbe thesocalledreferencecoordinates.Moreover,(,t)denotes thetransformationfromtox.Consideraquantityof inter-est,whichcanbeviewedequivalentlyinbothconfigurations throughtherelation:g(x,t)=g((,t),t)= ˜g(,t).Hence, ˜g(,t) representsthefunctionasattachedtothemovingmesh.In thisframework,onemayrelatethetimederivativeof ˜gtothat ofgthroughtheformula:
∂J˜g ∂t =J
∂g ∂t+∇·(gW)
(5)
whereW=∂/∂tisthemeshvelocityandJistheJacobianofthe transformation.Inthepresentcase,themeshmovementisa rotationandthusJ=1.Thegoverningequations,inarotating mesh,foraphasekhencetakethefollowingform:
∂ ∂t( ˜˛k˜k)+∇·(˛kk(Uk−W))=0 (6) ∂ ∂t
˜ ˛k˜kU˜k +∇·(˛kk(Uk−W)⊗Uk)=−˛k∇Pg+˛kkg +Ik−∇·(˙k) (7)
Eqs.(6)and(7)meanthatinthefinitevolumeprocedure themeshrotationwillessentiallyimpactthemassfluxand convectionterms.Thegeneralmethodforcomputingthese termsisbrieflyrecalledbelow.Fig.2showsthegeometrical
definitionsusedforthenumericalscheme.LandRdenotethe leftandrightcellrespectively,nLR istheunitnormaltothe
faceadjacenttoLandRandSLRisitsarea.Below,XYdenotes
thevectorjoiningagivenpointXtoagivenpointY.
Themassfluxatagiveninteriorfaceiscomputedusing thefollowingscheme:
[˛kkUk]F.SLRnLR=˛k,F(pk[kUk]A+(1−pk)[kUk]B)·SLRnLR (8)
where[kUk]Aisdefinedas:
[kUk]A=k,A
Uk,A+ 1 2min(˛k,A,˛k,B) ([∇Uk]A·AA+ [∇Uk]B·BB)) (9)
andtheponderationcoefficientpisdefinedas:
pk=
c˛k,A
c˛k,A+(1−c)˛k,B
(10)
wherec=|FB|/|AB|and˛k,Aisdefinedas:
˛k,A=˛k,A+ [∇˛]A·AA (11)
Notethatifthecellbelongstoarotatingpartthevelocity usedtocomputethemassfluxisUk−Winaccordancewith
Eq.(6).
As forthe convective fluxbetweencells L and R,noted
FLR(˛kkUk,Uk),itiscomputedasfollows:
FLR(˛kkUk,Uk)=([˛kkUk]F·SLRnLR)·Uk,F (12)
where Uk,F is defined depending on the chosen numerical
scheme.TheonebeingusedinthisworkistheSecond-Order LinearUpwindScheme(SOLU)whichisgivenby:
Uk,F={
Uk,A+(∇Uk)A·AF if[˛kkUk]F·SLRnLR≥0
Uk,B+(∇Uk)J·BF if[˛kkUk]F·SLRnLR<0
(13)
InEq.(12)thecomputationoftheconvectivefluxisbased onthemassflux.Therefore,ifthefacebelongstoarotating cell,nofurthermodificationisneededastherotationvelocity willalreadyhavebeenaccountedforinthemassflux.
In addition to taking the mesh rotation velocity into accountwhencomputingthemassandconvectivefluxes,the actualrotationofthemeshhastobeintegratedintothecode’s timemarchingscheme.Thetimemarchingschemeisbased onanellipticorientedfractionalstepmethod(Méchitouaetal., 2003). The method can be seen as a prediction/correction scheme.Theactualrotationofthemeshisperformedbetween thepredictionandthecorrectionsteps.Therefore,the algo-rithmcanbesummedupasfollows:
1. Velocitypredictionstep:Unk−→Uprk
2. Rotaterotormeshbyanangleωt(ωtheangularvelocity andtthecurrenttimestep).Then,matchtostatormesh usinganon-conformalmatchingalgorithm
3. Correction step: compute all variables at iteration n+1 through the pressure-mass-momentum-energy coupling step
Fig.3–Illustrationofinterfacejoining:theredfacesandblackfacesarejoinedbyintersectingtheiredgesandbuildingnew
faces(dottedlines).(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtotheweb
versionofthearticle.)
