Mixing performance in Split-And-Recombine Milli-Static Mixers—A numerical analysis

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Chemical Engineering Research and Design

j o u r n a l h o m e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / c h e r d

Mixing performance in Split-And-Recombine Milli-Static Mixers—A numerical analysis

Charbel Habchi

, Akram Ghanem

, Thierry Lemenand

, Dominique Della Valle

, Hassan Peerhossaini

aNotreDameUniversity—Louaize,ThermofluidsResearchGroup,P.O.Box:72ZoukMikael,ZoukMosbeh,Lebanon

bELISAAerospace,48RueRaspail,02100Saint-Quentin,France

cAngersUniversity,ISTIA,LARISEA7315,49000Angers,France

dONIRIS,44300Nantes,France

eUniversitéParisDiderot,SorbonneParisCitéUniversity—EnergyPhysicsGroup,AstroParticlesand Cosmology-CNRSUMR7164,75013Paris,France

fUniversityofWesternOntario,MechanicsofActiveFluidsLaboratory,Civil&EnvironmentalEngineeringand Mechanical&MaterialsEngineering,London,ONN6A3K7,Canada

a r t i c l e i n f o

Articlehistory:

Received24May2018 Receivedinrevisedform2 December2018

Accepted12December2018 Availableonline21December2018

Keywords:

Mixingenhancement CFDstudy

Staticmixer Creepingflow Baker’stransform

a bs t r a c t

VerylowReynoldsnumber laminar flowin Split-And-Recombine(SAR)static mixersis numericallyinvestigatedbyusingafinitevolumemethod.Intheseconfigurations,advective chaoticflowstructuresarecreatedbythepassivecontroloftheflow,mimickingthebaker’s transformation,throughaseriesofflowsplitting,rotations,andre-combinationsthatgen- erateintertwinedlamellarstructures.Thisprocessleadstoextrasurfacecreation,which ultimatelyintensifiesmasstransfer.Themainaimofthisstudyisfirstlytodemonstratethe interestofthistechniqueforenhancingmixing,whilekeepingpressuredropsmoderate, andsecondlytooptimizethistypeofgeometry.Forthelatterstake,twodifferentSARs, namelySAR1(Gray’sconfiguration)andSAR2(Chen’sconfiguration),arecomparedinterms ofdistributiveanddispersivemixingandenergyexpendituresevaluatedbythepressure drop.Theenhancementeffectofsplitting/recombinationisisolatedthroughthecompari- sonwithachannelcomposedofsmallstraightsectionsatrightanglebendswithalternate chiralities(referredas3D-Flow),withoutsplitting.Aplainsquare-sectionchannelisusedas base-linereferencegeometrytoassesstherelativemixingefficiencyofeachconfiguration.

ResultsshowthattheSARtechniqueiscapableofenhancingmasstransferincreepingflows comparedtonon-splitting-flowmixers/exchangers,thusallowingagaininresidencetime andmixersizeforthesamefinalresults.BetweenthetwoSARgeometries,thesuperiority ofChen’sconfigurationregardingtherelativemixingefficiencyisdemonstrated,with83%

mixingintensification,accompaniedbyacostincreaseof68%inthefrictioncoefficient.

©2018InstitutionofChemicalEngineers.PublishedbyElsevierB.V.Allrightsreserved.

1. Introduction

Laminarmixingisimportantformanyengineeringapplicationswhere turbulencecannotbeusedorgenerated.Thisisacommonissuein pharmaceutical,cosmetic,andbiologicalapplicationswherehighly viscousandfragilefluidsare frequentlyused. Inthiscase,mixing

∗ Correspondingauthorat:UniversitéParisDiderot,SorbonneParisCitéUniversity—EnergyPhysicsGroup,AstroParticlesandCosmology- CNRSUMR7164,75013Paris,France.

E-mailaddress:[email protected](H.Peerhossaini).

shouldbeenhancedatverylowReynoldsnumberswithoututilizing vortexgeneratorsorturbulencepromoters. Compactnessofmixers inlaminarregimeisalsoofgreatimportance(Neerincxetal.,2011) sincethemixingtime,i.e.thetimerequiredforachievingacertain degreeofhomogeneity,isusuallylargerthanthatinturbulentflow regimes.

https://doi.org/10.1016/j.cherd.2018.12.010

0263-8762/©2018InstitutionofChemicalEngineers.PublishedbyElsevierB.V.Allrightsreserved.

