Influence of the meandering channel geometry on the thermo-hydraulic performances of an intensified heatexchanger/reactor

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Influence of the meandering channel geometry on the thermo-hydraulic performances of an intensified

heatexchanger/reactor

Zoé Anxionnaz-Minvielle, Michel Cabassud, Christophe Gourdon, Patrice Tochon

To cite this version:

Zoé Anxionnaz-Minvielle, Michel Cabassud, Christophe Gourdon, Patrice Tochon. Influence of the meandering channel geometry on the thermo-hydraulic performances of an intensified heatex- changer/reactor. Chemical Engineering and Processing: Process Intensification, Elsevier, 2013, vol.

73, pp.67-80. �10.1016/j.cep.2013.06.012�. �hal-00905419�

This is an author-deposited version published in : http://oatao.univ-toulouse.fr/

Eprints ID : 9911

To link to this article : DOI:10.1016/j.cep.2013.06.012 URL : http://dx.doi.org/10.1016/j.cep.2013.06.012

To cite this version :

Anxionnaz-Minvielle, Zoé and Cabassud, Michel and Gourdon, Christophe and Tochon, Patrice Influence of the meandering channel geometry on the thermo-hydraulic performances of an intensified heatexchanger/reactor. (2013) Chemical Engineering and Processing: Process Intensification, vol. 73 . pp. 67-80. ISSN 0255-2701

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Influence of the meandering channel geometry on the thermo-hydraulic performances of an intensified heat exchanger/reactor

Zoé Anxionnaz-Minvielle

, Michel Cabassud

, Christophe Gourdon

, Patrice Tochon

aCEA,LITEN,LETH,17ruedesMartyrs,38054Grenoble,France

bUniversityofToulouse,LaboratoiredeGénieChimique,UMR5503,CNRS/INPT/UPS,31432Toulouse,France

Keywords:

Heatexchanger/reactor Wavychannel Corrugation Deannumber Scale-up

Processintensification

a b s t r a c t

Intheglobalcontextofprocessintensification,heatexchanger/reactorsarepromisingapparatusesto implementexothermicchemicalsyntheses.However,unlikeheatexchangeprocesses,theimplementa- tionofchemicalsynthesesrequirestocontroltheresidencetimetocompletethechemistry.Awayto combinethelaminarregime(i.e.enoughresidencetime)withaplugflowandtheintensificationofboth heatandmasstransfersisthecorrugationofthereactionpath.

Inthiswork,theexperimentalset-upisbasedonplateheatexchanger/reactortechnology.7milli- channelcorrugatedgeometriesvaryingthecorrugationangle,thecurvatureradius,thedevelopedlength, thehydraulicdiameterandtheaspectratiohavebeendesignedandexperimentallycharacterized(heat transfer,mixingtimes,pressuredrops,RTD).Theobjectivesweretoassesstheirrespectiveperformances toderivesomecorrelationsdependingonthechanneldesign.

Theresultsconfirmedthebenefitsofthereactionchannelcorrugation.Heatandmasstransfershave beenintensifiedwhilemaintainingaplugflowbehaviourintheusuallylaminarflowregime.Moreover, whateverthemeanderingchannel’scurvatureradius,theresultshighlightedtherelevanceofconsidering theDeannumberasthescale-upparameter.Thisdimensionlessnumber,morethantheReynoldsnumber, seemstogoverntheflowinthewavychannels.

1. Introduction

In the global context of process intensification, defined by StankiewiczandMoulijn[1]as“anychemicalengineeringdevelop- mentthatleadstoasubstantiallysmaller,safer,cleanerandmore energy-efficienttechnology”,heatexchanger/reactorsarepromis- ingapparatuses[2,3].Indeed,bycombiningareactorandaheat exchangerinonlyoneunittheheatgenerated (orabsorbed)by thereactionisremoved(orsupplied)muchmorerapidlythanina classicalbatchreactor.Asaconsequence,heatexchanger/reactors mayofferbettersafety(byabetterthermalcontrolofthereaction), betterselectivity(bya more controlledoperating temperature) andby-productsreduction(incaseoftemperature-dependentsec- ondaryreactionforinstance).