AdaptedfromEDF(2015).
Fig.4–Sketchoftherotatingdrumcase.
Asstatedinstep2anon-conformalmeshmatching algo-rithm is used after rotation. Indeed, even if the matching betweenmeshesisconformalatthefirstiteration,itwillnot bethecaseatthesubsequentones.Indeed,therotormesh willhaverotatedbyananglewhichdoesnotnecessarily coin-cidewiththeangulardiscretizationofthemeshes.Hence,in ordertosolvethegoverningequations,informationhastobe passedatthenon-conforminginterface betweenthestator andtherotor.Thisisdonethankstoameshjoiningalgorithm andhastobeperformedateachtimestep.Themainideaof thismethodistointersectthefacesofbothinterfacesand splitthemsoastobuildconformingmeshes.Anillustration ofthismethodisshowninFig.3.Moreinformationonthis procedurecanbefoundinEDF(2015).
Itshouldalsobenotedthattherotationimposesa restric-tiononthetimestep.Indeed,soastoremainconsistentwith respecttotheflowphysics,thecurvilineardistancetraveledby therotatingmeshatthejoininginterfaceduringagiventime stepshouldnotexceedthatofameshcell.Inotherwords: ωRjt≤sj,whereRjisthejoiningradiusbetweenrotorand
statorandsjisthecellsizeinthetangentialdirection.
4.
The
rotating
drum
(without
gravity)
Thegoalofthissectionistonumericallyvalidatethe rotat-ingmeshmethodwithasimpletestcase.Indeed,considera rotatingdrumasillustratedinFig.4.Thedrumisrotatedatω =1.21rads−1andinitiallyuniformlyfilledwithgasand parti-cleswithvolumefractionsrespectively˛g=0.56and˛p=0.44.
Moreover, as this case is purely numerical, gravity is not present.
Table3–CPUtimestosimulate100sofphysicaltime
(ononecore)
Computationtime
Rotor/Stator 815s
SlidingWall 690s
Inthiscase,thegasflowisconsideredlaminarwhile trans-port equations are solved forthe particle agitation kinetic energy q2
p (see Appendix A for further details). The
prob-lem is treated intwo dimensions. Hence, the lateral faces are treatedassymmetryboundary conditions.Thisimplies ahomogeneousNeumannconditionforscalarvariables.For thevelocities,thesymmetryboundaryconditionimposesa homogeneousDirichletconditiononthenormalcomponent and a homogeneous Neumannon the tangential one.The outercylinderboundaryistreatedasawall,whichmeansa no-slipboundaryconditionforvelocitiesandzerofluxforscalar variables, inparticularforthesolid random kineticenergy (Fedeetal.,2016).
Asthegeometryfeaturesrotationalsymmetry,thecasecan alsobesimulatedbyusingaslidingwallboundarycondition. Thisenablesapurelynumericalvalidationbycomparingthe resultsobtainedwithbothnumericalapproaches.Thesliding wallapproachconsistsinimposingatangentialvelocityatthe cylinder’sboundary. Inthisway,therotatingmotionofthe cylinderiseffectivelytakenintoaccount.Ontheotherhand, asexplainedinSection3,therotatingmeshmethodconsists inconsideringtwomeshes.Aninnerfixedmeshandanouter rotatingcrownmesh.Numericalvalidationisthenassessed bycomparingtheresultsobtainedwithbothmethods.Fig.5 illustratesthedifferencesbetweenthetwoapproaches.The mesh(Fig.5)iscomposedof1280cellsinthecrownand3840 cellsinthefixedarea,makingatotalof5120cells.
Thedensityofthegasissettog=1.2kgm−3andits
viscos-itytog=1.8510−5Pas.Table2givesthephysicalparameters
ofparticlephase(glassbeads).
Astheflowexhibitsarotationalsymmetry,theresultsare extractedalong theradialdirection.InFig.6theprofilesof solidvolumefractionobtainedatdifferenttimes(t=10s,20s, 30sand40s)withbothapproachesareplottedandcompared. Averygoodagreementbetweenbothmethodsisachieved.