Table1–Characteristicsoftheflowconfigurations.

Plainchannel 3D-Flow SAR1(Grayetal.,1999) SAR2(ChenandMeiners,2004)

Crosssection(mm²) 3×3 3×3 3×3 3×3

Numberofelements – 18 13 10

Elementdevelopedlength(mm) – 39 54 69

Totaldevelopedlength(mm) 700 702 702 690

Pre-conditionerlength(mm) – 30 30 30

Post-conditionerlength(mm) – 30 30 30

Thischallengehasledtothedevelopmentandoptimizationofthe so-called“SplitAndRecombine”(SAR)staticmixers(Grayetal.,1999;

ChenandMeiners,2004;Jarrahietal.,2016;HirataandOhkawa,2016).

SARmixerconsistsofanetworkofdividedandthenrecombinedchan- nelsinwhichseveralfluidsareintroducedseparatelyandmixedby amulti-laminationprocess.ThisSARtopologyperformsaseriesof baker’stransformsontheconcentrationprofile(Carriere,2007).The flowstream-tubesare split out,rotatedin oppositedirections and thenrecombined,foldingovertheconcentrationprofileanddoubling thetransversgradient.Successiveverticalseparationandhorizontal reunitingoffluidstreamsincreasethenumberoflaminateswitheach stage;hencethecontact-surfaceareabetweenthetwofluidsisexpo- nentiallyincreased,resultinginfastermixing(Ghanemetal.,2014;

HossainandKim,2015).Recentinvestigations(Carriere,2007;Ghanem etal.,2013a;Creysselsetal.,2015)revealedthechaoticnatureofthe SARflowandunderfavorableconditions,themaximumvalueofLya- punovexponentofln (2) wasrecordedintheSARstaticmixerdesigned byGrayetal.(1999).

SARmixerscanbeclassifiedintwotypes:theplanaror2Dmixers, andthe3Dconfigurations.AnsariandKim(2010)investigatednumer- icallythe mixingprocessin unbalancedsplitsandcross-collisions planarcircularandrhombicmicro-channels forReynoldsnumbers rangingfrom1to80.Theyshowedthatthelowestmixingperformance isobservedinthebalancedcollisionconfigurationswhereasthehigh- estmixingindexwasobtainedinthecircularmicro-channelsinwhich Deanroll-cellsaregeneratedatReynoldsnumbershigherthan10.Chen etal.(2011)studiedthemixingprocessinstaggeredplanarDeanvor- texmicromixersforReynoldsnumbersrangingfrom0.5to50.Itwas foundthatforReynoldsnumberslowerthan5,diffusionisdominant overadvectivetransport,whileforReynoldsnumbershigherthan10 mixingisgovernedbyDeanroll-cells.

Anxionnaz-Minvielle et al. (2017) studied experimentally heat transferinthreedifferentconfigurationsof3DSARstaticmixersand comparedtheirefficiencytoplanarcorrugatedchannels.Itwasfound thattheenergyefficiencyofSARmixersincreasescomparedtothat of2Dcorrugatedchannelswithincreasingviscosities,especiallyfor Reynoldsnumberslowerthan50.Ghanemetal.(2013a,2013b)used bothnumericalandexperimentaltechniquestostudythemassand heattransferindifferentmilli-3DSARflowconfigurationsfordiffer- entflowregimes.Comparisonoftherelativemixingefficiencyandthe energyconsumptionamongthedifferentflowconfigurationsshowed thatthemixerproposedbyJarrahietal.(2016)ismoreefficientinthe laminarandmoderatelyturbulentflowregimes,forReynoldsnumbers rangingfrom40to5000Grayetal.(1999).Fromtheheattransferpoint ofview,theSARconfigurationproposedbyChenandMeiners(2004) showsanadvantageovertheotherconfigurationsforcreepingand deeplylaminarflows,withReynoldsnumbersbetween10⁻⁴ and10 (Ghanemetal.,2013a,2013b).