Themaininterestingwaystointensifyheatandmasstransfers inheatexchanger/reactorsaretoinsert3Delementslikemetal- licfoamsorfins[4–6]ortostructuretheflowpath(2Dor3D) [7–13].Suchapparatuseshavetoaddressmanypointsintermsof

∗Correspondingauthor.Tel.:+330438783567;fax:+330438785161.

E-mailaddress:zoe.minvielle@cea.fr(Z.Anxionnaz-Minvielle).

performances,but alsointerms ofpolyvalence (the technology shouldnotbededicatedtoonlyoneapplication),flexibility(toeas- ilyswitchfromoneflowregimetoanother),competitiveness(too complexgeometriesleadtotoohighmanufacturingandoperating costs)andscale-up(performanceshavetobemaintainedduring thescale-upprocedure).

Tooptimizetheheatexchanger/reactorperformancesvs.the investment capacity, the first step is to characterize the flow behaviourandthetransfermechanisms[8,14].Thisisoneofthe objectivesofthiswork.A2D-structuredmillimetricandmeander- ingprocesschannelisconsidered.Contrarytotheflowinastraight channel,thestreamlinesinacorrugatedflow arenotparallelto theflowaxis.Thecentrifugalforceencounteredineachbendand theimbalancebetweenthisforceandthepressuregradientgen- eratecounter-rotatingvorticesinthechannelcrosssection.W.R.

Dean[15]wasthefirsttosolvetheflowsolutioninacurvedduct.

HedefinedtheDean number,De, whichtakesinto accountthe secondaryloopsgeneratedinacorrugatedchannel:

De=Re×

r

Rc (1)

0255-2701/$–seefrontmatter© 2013 Elsevier B.V. All rights reserved.

http://dx.doi.org/10.1016/j.cep.2013.06.012

Fig.1.Visualizationofthesecondaryloopsinasquareductcrosssection[16].

ReistheReynoldsnumber, Re= ×u×dh

(2)

,anduarerespectivelytheflowdensity(kgm⁻³),viscosity(Pas) andvelocity(ms⁻¹).d_histhechannelhydraulicdiameter(m)and Rcisthecurvatureradius(m).

AnincreaseoftheDeannumbertendstomovetheaxialvelocity peakfromthecentretotheoutwardductwall.Aboveacriticalvalue oftheDeannumber,becauseoftheflowinstability,twoadditional vorticesappear.TheyarecalledtheDeanvortices.Fig.1illustrates thisflowphenomenon.

ThecriticalDeannumberrangesbetween100and250according totheemployedDeanvorticesdetectionmethod[16–18].Down- streameachbend,theflowtendstoalaminarflowinthestraight length.Byalternatingthecorrugations,theflowinstabilitiesare reactivatedineachbendwhichpromotestheradialhomogeniza- tionofthevelocity,temperatureandmomentumfields[19].

Ontheotherhand,thisexpectedgradientshomogenizationhas tobebalancedwithapressuredropincrease.Thegoalofthiswork isthustooptimizethecorrugatedchannelgeometryinorderto intensifytheheatandmasstransferswhilegettingalowReynolds numberflow.Thislastpointisrequiredtohavereasonableresi- dencetime(tocompletethechemistry)andlowpressuredrop.

Moreover to expect a future industrialization of heat exchanger/reactor technologies,the scalabilityis anothermajor steptostudy.Indeed,duringthescale-upprocedure,eachsimil- itude law (hydrodynamics, chemistry, geometry,...) should be verifiedinordertomaintaintheperformances.Howevereachlaw ischaracterizedbyitsowninvariantparameter(time,length,...) anditisoftenimpossibletoverifytheprincipleofsimilitude:

Gmock−up=k×Gpilot,with k a constant parameter (3) Itisthusrequiredtocharacterizetheeffectofthescale-uponthe performancesofoursystem.The2ndobjectiveofthisworkisthen todeducerulesandcorrelationswhichwillbeusedtopredictthe performancesofthemeanderingchannelatapilotscale.