Fig.7showsacomparisonbetweentangentialvelocities forbothapproachesandatvarioustimes.Averygood agree-mentisobserved.Startingoutfromthecenterofthedrum, thevelocityprofilefirstpresentsalowbutalmostlinearslope beforetransitionningtohighervaluesandahigherslope.
Fig.5– Illustrationoftheslidingwallandrotor/statormeshapproaches.
Fig.6–Comparisonofsolidvolumefractionsobtainedatdifferentphysicaltimes.
Steadystateisreachedforthiscaseafterapproximately 90s.AsshowninFig.8,inthisstate,alltheparticlesarepacked nearthewallandbehaveasarigidbody.
Theabove resultsshow thatthe rotatingmesh method isfully able ofcorrectly reproducingthe numerical results obtainedwithaconventionalslidingwallapproach.The sim-ulationswereperformedwithanadaptivetimestep.Forboth casesthe time stepsatevery iterationwere veryclose. At steadystate,thetimestepwas0.018sandlimitedbya maxi-mumCFLofone.However,therotatingmeshmethodismore computationallyexpensiveduetothe non-conformalmesh matchingwhich isrequiredat each timeiteration.Table 3 providesanideaofthisadditionalcost.
Also,itisinterestingtonotethatchanginglocationsforthe meshmatchinginterfacedoesnotchangetheresults.Indeed differentlocationsweretestedbutnotreportedherefor con-ciseness.
5.
The
flat
bladed
stirrer
Withtherotatingmeshmethodnowvalidated onasimple case,theroadisopentotheinvestigationofmorecomplicated cases.Inordertofurtherassessthemethod,averyinteresting caseisthatoftheflatbladedstirrer,investigated experimen-tallybyStewartet al.(2001). Inthissection,the flatbladed stirrercaseissimulatedwithboththerotatingmeshmethod andtherotatingframemethod.Indeed,thegeometryofthe flatbladedstirrer(describedbelow)hasarotationalsymmetry withrespecttoitsverticalaxis.Onemayhencealsosimulate thiscaseinarotatingframeofreference,whichrequiresthe additionoftheinertialforces(Coriolisandcentrifugal).
5.1. Geometryandmesh
Thegeometryofthestirrerconsistsofaverticalshafttowhich areattachedtwoopposedflatblades.Thisassemblyis
con-Fig.7–Comparisonofthetangentialvelocitiesobtainedatdifferentphysicaltimes.
Fig.8–Comparisonoftangentialvelocities(atsteadystate)
witharigidbodyvelocity(rω).
tainedwithinacylindricalvesselwithinwhichsolidparticles (glassbeads)arepresent.Rotationofthebladesthen gener-atesagranularflow.Fig.9(b)givesthedimensionsofthestirrer whileFig.9(a)showstheaxeswherethedataisextracted.It shouldbenotedthatagapof2mmwasleftbetweentheblade tipsandthevesselwallsinordertobeabletousetherotating meshmethod.
Fig.10(a)showstheskinmeshontheshaftandtheblades. Thestatormeshconsistedin240000elementswhiletherotor wasdiscretizedwith125000elements.
For the present study the filling is of2.8kgofparticles (approximately200000).Theshaftisrotatedat20rpm.The physicalpropertiesofthegasandparticlephasesarethesame asinsection4exceptfortheparticlediameterwhichissetto 2.2mm(seeTable2forparticlephysicalpropertiesand1for thefrictionalviscosityparameters).Forthiscase,all bound-ariesarewalls.
Fig.10–Stirrermeshandrotor/stator.
Fig.11–Bedheightatr=0.08m.Theverticalblacklinesrepresenttheblades.
Fig.12–Tangentialvelocityalongthez-axisatr=0.08mandtwodifferentangles.