Inthepresentpaper,numericalsimulationsareconductedtoana- lyzepassivescalarmixingindifferent3DSARflowconfigurationsfor averylowReynoldsnumbervalueof10⁻³,basedontheinletmean flowvelocity.ThislowReynoldsnumberallowstohighlightthebaker’s transformsoccurringinSARconfigurationsandtoexhibititseffectson mixingversustheincreaseinthepressuredrop.Thisanalysisallows establishingahierarchyofmixersbasedandtheirmixingefficiency andopensintoanoptimizationprocess.

Thepaperisorganizedasfollows:Section2isdevotedtoproblem descriptionandnumericalmodelused,theresultsarediscussedin Section3,andfinallytheconcludingremarksaregiveninSection4.

2. Problem description

For an incompressible,steady, Newtonianfluid, the flowis governedbythecontinuity(Eq.(1))andNavier-Stokes(Eq.(2)) equations:

∇·u=0 (1)

u·∇u=−∇p

+∇²u (2)

whereuisthe velocityvector,and arerespectively the densityandkinematicviscosityoftheworkingfluid,andpis thepressure.

Themixingprocessisanalyzedbytrackingapassivescalar Cthatwillallowcomputingthevariancedestructiondown- streamandtocharacterizemixing.Themassbalanceshould betreatedinapureconvectivetransporttoquantifythekine- maticmixing.Actually,CFDsimulationsgenerateanumerical diffusionthatdependsonthenumericalscheme,thesolver, and mostlythemeshing.Inordertoobviatenumericaldif- fusion,wehaveintroducedaphysicaldiffusioncoefficient˛, whichindeeddampsthepureSARmixingeffect(butremains identicalinallsystemsfortheglobalmixingcomparison),and couldbeclosertotheeventualexperimentalresults.Themass transferequationishencegiveninEq.(3):

∇·(uC)=␣∇²C (3)

2.2. Numericalmethod

The computational fluid dynamics (CFD) code ANSYS Flu- ent15 (ANSYS,2018)isusedforthepresent simulations.It isbasedonanEulerianapproachtocomputeNavier–Stokes equationsthroughcell-centeredfinitevolumediscretization.

Theflowandscalarequations aresolvedsequentiallywith doubleprecision.ThirdorderMUSCLschemeisusedforspa- tialdiscretizationofthelinearmomentumandmasstransfer equations.Diffusiontermsarecentral-differenceandsecond- orderaccurate.Pressure-velocitycouplingisachievedbythe SIMPLEalgorithmproposedbyPatankar(1980).

2.3. Flowconfigurationsandoperatingconditions

Numericalsimulationsareperformedforfourdifferentflow configurationssummarizedinTable1:Plainstraightchannel, 3D-Flow,SAR1andSAR2geometries.Allconfigurationshave thesameflowcross-sectionalarea.Thenumberofunitsper mixerischosensothatallconfigurations leadtothesame residencetime(definedbythereactivevolumeovertheflow rateratio).Foreachconfiguration,pre-andpost-conditioners areaddedtoattenuatetheentranceandexiteffectsonthe

Table2–Workingfluidpropertiesandoperating conditions.

(Pas) 100

(kg/m³) 1000

Pe 10⁶

Re 10⁻³

˛(m²/s) 10⁻¹⁰

resultsandtoobtainafullydevelopedflowattheinletofthe mixerelements.

The first flow configuration, plain channel, is a plain straightduct ofsquarecross section.The3D-Flowconfigu- rationconsistsofacontinuousductflowwithalternatebends atrightangles.ThetwoSARconfigurations are referredas SAR1 and SAR2. Basic elements ofthe 3D-Flow, SAR1 and SAR2configurationsareshowninFig.1.SAR1isthegeometry describedbyGrayetal.(1999),wedenotedthisgeometrySAR1 since there is one separation/recombination per element.

SAR2 is the configuration described by Chen and Meiners (2004),wedenoted thisgeometrySAR2sincethereare two separations/recombinationsperelement.Resultsarefurther normalizedbythoseobtainedintheplainsquareductusedas thereferencegeometry.