In the present work, a plate-type heat exchanger/reactor, includingoneprocessplateandonecoolingplate,hasbeenstud- ied.On eachplate,awavymilli-channelhasbeenmechanically etched.Thisworkaims atcharacterizingtheflowperformances in several geometries in terms of residence time distribution,

Fig.2. Viewoftheprocesschannelswithouttheutilityplate(rightside)andview oftheutilitychannels((leftside).

mixingtimes,heattransferandpressuredrops.Boththeinfluence ofthechannelgeometryandtheinfluenceofthechannelcharacter- isticsizes(hydraulicdiameterandaspectratio)havebeenstudied.

Sevengeometrieshavebeenconsideredandamock-uphasbeen builtforeachone.

2. Materialsandmethods

2.1. Experimentalmock-ups

Themock-upsaremadeofthreeplates:acoolingplatesand- wichedbetween a closing plate and a ‘process plate’ in which the reactants are flowing co-currently. The cooling plate is a 7mm thickness aluminium plate with a highthermal conduc- tivity(=247Wm⁻¹K⁻¹).Itisusedduringthermalexperiments andisremovedforthehydrodynamiccharacterizations.Boththe closingand‘process’platesaremadeofPolyMethylMethAcrylate (PMMA),whose thermalconductivity,coupledtoa thicknessof 20mm,islowenoughtoavoidthermallosses(l=0.19Wm⁻¹K⁻¹ and/e=8.5Wm⁻²K⁻¹;negligibleincomparisonwiththelocal heattransfercoefficients).Thetransparencyoftheseplatesisnec- essaryinordertovisualizetheflowinthechannelduringmixing experiments.Totestvariouschanneldesigns,themock-upshaveto beeasilyandquicklyimplemented.Asaconsequence,eachplate isetchedinthelaboratorywithanumericallycontrolledmilling machine.Then,theplatesareassembledwith16clampingscrews forthethermalexperiments(seeFig.2),orchemicallybondedfor thehydrodynamictests.

Eachmock-upisconnectedtoatestbenchequippedwith2mag- neticdrivepumps(Verder,0–5and0–10Lh⁻¹),threemassflow metres (Micromotion), a differential pressure measuring trans- ducer(Rosemount,0–5bars),12temperatureprobes(Pt100and thermocouples),twospectrophotometers(AvaSpec2048,AvaLight DHcandIn-lineflowcellsAvantes)andasyringepump.Experi- mentaldata(temperature,flowrate,pressureandabsorbance)are recordedbyanon-linedatastoragesystem.

2.2. Geometricalparametersofthewavychannels

Twomainchallengeshavetobeaddressedwhenstructuringthe wavychannels:

-Howcanaplugflowandasufficientresidencetimeforthechem- istrybecombined?

Fig.3. Schematicviewofthegeometricalparameters.

-Howcanalaminarflow(inordertogetasufficientresidencetime andlowpressuredrop)andtheintensificationofmassandheat transfersbecombined?

Inordertooptimizethegeometrywithrespecttothesecrite- ria,theinfluenceofthegeometricalparametersonresidencetime distribution,mixing,thermalperformancesandpressuredropshas beeninvestigatedinthiswork.

Thegeometricalparameterswerechosenwithregardstothose commonlystudiedintheliterature.Chengetal.[20]andFellouah etal.[16]studiedtheinfluenceofboththecurvatureradiusandthe aspectratioonthecriticalDeannumber.Theaspectratioisdefined astheratiobetweentheductheightanditswidth.Worksabout itsinfluenceonboththeNusseltnumber[21,22]andthefriction factor[23]havealsobeendone.Finallytheeffectsofcorrugation angle[24,25]andcorrugationpitch[25,26]havebeenstudied.

Fromtheseworks,fourparametersappeartobeimportant(see Fig.3):theaspectratio,a/bwithaandbrespectivelythechannel heightandwidth;thecurvatureradius,Rc;thecorrugationangle, andthestraightlength,L_dwhichcorrespondstothelengthbetween twobends.Thecorrugationpitchandthecorrugationheightboth dependonthecorrugationangleandthestraightlength.