5.2. Resultsanddiscussion
Inthissection,theresultsobtainedusingtherotatingmesh method are compared tothose obtainedwith the rotating framemethod.Forthispurpose,allresultsarepresentedinthe rotatingframeofreference.Itshouldbenotedthatalthough someaspectslinkedtothedynamicsoftheflowarepointed out,themainfocusisthecomparisonofbothapproachesso astoassesstherotatingmeshmethod.Itishowever interest-ingtonotethatallthecomputedflowpatternsareconsistent withtheobservationsofStewartetal.(2001).
Startingwiththeflowpatterninaradialcut,Fig.11shows acomparisonoftheparticlebedheights.Theyareplottedas afunctionofthecurvilinearabscissa(s)andalongthefull cir-cumferenceatr=0.08m.Thepredictedbedheightscompare verywell.Onemayalsonotethataheapiscaptured,whichis aprimaryfeatureoftheflow.
Fig.12 showsthe tangentialvelocityalong thez-axisat
r=0.08mandtwodifferentangles( =37.5◦ and =60◦).The agreementbetweenthepredictedvelocitiesisvery satisfac-tory.Moreover,thechangeinsignshowsthatinthehigher partoftheheaptheparticlesarepushedback.
Fig.13–Tangentialprojectionoftheparticlevelocityfield
onaradialcutatr=0.08m(rotatingmesh).
Coherently, a recirculation pattern in the heap can be observedinFig.13.Itshowsthetangentialprojectionofthe particlevelocityfieldonaradialcutatr=0.08m.
Concerningtheflowinverticalcuts,Fig.14showsthe tan-gential velocityalongther-axisattwodifferentanglesand heights. Theagreement betweenthepredicted velocitiesis excellent.TheprofilesinFig.14showthattheparticlemotion istowardsthebladenearthewallsandawayfromtheblade inthenearshaftarea.
Fig.14–Tangentialvelocityalongther-axisattwodifferentanglesandheights.
Table4–CPUtimestosimulate2.5sofphysicaltime(on
fortycores)
Computationtime
Rotatingmesh 2h
Rotatingframe 1.4h
Fig.15–Tangentialprojectionoftheparticlevelocityfield onaverticalcutatz=0.026m(rotatingmesh).
ThispatternbecomesclearerinFig.15,whichshowsthe tangentialprojectionoftheparticlevelocityfieldonavertical cutatz=0.026m.Theglobalflowoftheparticlesinthisvertical cutistowardstoblades.However,inthenearshaftregion,a smallrecirculationdevelops.
Asforthepreviouscase,thetimestepwasadaptiveand remainedverycloseforbothcasesateveryiteration.Atsteady state,thetimestepwas0.001sandlimitedbyamaximumCFL ofone.Table4providesanideaoftheadditionalcostwhen usingtherotatingmeshmethod.
6.
Towards
industrial
scale
applications:
prototype
of
a
horizontally
stirred
reactor
Inthis final section, theapplicability of the rotatingmesh method to full scale industrial problems is demonstrated. Indeed,itisforthesekindsofcomplexapplicationsthatthe methodproves tobe the mostusefull. In this section,the hydrodynamicsofagas-particlesflowinsideanumerical pro-totypeofahorizontallystirredreactorissimulated.Asshown inFig.16,thenumericalprototypehasachimney.Thismeans thattheuseofmethodsbasedonrotationalsymmetrieswould nothavebeenpossible.
6.1. Geometryandmesh
Thegeometryofthereactorisstronglyinspiredfromthe illus-trationsthatcanbefoundin(SoaresandMcKenna,2013).Itis composedofahorizontalstirrerandachimney,asshownin Fig.16(a).AsstatedbySoaresandMcKenna(2013),the hori-zontalvesselisusually10–15mlongand1.5–4mindiameter. Therefore,inthecaseofournumericalprototype,thevesselis 11mlongand4mindiameter.Also,thechimneyis6mhigh. Therotatingpartistheaxialstirrercontainedwithinthe horizontalvesselwhereasthestatic(stator)partistherest ofthe vesseland thechimney(see Fig.16(b)).Itshouldbe notedthatthisisnotareproductionofanactualreactor.It isanumericalprototypedesignedonlytodemonstratethe applicabilityofthemethodtoindustrialscaleproblems.
Themeshwas composed of1.3millionhexahedral ele-ments.Therotatingpartwasmeshedwith600000elements andthestaticpartwith700000elements.Theresultingmesh isshowninFig.17.