No-slip boundary conditions are prescribed on all wall boundaries.Auniformvelocityprofilewithaveragevelocity equal to 3.33cm/s is imposed at the inlet of the pre- conditioner. The scalar field at the inlet cross-section is dividedintotwoequalparts,leftandright,withC_left=0and C_right=1.Attheoutlet,streamwisegradientsofallthevariables are settozero. Theworking fluidpropertiesandoperating conditionsareshowninTable2.TheReynoldsnumberRe,is calculatedbasedonthehydraulicdiameteroftheductcross- sectionandtheinletmeanflowvelocity.Theviscosityofthe workingfluidischosentomatchthatofviscousfluidproducts encounteredinindustrialapplicationswhereSARdevicesare usuallyusedforcreepingflows(Thakuretal.,2003).Numeri- calsimulationsarecarriedoutinisothermalconditions,with alowmassdiffusivitycoefficient(highPécletnumber)sothat diffusioncannotdominateoveradvectivemixing.InTable2,Pe isthePécletnumberrepresentingtheratiobetweenadvective anddiffusivetransportrates.Asnoticed,theadvectivetrans- portisdominantoverdiffusionsincethediffusivitycoefficient

˛isverysmall.

Scaled residualvalues of10⁻⁸ are set as aconvergence criterionforthe solutionofmomentumand passive scalar equations.

2.4. Mesh

SinceintheSARconfigurationsmixingelementsarerepeated, aseriesofnumericalsimulationswereconductedforasingle elementinSAR1configurationwithdifferentmeshdensities inordertodeterminetheappropriatefinalmeshtobeused inthestudy.Themeshsensitivityanalysisisdoneaccording totheprocedureproposedbyCeliketal.(2008);fivedifferent meshdensitieswereconsideredasshowninTable3.Thegrid sizeratioisthegridsizeofmeshi+1dividedbythatofmesh i;itisrecommendedtobeatleastequalto1.3(Celiketal., 2008).Forallcasesthemeshisstructuredincubicalcellsto reducenumericaldiffusion.Thecriteriaforthemeshsensitiv- ityarethepressuredrop,i.e.thepressuredifferencebetween theinletandoutletoftheSARgeometry,andthenormalized

standarddeviationofthecomputedscalarvaluewhichisused toquantifymixinginthedifferentgeometriesinthispaper:

0=

C²−C¯²

C²₀−C0 2

(4)

where ¯C0correspondstotheaveragescalarconcentrationata givencrosssectioncomputedinplainductflow.

Table3representsthemeshsensitivityanalysisperformed ononeelementintheSAR1configuration.Asshowninthis table, thevalue ofGCI on/₀ is0.33% and it is0.48%for pforthe finermesh(meshnumber 5).TheGCIissimilar todiscretizationerrorandcouldbeusedaserrorbarwhen presentingtheresults.Finally,thefinermesh5isadoptedfor allgeometries.

3. Results and discussion

Theflowstreamlinesthroughthelastelementinthethree different geometries are shown in Fig. 2. In the 3D-Flow configuration (Fig. 2(a)), the streamlinesare simplytwisted periodicallywhilecirculatinginthesuccessivebends.Since the flow inertial forces are small relative to the viscous forces,thereisnogenerationofhydrodynamicinstabilityor vortices and thus the only irreversible mixing mechanism presentisthemoleculardiffusion.IntheSAR2configuration (Fig.2(c)),thestreamlinesareevenlysplitintotwoperpendic- ulardirections,thenrotatedby90^◦twotimesbeforetheyare recombined.Thisprocessisrepeatedtwotimesineachele- ment.Herealsotherearenovorticesorinstabilities dueto highfluidviscosity.However,itwillbeshownlaterthatthe mixingoccursaccordingtobaker’stransformationinthesuc- cessiveelements.Theflowstructureinthethirdconfiguration SAR1(Fig.2(b))issomehowmorecomplexthanthefirsttwo;

streamlinesareinitiallysplitintwooppositedirectionsthen rotatedby90^◦fourtimesinoppositedirectionsbeforetheyare recombinedtogether.Vorticesarealsoabsentinthisconfig- urationduetothelowReynoldsnumber.Inthenextsection, weexamineindetailshowthis successionofthesplitand recombineprocessesaffectspassivescalarmixing.