Thevalueofthecurvatureradiusdependsonthechannelwidth asillustratedontheFig.4.

ThechannelwidthinFig.4isequalto4.02mm.Threecurvature radiusvaluescoexist.Thecurvatureradiusdefinedalongtheneu- tralaxisequals4mmwhereastheexternalcurvatureradius(R_c,e) andtheinternalcurvatureradius(R_c,i)arerespectivelyequalto6 and2mm.Theyaredefinedby:

Rc,e=Rc+b

2 (4)

and Rc,i=Rc−b

2 (5)

withbthechannelwidth.

TheDean number is defined asDe=Re×

q

R_c.Asa conse- quence,three differentDeannumberscanbeconsideredinthe channel.Thelargestone,i.e.theoneresultingfromthesmallest

Fig.4.Schematicviewofthecorrugatedchannelof4.02mmwidth.

curvatureradius(theinternalcurvatureradius),willbenamedthe internalDeannumber,De_i.

In order to identifythe influenceof each parameteron the thermo-hydraulicperformances,onlyoneparametervariesfrom onemock-uptoanother.ThegeometricaldataaregiveninTable1 andFig.5illustratesthem.

Themock-up‘a’ismadeof37rowsand36180^◦turningbends.

2rowsareillustratedinFig.5.Eachrowis90mmlong.Then,from mock-up‘b’tomock-up‘g’,onlyoneparameterchangesfroma mock-up toanother.Asaconsequencetounderstandtheinflu- enceofeachgeometricalparameter,itisimportanttocomparethe rightmock-ups.Forinstance,tostudytheinfluenceofthecurva- tureradius,onlytheperformancesofmock-ups‘d’and‘e’willbe compared.Thehydraulicdiameterofmock-up‘g’(4mm)hasbeen chosensoastohavearangeofReynoldsnumberincommonwith the2mmmock-upswithoutanymajorchangeofthepumporflow metreequipment.

2.3. Experimentalcharacterizations

Themock-ups have beencharacterizedin termsof pressure drops,ResidenceTimeDistribution(RTD),thermalperformances andmixing[27].Thesecharacterizationsaimatbothstudyingthe flowbehaviouraccordingtothegeometricalparametersandcor- relatingtheperformancestoascale-upparameter.

Pressuredropsaremeasuredwithadifferentialpressuresen- sor.Itsmeasurementrangevariesfrom0to5barwithanaccuracy

Fig.5.Schematicviewof2rowsofthecorrugatedchannels.

Table1

Geometricaldataofthecorrugationparameters.

Mock-upn^◦ a/b dh,eq u Rc,i(mm) Ld(mm) Ltot(m) Vtot(cm³) Nb/L(bends/m)

a 1(2/2mm) 2 – – 90 3,6 13,6 20

b 1(2/2mm) 2 75^◦ 0.5 6.94 5.0 20.0 90

c 1(2/2mm) 2 45^◦ 0.5 6.94 3.5 14.0 106

d 1(2/2mm) 2 45^◦ 0.5 20.28 3.5 14.0 44

e 1(2/2mm) 2 45^◦ 3 20.28 3.4 13.6 38

f 0.5(2/4mm) 2.67 45^◦ 3 20.28 3.4 27.2 38

g 1(4/4mm) 4 45^◦ 2 20.28 3.4 54.4 38

of±5%.Pressureprobesaresetattheinletandtheoutletofthe mock-upswheretemperaturesensorsarealsoconnected.Thetem- peratureis,indeed,animportantdatatoassessthefluidproperties (densityandviscosity).Theworkingfluidsaredistilledwaterand 67%w.glycerol.Fromtheexperimentallymeasuredpressuredrop, theDarcy coefficient,3, isassessed accordingtothe following expression:

= 1P

L

d_h××^u₂²

(6)

withthefluiddensity(kgm⁻³),uitsvelocity(ms⁻¹);Landd_h respectivelythelength(m)andthehydraulicdiameter(m)ofthe channeland1Pthepressuredrop(Pa).