6.2. Resultsanddiscussion
In this case, the reactor is filled with 25 tonnes of par-ticles (located in the horizontal vessel). Their density is p=500kgm−3andtheyhaveadiameterofdp=0.5mm.This
isequivalenttomorethan1010particlesinthereactor.The
physicalpropertiesofthegasarechosensoastobemore rep-resentativeofindustrialprocesses.Valuesfromtheliterature whereusedasanorderofmagnitude(seeforexample(Kaneko etal.,1999)).Forthiscase,thedensityisg=45kgm−3andthe
viscosityg=1.1×10−5Pas.Concerningtheboundary
condi-tions,thetopofthechimneyisanoutletwhereasallother boundariesarewalls.
Itshouldbenotedthatforthiscase,theflowofthegas phasecannolongerbeassumedlaminar.Hence,ak− model whichincludesadditionaltermsaccountingfortheinfluence oftheparticlesonthefluidturbulenceisused(Vermoreletal., 2003).
In terms ofcomputational cost, on 100 cores, 4.3h are neededtosimulate20sofphysicaltime.
Thehydrodynamicsmayfirstbeinvestigatedbylookingat thesolidvolumefraction.Fig.18showssolidvolumefractions attheskinofthereactor.Akindofwavystructureisobserved whichisduetothebladespassingandprojectingparticles towardstheboundaries.Moreover,thepeaksofthewavesare shiftedcoherentlywiththebladesthatarepassingby.
Fig.16–Viewofthegeometry.
Fig.17–Meshcomposedofhexahedralcells:600000elementsfortherotatingpartand700000elementsforthestaticpart.
Fig.18–Solidvolumefractionattheboundariesaftertwoconsecutivebladepasses.
Fig.18 showssolid volumefractions inanYZ cut.Here again,structuresareformedbytheblademotionwhichsends theparticlesupwards.
Itisalsoofinteresttolookattheparticlevelocityfields. Let’sfirstconsideracutintheYZplane,asshowninFig.20(a). Thecutisdonejustafterasetofbladeshaspassedthrough theplane.Intheupperhalf,thevelocityvectorsshowa com-plexpattern.Wherethebladehasjustpassedtheparticlesare pushedupwards.Ontheotherhand,theparticlesinthemost upperpartofthevesselaremovingbackdown,creatingthe kindofcircularpatternwhichisobserved.
Secondly,let’sconsideracutintheXYplane(Fig.20(b)). ThecutisdoneatthesametimeastheoneintheYZplane. However,notethatthecutisperformedsoastointersectwith abladethatisalmostat0◦.Herealso,complexpatternscanbe observed,withparticlesbeingpushedupwards,othersfalling backdownandthebulkofthebedbeingstirredatthebottom.
7.
Conclusions
and
perspectives
Thegoalofthispaperwastopresentarotatingmeshmethod forsimulatingdensegas-particleflowsinrotatinggeometries. Thisworkfocusesonthevalidationofthemethodby
com-Fig.19–YZcut:solidvolumefractionaftertwoconsecutivebladepasses.
Fig.20–SolidvolumefractionandparticlevelocityvectorfieldsinanYZandXYcut(seeFig.16(a)foraxes).
paringit with moreconventional approaches (slidingwall, rotatingframe).WithintheframeworkofanEuleriann-fluid modeling approach,this methodenablestotreat problems featuringcomplexgeometries.
After having presented the physical modelling and the numericalmethod,arotatingdrumcasewasconsidered.For thepurposeofthiscase,gravitywasnotpresentandthedrum wasinitiallyfilledwithauniformgas-particlemixture.The problemwasalsotreatedusingaslidingwallapproach.The numericalresultsobtainedwithbothmethodswerecompared andshowedexcellentagreement.Fromthisstep,itappeared thattherotatingmeshmethodisequivalenttothe conven-tionalslidingwallapproachandisfullyableofcapturingthe mainfeaturesoftheflow.