3.2. EquivalentPoincarésectionsanddistributive mixing

Chaoticpropertiesoftheflowcanbecharacterizedbymeans of Poincaré sections. For this purpose, the outlet section- plane isconsidered for each ofthe four configurations.At thecenteroftheinlet-plane,around1000inertparticlesare releasedstartingfromaninitialpositionintheformofadisk of0.15mmdiameter.Thetrajectoriesoftheindividualparti- clesaretrackedandtheircorrespondingpositionsattheoutlet sections are exported.Theset ofpositions ofthe particles inthecomputationaldomainoutletconstitutethePoincaré sections, which are shown in Fig. 3. For the straight plain duct, the particles leavethe outletatthesame positionin whichtheywerereleasedattheinlet-sectionsincetheflow islaminarandthestreamlinesarestraight.Therefore,dueto highfluidviscosityanddeeplaminarflow,fluidparticlesare organizedinuniformparallelstratathatmovealongthelon- gitudinaldirectionwithnoradialdeviationinthetrajectories.

Undertheseconditions,the3D-Flowconfigurationalsoshows

Fig.1–Isometricviewofoneelementfromeachflowconfiguration:(a)3D-Flowconfiguration,(b)SAR1(Grayconfiguration (Grayetal.,1999)),and(c)SAR2(Chenconfiguration(ChenandMeiners,2004)).

Table3–MeshsensitivityanalysisperformedononeelementintheSAR1geometry.

Meshnumber 1 2 3 4 5

Cellnumber 1.07×10⁴ 7.09×10⁴ 5.67×10⁵ 1.15×10⁶ 2.25×10⁶

Gridsize(mm) 0.363 0.193 0.097 0.076 0.061

Gridsizeratio 1.9 2.0 1.3 1.3

/0 0.7683 0.8373 0.9227 0.9344 0.9351

GCI(%) 13.17 1.57 0.33

p (bar) 1.632 1.749 1.806 1.969 1.971

GCIp(%) 4.49 10.35 0.48

Fig.2–Streamlinesthroughthelastelementof(a)3D-Flowconfiguration,(b)SAR1and(c)SAR2.

apoordispersingperformanceclosetothestraightduct.The SARconfigurationshoweverexhibit,asexpected,aneffective signofchaoticadvection.Theparticlesarelaterallydispersed andarecapableofvisitingvariousradialpositionsintheflow cross-section,revealingthemixingenhancementbytheSAR mechanism.Atafirstglance,theSAR2configurationshows thebestperformance:particlescoverthelargestareaofthe flowcross-sectionandextendtowardsthewalls,comparedto SAR1geometryinwhichwidergapscanbeseen.

For a better quantitativecomparison ofthe distributive mixingeffects(SomanandMadhuranthakam,2017;Parketal., 2018),thecoefficientofvariationoftheparticle locationsis computedusingtheradiusasthecharacteristiclength:

CoVr=r

¯

r (5)

where ¯ristheaverageradiusandrthestandarddeviation obtainedasfollows:

r=

i=1

(r_i−r)¯²

N−1 (6)

Fig.4comparesthecoefficientofvariationofthedistribu- tivemixingamongallthe geometriesstudiedinthiswork.

Theplainpiperepresentsthelowestdistributivemixingper- formance with a CoVr almost equal to zero. The 3D-Flow configuration comessecondwithaverylowCoVr,whichis consistentwiththePoincarésectionshowninFig.3(b).The SAR2 geometry shows the highest performancewith CoV_r around 0.347comparedto0.299 intheSAR1 configuration, showingaround17%relativeincreasecomparedtothatofthe

Fig.3–Poincarésectionsforthestudiedflowconfigurations.Theinitialpositionisadisk(inblue)ofradius0.15mmatthe inletcenter:(a)Plainchannel,(b)3D-Flow,(c)SAR1,and(d)SAR2configurations.(Forinterpretationofthereferencestocolor inthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

Fig.4–Distributivemixinganalysisusingthecoefficientof variationoftheparticlelocationascharacteristicdistance.

SAR1.Inthenextsectionthemixingperformanceisstudied bytheanalysisofpassivescalarmixinginthestudiedgeome- tries.