Thethermalstudyisbasedonexperimentswhichaimatcool- ingthefluidflowingintheprocessplate.Processfluidisdistilled waterheatedat75^◦Cwhereasutilityfluidisrawwateratabout 13^◦C.Foreachexperiment,theutilityflowrateissetto100Lh⁻¹ whereastheprocessflowratevariesfrom1to15Lh⁻¹.Inorderto avoidtemperaturepinch,temperatureismeasuredallalongthe channel:6temperaturesensorsaresetbetweentheinletandthe outletoftheprocessplateand2temperaturesensorsaresetatthe inletandtheoutletofthecoolingplate.Thetemperatureprofileall alongtheprocesschannelshowsthatbetween40and60%ofthe totalchannellengthissufficienttoexchangetheheatduty.Asa consequence,thethermalbalanceshavebeenmadebetweenthe processchannelinletand thefirsttemperaturesensor,which is locatedonthe2ndrow(0.26mfromtheinlet).Fromthemeasured temperaturedifference,1T,theexchangedthermalpower,P(W), isassessed:

P=m˙ ×Cp×1T (7)

m˙ andC_parerespectivelytheflowrate(kgs⁻¹)andthespecificheat ofthefluid(Jkg⁻¹K⁻¹).

Theglobalheattransfercoefficient,U(Wm⁻²K⁻¹),isthencal- culatedwith:

U= P A×1Tml

(8) 1T_mlisthelogarithmicmeantemperaturedifferenceandAtheheat exchangearea.Aspreviouslydescribed,thechannelisetchedina PMMAplateandclosedwithanaluminiumplate.Asaconsequence theheatexchangeareacorrespondstotheareaincontactwiththe aluminiumplate.

TheNusseltnumberiswrittenas:

Nu=hp×dh

p

(9) d_histhechannelhydraulicdiameterandpisthefluidthermal conductivity(Wm⁻¹K⁻¹).h_pisthelocalheattransfercoefficient oftheprocessfluid(Wm⁻²K⁻¹)andisdeducedfromthefollowing equation:s

1

U×A= 1 hp×Ap

+ e + 1

hu×Au

(10)

eandarerespectivelythethicknessandthethermal conduc- tivity of thealuminium plate (Wm⁻¹K⁻¹).hu is thelocal heat transfercoefficientofthecoolingside.Ithasbeenassessedtobe 27kWm⁻²K⁻¹inourgeometry.Thisvalueishighenoughtocon- siderthattheheattransfermechanismonthecoolingsideisnota limitingparameter.

Finally,tocompareeach geometry,theNusseltnumbersare plottedagainsttheReynoldsnumberintheresultssection.

Tocharacterizetheglobalmixinglevelofeachmock-up,the reactionofdiscolourationofaniodinesolutionwithsodiumthio- sulfateisimplementedaccordingtothefollowingreactionscheme:

I2(brown)+2Na2S2O3(colourless)→ 2NaI(colourless) +Na₂S₄O₆(colourless)

Thiosulfateandiodineconcentrationsarerespectivelyequalto 1.4×10⁻²molL⁻¹and 5×10⁻³molL⁻¹.Theratioofflowratesis equalto1[28].

This homogeneous reaction is instantaneous and mixing limited. As a consequence, at steady state, by measuring the requiredlengthtocompletethediscolouringofiodine,themixing timecanbeeasilydetermined.Themixingtimesarethencompared accordingtoboththeflowregimeandthechannelgeometry.

Finally, RTD experiments have been carried out in order to characterize the hydrodynamic behaviour in each geometry. A spectro-photometrictechniquethatentailsacolouredtracerhas beenused.Twomeasuringprobesaresetupattheinletandat theoutletofthemock-up. Theacquisitiontime issetto0.12s.