Thirdly,thedensegranularflowinaflatbladedstirrerwas simulatedusing the rotatingmesh method and a rotating frameapproach.Theresultsfrom bothmethods compared very well. In addition, the method captured recirculation regions,whichisanimportantfeatureofthisflow.Thisshows thatthenumericalmethodisabletotreatflowswithmore complicatedpatterns.
Finally,theapplicabilityofthemethodtoindustrialsize complex problems was demonstrated. Indeed,the method wasusedtoinvestigatethehydrodynamicsofthegas-particle flowinsideaprototypeofahorizontallystirredreactor.This enabled toidentify several flowpatterns and hence opens thedoorstointerestinginvestigationsofgas-particleflowsin
complexgeometries.Hence,themethodhasstrongpotential forapplications toindustrial reactordesignand the inves-tigation ofchemicallyreactingparticulateflowsinrotating geometries.
Overall, the rotatingmesh methodtherefore presents a promisingperspective fortheEulerian simulationofdense granular flowsin rotatinggeometries.This willenable the assessmentandvalidationofphysicalmodelsbycomparing simulations toexperimentsmaking useofarotating appa-ratus (suchasrotatingdrums).Moreover,the gaininusing thismethodisthatitallowstosimulatecomplexproblems ofindustrialsize.Forexample,aconfigurationofinterestis thesimulationofafullscalechemicalreactorfeaturing chim-neysandothercomplexparts,andincludingthemodellingof chemicalreactions.Theseaspectsarepartofongoingwork.
Acknowledgements
ThisworkwasgrantedaccesstotheHPCresourcesofCALMIP supercomputingcenterundertheallocation2015-0111.This workwasperformedusingHPCresourcesfromGENCI-CINES (Grant2015-c20152b6012).
Appendix
A.
Mathematical
Model
Thisappendixgivesthesetofequationsofthen-fluidEulerian model.TheEuleriann-fluidapproachusedisahybridmethod,
inwhichthetrans-portequationsarederivedbyusingphase ensembleaverage forthe fluidphase and by usingkinetic theoryofgranularowssupplementedbyfluideffectsforthe dispersedphase,thankstojointfluid-particleprobability den-sityfunction(PDF)approach(Simonin,1996).Inthefollowing equations,subscriptk=greferstothegasphaseandk=pto theparticle phase. ˛pp inthe particle transportequations
representsnpmpwherenpisthenumberdensity ofparticle
centersandmp isthemassofasingleparticle:˛p=npmp/p
isanapproximationofthelocalvolumefractionofparticlep.
Hence,gasandparticlevolumefractions,˛gand˛phaveto
satisfy:
˛p+˛g=1 (A.1)
Appendix
A.1.
Volume
Fraction
Transport
Equation
Themassbalanceequationwrites
∂ ∂t˛kk+
∂ ∂xj
˛kkUk,j=0 (A.2)
where˛kisthevolumefractionofthephasek,kthematerial
densityandUk,i theith componentofthemean velocity.In
Eq.(A.2)theright-hand-sideisequaltozerobecausenomass transfertakesplace.
Appendix
A.2.
Mean
Momentum
Transport
Equation
Themeanmomentumtransportequationwrites
˛kk
∂ ∂t+Uk,j ∂ ∂xj Uk,i=−␣k ∂Pg ∂xi +␣k kgi+Ik,i− ∂k,i,j ∂xj (A.3)wherePgisthemeangaspressureandgithegravity
accelera-tion.k,ijtheeffectivestresstensorwhichiscomposedoftwo
parts:
k,ij=˛kk uk,iuk,j +k,ij (A.4)
whereuk,iisthevelocityfluctation uk,iuk,j aretheturbulent
orkineticstressesandk,ijtheviscousorcollisionalstresses
forthegasphaseortheparticulatephaserespectively.
Gas-ParticleInterphaseMomentumTransfer.InEq.(A.3),Ip,iis
themeangas-particleinterphasemomentumtransfer with-outthemeangaspressurecontribution.Accordingtoalarge particle to gas density ratio only the drag force must be accountedforinthemeangas-particleinterphasemomentum transferterm,whichiswrittenas:
Ip,i=−˛pp
Vr,i
gpF
and Ig,i=−Ip,i. (A.5)
Themeanfluid-particle relativevelocity,Vr,i, isgivenin
termsofthemeangasandsolidvelocities:
Vr,i=Up,i−Uf,i−Vd,i (A.6)
whereVdisthedriftvelocity,whichcanappeardueto
turbu-lence(Gobinetal.,2003)orsubgrideffect(Parmentieretal., 2012).