3.3. Scalarmixing

Pure kinematic mixing produced by the baker’s transform leads tothe formationof parallelfluid layerswithvarying propertiesfollowingtheinitial(inlet-section)stateofsegre- gation(Carriere,2007).However,usingfinitevolumemethod resultsinnumerical diffusionduetotheinterpolation and discretizationschemesadopted(HossainandKim,2015).To further reduce the numerical diffusion, in addition to the meshsensitivityanalysis,weusehigherorderdiscretization schemes.Thereforea3rdorderMUSCLschemewasadopted fortheconvectivetermsintheNavier–Stokesequations.

Scalar mixingis studiedbydividingthe inlet-sectionof each configurationintotwopartsasshowninFig.5where, theleftpartcorrespondstoanarbitraryscalarvalueequalto 0andtherightpartcorrespondstoascalarvalueattheexit- planeofthe firstfiveelementsofthe3D-Flow andthetwo

Fig.5–Scalarmixingattheoutlet-sectionofeachofthefirstfiveelementsfor(a)the3D-Flow,(b)theGray(SAR1),and(c) theChen(SAR2)configurations.

SARconfigurations.Inspiteofthelowfluiddiffusivity,with thelowvelocityand relativelylarge contacttime intervals, smalldiffusionoccursattheinterfacebetweendifferentscalar layersproducingratherintertwinedlamellarstructures.Yet thesignatureoftheSARmechanismcanbeclearlydetected andthearrangementofscalarstratafollowingthetheoret-

ical predictions of the baker’s transform is visible, though accompaniedbythedistortioncausedbymoleculardiffusion asshowninFig.5(b)and(c).FromthecontoursinFig.5(a),it canbenoticedthatforthisflowregimethebendsanddirec- tionchangesinthe3D-Flowconfigurationhavenoimportant effectonthemixingofthefluidstreams.Theonlyvisibleinflu-

Fig.6–Normalizedscalarstandarddeviation(a)atthe outletofeachelementand(b)versustheresidencelength forthethreeconfigurations:3D-Flow,SAR1,andSAR2.

enceisthatoftheinterfacialdiffusionspreadingsidewaysas thecontacttimebetweenthedifferentscalarlayersincreases downstreamintheflows.Bycomparingqualitativelythetwo SARconfigurations,itcanbeobservedthattheSAR2shows abettermixingperformance,wherethescalargradientsare rapidlyhomogenizedcomparedtoSAR1andaquasi-uniform distributionofscalarisobtainedattheoutletcross-sectionof thefifthelement.

Variationofthenormalizedstandarddeviation/₀ofthe scalarattheexitofeachelementinthethreedifferentconfigu- rationsisshowninFig.6,plottedagainsttheelementnumber inFig.6(a)andtheresidencelengthinFig.6(b),sincetheele- mentsofthedifferentconfigurationsdonotsharethesame developedlength. The 3D-Flow configuration shows quasi- constantlevelsofsegregationovertheentirelength,witha relativestandarddeviationequalto1,meaningafully seg- regated state. Conversely, the segregation index decreases continuously in the SAR configurations, as the fluid flows through the different elements. As observed in the scalar distribution,SAR2geometryclearlyexhibitsthebestmixing quality,withafasterdestructionratethanthatoftheSAR1 configuration.Startingfromthe3rdelement,themagnitude ofthecurve’sslopeincreases;thegradientsarerapidlydissi-

Fig.7–(a)Normalizedpressuredropand(b)normalized Fanningfrictioncoefficientforthethreeconfigurations:

3D-Flow,SAR1,andSAR2.

patedbyvirtueoftheexponentialincreaseintheinterfacial area.Theexponentialfittingafterthe3rdelementshowsthat thedecayin/₀fortheSAR2isalmost2.4timesgreaterthan thatinSAR1.Attheinlet-sectionofthe5thelementinSAR2 geometry,thedeviationfromthemeanbecomesinferiorto 1%,avaluelargelyacceptableforawiderangeofprocessengi- neeringapplications,whilethedeviationisequalto27%inthe SAR1configurationatthesamelocation.