Eachexperimenthasbeenperformedatroomtemperaturewith aflowrate varyingfrom5to18kgh⁻¹ (Reynoldsnumber range from450to2500).TheNETR(NumberofEquivalentstirredTank Reactors)hasbeenassessedfromtheexperimentalRTD[29].The higherNETRis,themoretheflowbehaveslikeaplugflow,which isnecessarytoperformsafelyandefficientlychemicalsyntheses.

2.4. Numericalsimulations

Tocompletetheexperimentalresultsandunderstandtheflow mechanisms in the corrugated channel, numerical simulations havebeenimplementedwithacommercialCFDcode(Fluent).A schematicviewofthemodelledcorrugatedchannelisillustrated inFig.6.Thegreencross-sectionshavebeendefinedtovisualize theflowallalongthestraightandthecurvedlengths.

Thenumericalresultsinthecrosssectionslocateddownstream the7thbendhavebeenconsideredtoestablishthevelocitypro- filesandstudytheflowmechanisms.Thevelocityprofilesofthe Fig.6havebeenplottedversustheradialpositionupstreamthe 3rdbendandupstreamthe7thone.Theyperfectlymatchandthus demonstratethattheflowisestablishedintheportionofinterest.

A RNG (Renormalization Group) k-epsilon model has been consideredtorunthesimulationsandthemeshsizehasbeenopti- mizedto6×10⁵cells.Abovethisvalue,thenumericallymeasured pressuredropsareindependentofthemeshsize.

Fig.6.Meshedgeometryforthenumericalsimulationsandvelocityprofilesaccordingtotheradialposition.

3. Resultsanddiscussion

Theresultsaredividedintotwosections.Thefirstoneconcerns withthe influence of thecorrugation’s geometrical parameters (curvatureradius,straightlengthandbendangle)onboththeflow behaviourand thetransfer mechanisms.It concernsthe mock- ups ‘a’, ‘b’, ‘c’,‘d’ and ‘e’. Thesecond section is about thefirst stepstowardsthescale-upofcorrugatedchannels.Theobjective istopredictthemeanderingflowbehaviourduringthescale-up procedure.Threemock-ups(mock-ups‘e’,‘f’and ‘g’)havebeen experimentallycharacterizedandcomparedforthatpurposeand theinfluenceofthehydraulicdiameterandaspectratiohasbeen investigated.

3.1. Influenceofthecorrugationgeometry 3.1.1. Pressuredrops

Foreachgeometry(frommock-up‘a’tomock-up‘e’),pressure dropsperunit oflengthversusReynoldsnumber areplottedin Fig.7.

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0 500 1000 1500 2000 2500 3000 3500

ΔP/L (bar.m-1)

Reynolds number

Mock-up a Mock-up b Mock-up c Mock-up d Mock-up e

Fig.7. PressuredropperunitoflengthversusReynoldsnumber(waterat25^◦C).

Thisgraphhighlightstheincreaseofpressuresdropswhenthe numberofbendsperunitoflengthincreases.Thus,asexpected, pressuredropsinmock-up‘a’arethelowestones.Thisisalsothe casewhengeometries‘d’(44bends/m)and‘c’(106bends/m)are compared.

Ifwecomparethemock-ups‘b’and‘c’,thenumberofbendsper unitoflengthisveryclose(90and106bends/m),whereasthebend angleisdifferent(75and45^◦).Asaconsequenceanincreaseofthe bendangleseemstoincreasethepressuredrops.

Theincreaseofthecurvatureradiusalsoincreasesthepressure drops.Thisisthecasewhenmock-ups‘d’and‘e’arecompared.

Thenumberofbendsperunitoflengthisquitesimilar(44and38 bends/m)whereastheinternalcurvatureradiusincreasesfrom0.5 to3mm.

TheDarcycoefficient,whichischaracteristicofthesizeandof thegeometryofthechannel,hasbeenplottedagainsttheReynolds numberinFig.8.