FollowingGobinetal.(2003)thedriftvelocitydueto turbu-lenceismodelledby:
Vd,i=−Dgp,ij[ 1 ˛p ∂˛p ∂xj − 1 ˛g ∂␣g ∂xj ]with Dgp,ij= 1 3 t gpqgpıij (A.7) where t
gpthecharacteristictimescaleofturbulence‘seen’by
the particles(includingcrossingtrajectoryeffects)and qgp=
ug,iup,i thegas-particlecovariance.
Theparticlerelaxationtimescalewrites
1 F gp =43g p |vr| dp Cd (A.8)
whereCdisthedragcoefficientandwith
|vr| ≈
v2r =
Vr,iVr,i+ vr,ivr,i (A.9)
Thevarianceofthefluid-particlerelativeveloticyisgiven by:
v
r,ivr,i =2q 2
p+2kg−2qgp (A.10)
Totakeintoaccounttheeffectoflargesolidvolumefraction Gobinetal.(2003)proposedthefollowingcorrelationforthe dragcoefficient
Cd=
min(Cd,Erg,Cd,WY) if˛p>0.3
Cd,WY otherwise
(A.11)
whereCd,ErgisthedragcoefficientproposedbyErgun(1952):
Cd,Erg=200
˛p
Rep+
7
3 (A.12)
andCd,WYbyWenandYu(1965):
Cd,WY =
⎧
⎨
⎩
0.44˛−1.7g ifRep≥1000 24 Rep1+0.15Re0.687p ˛−1.7g otherwise . (A.13)
TheparticleReynoldsnumberisgivenby
Rep=˛g
g |Vr| dp
g . (A.14)
Moreover, t
gpisrelatedtothecharacteristictimescaleof
turbulence, t
g,throughthefollowingrelation:
tgp= tg
1+Cˇr2 with 2 r= Vr,iVr,i 2/3kg (A.15)withCˇ=1.8andwhere gtisgivenbytheturbulencemodel. SolidStressTensor.Thesolidstresstensorwritesp,ij=gp,ij+
fp,ij,wherefp,ijisthefrictionalstresstensordescribedin sec-tion 2.In theframeofthe kinetictheory ofrapidgranular owsmodifiedbythedragforce(Boëlleetal.,1995)the gran-ularstresstensorcanbecomputedfromseparatetransport equations.However,fornumericalimplementationweusethe
followingapproximatedformforthe granularstress(Boëlle etal.,1995;Balzeretal.,1995;Balzer,2000)
gp,ij=
Pp−p ∂Up,k ∂xk ıij−pSp,ij (A.16)wherethestrainratetensorSpisdefinedbyEq.(2).The
gran-ularpressure,viscositiesandmodelcoefficientsaregivenby
Pp= 2 3˛ppq 2 p
1+2˛pg0(1+ec) (A.17) p= 4 3˛ 2 ppdpg0(1+ec) 2 3 q2 p (A.18)p=˛pppkin+˛ppcolp (A.19)
pkin=
1 3qgp t gp+ 1 2 F gp 2 3q 2 p(1+˛pgoc)
/ 1+ 2 gpF c (A.20) pcol= 4 5˛pg0(1+ec) kinp +dp 2 3 q2 p (A.21) c= 2 5(1+ec) (3ec−1) (A.22) = 1 5(1+ec) (3−ec) . (A.23) Thecollisiontimescale cisgivenby(Simoninetal.,2002):
1 c= 4g0nqd2p
2 3q2p1−gp (A.24)
wheregpisafluid-particlecorrelationcoefficientthattakes
intoaccounttheeffectofturbulenceoncollisionsandisgiven by: gp= q2 gp 4q2 pk (A.25)
Moreover,theradialdistributionfunction,g0,iscomputed
accordingtoLunandSavage(1986)as
g0(˛p)=
1− ˛p ˛max
−2.5␣max (A.26)
where˛max=0.64istheclosestrandompacking.