3.4. Pressurelossesandfrictionfactors

Fig.7(a)representsthenormalizedpressuredropbetweenthe inletandtheoutletofeachmixingelementinthethreecon- figurations,wherep0correspondstothepressuredropinthe plainductflow.Theseeminglysurprisingresultfromthisfig- ureisthatthepressuredropinthe3D-Flowconfigurationis largerthanthatintheSARconfigurations andthattherel- ative pressure dropin thelattercaseislower than 1.This implies thatthepressure lossesacrosstheSARmixingele-

mentsaresmallerthanthoseinthestraightplainductwith thesameinletmeanflowvelocity.However,thereasonbehind this behavioristhat inthe SARconfigurations,the flow is dividedintotwochannelsofthesamecrosssection,sothat theflowrateisdividedby2ineachchannel,inducinglower pressuredropasitcanbeobservedfromthestreamlinescol- oredbythemeanflowvelocityinFig.2.Therefore,toaccount forthisissue,itispreferabletocomparethefrictionfactors thatcanscaleouttheflowrateissue.

TheFanningfrictioncoefficientisobtainedfromthepres- suredropacrossthemixerinletandoutletviathefollowing relation:

f= pD_h

2Lu²_m (7)

whereumisthemeanflowvelocity.

Inthestraightplainpipeand3D-Flowconfigurations,umis simplythemeanflowvelocity,whichisuniformthroughout theentiremixerlength.However,intheSARconfigurations, thecalculationofthefrictioncoefficientshouldbeweighted bythecorrespondingflowvelocity.InSAR1configuration,10%

oftheflowvolumecorrespondstoameanflowvelocityum

while90%correspondstothehalfofthemeanflowvelocity um/2.InSAR2,thevolumecorrespondingtoumis13%while thatcorrespondingtoum/2is87%.Letusdenoteˇthefraction ofthevolumecorrespondingtoum/2;thefrictioncoefficient wouldbecalculatedusingthefollowingexpression:

fSAR= pDh

2Lu²_m(1−ˇ)+ pDh

2L

2m

2ˇ (8)

whichleadstothefollowingequation:

fSAR= pD_h

2Lu²_m(1+3ˇ) (9)

ThenormalizedfrictioncoefficientsareshowninFig.7(b) wheref0correspondstothefictioncoefficientoftheplainduct flow.Thefrictionratios aregreater than1,sincethelosses instraightductarealwaysthelowest.Moreover,thefriction inthe3D-FlowconfigurationissmallerthanintheSARcon- figurationsduetosmallernumberofbends.Theincreasein thefrictionlossesinthe3D-Flowrelativetothestraightduct isabout11%,whileit islargerinthe SARcases:SAR1 has anincreaseofabout65%andSAR2hasanincreaseofabout 68%,whichisquitesimilaralthoughslightlyhigherthanSAR1 configuration.

3.5. Relativemixingefficiency

Analternativewayofassessingthemixingperformanceisby comparingtherelativemixingefficiencyofthedifferentcon- figurationsatconstantpumpingpower.Therelativemixing efficiencyisdefinedastherelativedifferenceinthestandard deviationsbetweenthemixerandthatofastraightductat constantpumpingpower:

= 0− ₀ |

pp

(10)

Forequalpumpingpower:

V˙0p0=V˙ (11)

Fig.8–Relativemixingefficiencyversustheresidence lengthinthethreeconfigurations:3D-Flow,SAR1,and SAR2.

where ˙Visthevolumetricflowrate.

Substitutingpbyanexpressionintermsoffandsubsti- tuting ˙VbyanexpressionintermsofReynoldsnumberRe,it followsthat:

Re Re0 =

f0

(12)

Thusatconstantpumpingpower,therelativemixingeffi- ciencybecomes:

=

0− 0

f0

(13)

Fig.8representsthevariationoftherelativemixingeffi- ciency versus the residence length for the three different configurations.TheSAR2geometryhashigherrelativemix- ingefficiencythanSAR1and3D-Flowconfigurations.Inthe 3D-Flow,theefficiencyincreasesveryslowlytoreachamax- imumvalueofabout22%atthemixerexitsincethemixing inthisconfigurationisalmostentirelyduetomoleculardiffu- sion.IntheSARconfigurations,theefficiencyincreasesrapidly toreachaplateauafterabout300mmfromtheinlet.Themax- imumefficiencyinSAR1andSAR2geometriesisalmostthe same;itisaround83%.However,mixingincreasesfasterin theSAR2comparedtoSAR1.

4. Conclusions

Mixing quality in chaotic flux recombination reactors is numerically investigated. The split/recombination mecha- nism isapractical realizationofthe mathematical baker’s transformthatgenerateschaoticstructuresandexponentially increasestheinterfacialareabetweenfluidlayers.Thispro- cess significantly promotes mixing by molecular diffusion.

Between two subsequent split and recombination phases, several rotations and flow direction changes are passively imposedontheflowcontributingtothecontinuousstretching andfoldingoftheviscousfluid.

ThestudiedcompactSARconfigurations(SAR1andSAR2), first proposed byGray et al. (1999) and Chen and Meiners (2004)respectively,arereproducedonamilli-scaleexploitable

in the industrial applications of handling viscous fluids and deeply laminar flows. A three-dimensional flow con- figuration with no split/recombination (3D-Flow) is also studied in terms of relative mixing efficiency and pres- sure drop, the baseline geometry being a plain square channel.

Thegoverningmomentum andpassive scalarconserva- tion equationsare solved usingaEulerian approachand a finitevolumemethodwiththeSIMPLEalgorithmforpressure- velocitycoupling.

Astheflowisdeeplylaminarandthefluidishighlyviscous, the3D-Flowconfigurationshowslittlemixingenhancement withrespecttoplainpipeflowandgeneratesahigherpres- suredrop.Thefluxrecombinationmixershowever,showan interestingrapiddecreaseinscalarconcentrationgradients.In SAR2configuration,forexample,thenormalizedscalarcon- centrationstandarddeviationfromthemeanattainsvalues aslowas1%asearlyasattheinletofthefifthelement.At thesamelocation,thoseforSAR1geometryarecalculatedat 27%.Indeed,SAR2mixershowssuperiormixingperformance comparedtotheotherconfigurations.ThePoincarésections, producedbyreleasinginertparticlesattheinletandfollowing theirpathsthroughthemixeroutlet,showthecapacityofSAR configurationstogeneratechaotic-typebehaviorinlaminar flow.Particlesremaingroupedinplainpipeand3D-flowwhile theyget dispersedand visit severallocations in thecross- sectionoftheSARconfigurations,especiallyinSAR2mixer, confirmingtheconclusionsbasedonscalarstandarddeviation values.

Toproperlyevaluatetheinterestofthesedevicesinprocess industry,energyexpenditures behind the mixingenhance- mentareevaluatedintermsofpressuredrop.Forsimilarinlet Reynoldsnumber,SARmixersgeneratelowerpressuredrops than3D-Flowandtheplainchannelduetothefactthatthe fluidflowsinthe majorityoftheir volumewithhalfofthe inletvelocitydueto splitting,thus decreasinghead losses.

Toaccountforthis point,the normalizednon-dimensional longitudinal Fanning friction coefficient is calculated and compared:3D-Flowshowsvalues11%increasecomparedto theplainchannel;theincreaseinSAR1andSAR2mixersbeing 65%and68%respectively.

Mixingperformanceandenergyconsumptionaresimulta- neouslyevaluatedintherelativemixingefficiencyparameter which gives the relative increase in mixing quality with respecttoaplainsquarechannelatconstantpumpingpower.

Whilethemaximumefficiencyin3D-Flowremainslimitedto 22%,SAR1and SAR2configurationsshowvaluesashigh as 83%withtheSAR2mixerfinallymanifestingthefastestmix- ingrates,dissipating concentrationgradientswithminimal residencetimeandmoderatepressuredrops.

Inthiscontext,thenumericalsimulationsofferapower- fuldesignandoptimizationtoolgivinginsighttotheoptimal numberofelementsasacompromisebetweenproductquality andenergyexpenditures.

Acknowledgments

C. Habchi and T. Lemenand would like to thank the PHC CEDREprogramandtheResearchCommissionofAngersUni- versityforthegrantsallowingresearcherexchangebetween FranceandLebanon.Theauthorsdedicatethisworktotheir dearcolleaguePhilippeCarrière,seniorscientistintheCNRS,

deceasedinFebruary2015.Amongmanyothercontributions, hehasintroducedustotheartofthebaker’stransform,inthe frameworkoftheEnergyProgram2006ACPR1-2-22.

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