Appendix
A.3.
k
–
Model
Adapted
to
Gas-Particle
Flows
Thetransportequationforthegasturbulentkineticenergy reads: ˛gg
∂kg ∂t +Ug,j ∂kg ∂xj = ∂ ∂xj ˛gg t g k ∂kg ∂xj − u g,iug,j ∂Ug,i ∂xj −˛gg +kg p→g (A.27)wherek=1andwith
kg p→g= ˛pp ˛gg 1 F gp
qgp−2kg+Vd,iVr,i (A.28) kgp→grepresentstheeffectofparticles.Notethatthe
produc-tionofagitationbythewakeofaparticleisassumednegligible compared to the large scale turbulent agitation (Vermorel etal.,2003).
TheBoussinesqhypothesisisusedtomodeltheReynolds stresstensor,yielding:
u g,iug,i =−tg
∂Ug,i ∂xj + ∂Ug,i ∂xj +23 kg+tg ∂Ug,k ∂xk ıij (A.29)wheretheturbulentviscosityisgivenby:
tg= 2 3kg t g
1+C12␣p p ␣gg t gp F gp 1−qgp 2kg −1 (A.30) withC12=0.34. Moreover t gisgivenby: gt=C 3 2 kg (A.31) whereC=0.09.Thetransportequationfortheturbulentdissipationreads:
˛gg
∂ ∂t+Ug,j ∂ ∂xj
=−˛ggkg
C ,1 u g,iug,i ∂Ug,i ∂xj +C ,2
+ ∂ ∂xj ˛gg t g ∂ ∂xj + p→g (A.32)
with =1.3andwhere
p→g=C ,3 kg kg p→g (A.33) withCε,3=1.2.
Appendix
A.4.
Solid
Random
Kinetic
Energy
Transport
Equation.
Thesolidrandomkineticenergytransportequationwrites:
˛pp
∂q2 p ∂t +Up,j ∂q2 p ∂xj =− ∂ ∂xj ˛ppKkinp +Kcolp ∂q2 p ∂xj −g p,ij ∂Up,i ∂xj −˛pp F gp 2q2p−qgp −˛pp1 2 1−e2 c c 2 3q 2 p (A.34)
InEq.(A.34),thefirsttermontheright-hand-sideisthe tur-bulenttransportoftheparticleagitation.Thattermiswritten byintroducingthediffusivitycoefficients:
Kkin p =
1 3qgp t gp+ 2 3q 2 p 5 9 F gp(1+˛pg0c)
/
1+5 9 F gp c c
(A.35) Kcol p =˛pg0(1+ec) 6 5K kin p + 4 3dp 2 3 q2 p (A.36)
c= 3 5(1+ec) 2 (2ec−1) (A.37) c=(1+ec) (49−33ec) 100 (A.38)
Thesecondtermontheright-hand-sideofEq.(A.34) rep-resentstheproductionofparticleagitationbythegradients ofthemeansolidvelocity.Thethirdtermistheinteraction withthegas.Finallythefourthtermistheparticleagitation dissipationbyinelasticcollisions.
The transport equation for the gas-particle covariance reads: ˛pp
∂qgp ∂t +Up,j ∂qgp ∂xj = ∂ ∂xj ˛pp t gp k ∂qgp ∂xj −˛pp ug,iup,j p ∂Up,i ∂xj −˛pp ug,jup,i p ∂Ug,i ∂xj −qgp −˛pp qgp t gp (A.39)wheretheinversecouplingtermeisgivenby:
qgp= ˛pp F gp
1+˛pp ˛gg qgp−2kg−2 ˛pp ˛ggq 2 p (A.40)The Boussinesq hypothesis is used to model the gas-particlecorrelations,yielding:
u g,iup,j p=−tgp
∂Ug,i ∂xj + ∂Up,j ∂xj
+1 3
qgp+t gp ∂Ug,k ∂xk + t gp ∂Up,k ∂xk
ıij (A.41)
wheretheturbulentviscosity,t
gp,isgivenby t gp= 1 3qgp t gp (A.42)
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