Whateverthegeometry,thegraphisdividedintotwozoneson bothsidesofRe=200.BelowthisvaluetheDarcycoefficientsare

0.01 0.1 1 10

10 100 1000 10000

Darcy coefficient

Reynolds number

Mock-up a Mock-up b Mock-up c Mock-up d Mock-up e

square straight channel-experimental data

Fig.8.DarcycoefficientversusReynoldsnumber,experimentswithwaterandglyc- erol(67w%)at25^◦C.

Table2

Comparisonbetweenliteraturedataandnumericalresultsaboutthepressuredropinasinglebendandinastraightlength(Re=686).

Theoreticalresults(literaturecorrelation)[30] Numericalresults

Bend Darcycoefficient,3 0,27 –

Pressuredrop,1P(Pa) 16,77 16,76

Straightlength Darcycoefficient,3 0,084 –

Pressuredrop,1P(Pa) 16,82 45,53

closetothecharacteristicvalueofastraightchannel.Inthisregime (viscousfluidsapplicationsforinstance),theeffectofthebendson thepressuredropsisquitenegligible.Thetransitionbeginsaround Re=200. Thismay belinkedto theformation of Dean vortices aroundDe=150[16].ThiscriticalvaluecorrespondstoaReynolds numberof130inmock-ups‘b’,‘c’and‘d’and212inmock-up‘e’.

AboveRe=200andasalreadyobservedinFig.7,themainparam- etersarethecorrugationangle(mock-up‘b’vs.mock-up‘c’),the numberofbendsperunitoflength(mock-up‘c’vs.mock-up‘d’) andthecurvatureradius(mock-up‘d’vs.mock-up‘e’).Thesethree parametersareactuallyresponsiblefortheincreaseofthecurved length/straightlengthratiowhichpromotespressuredrops.

Foreachmock-up,thestraightlengthisgiveninTable1whereas thecurvedlengthforonebendisassessedfromthefollowingequa- tion:

Lcurved=×Rc

Fromthepreviousexperimentalresults,thepressuredropof onebendhasbeendeducedforeachgeometry.Itincludesthepres- suredropduetothecurvedlengthitselfandtheoneduetothe propagationofthebendvorticesalongthedownstreamstraight length.Thesimulationresultshighlightthesetwopressuredrop componentsinTable2.

Thepressuredropassessedinabendisexactlythesamegivenby literaturecorrelationorsimulationresults(16.77Pa).Theupstream corrugationshaveactuallynoinfluenceontheconsideredbendin thesesimulationflowconditions.Theflowislaminaratthebend

inlet.Thepressuredropinthestraightlengthhasbeentheoreti- callyassessedconsideringnoupstreambend(3=0.9×64/Re).Itis 3timeslowerthaninthesimulatedstraightlengthdownstream thebend.Thishighlightsthepressuredropcomponentduetothe vorticespropagation.Fig.9illustratesthisphenomenon.

The highest intensity of the counter-rotating vortices is observed at the bend outlet (S1 onFig. 9). Then theintensity decreases all along the channel length before disappearing at S4.S5showsthattheflowbecamelaminaragainwiththeaxial velocitypeakslightlyoff-centredbecauseofthebendcentrifugal force.

Thefollowinggraph(Fig.10)showsthepressuredropperbend foreachgeometry.Ithasbeenassessedbyremovingthetheoretical pressuredropduetothestraightlengthsfromtheglobalpressure drop.Then,thisremainingpressuredrophasbeendividedbythe numberofbends.Asaconsequence,thepressuredropperbend takesintoaccountthepressuredropduetoboththecurvedlength ofthebendandthevorticespropagation.

Thepressuredropperbendishigherinthemock-up‘d’thanin themock-up‘c’whereastheoppositetrendwasobservedinFigs7 and8.Sincethegeometricalparametersofthebends(Rcand)are thesame,thecurvedlengthgeneratesthesamepressuredrop.The differencecomesactuallyfromthestraightlengthalongwhichthe vorticesarepropagated.Thislengthisindeedthreetimeshigher inthemock-up‘d’thaninthemock-up‘c’(20.28mmvs.6.94mm).

Thus,thepressuredropduetothepropagationofthebendvortices increasesthepressuredropperbendofmock-up‘d’.

Fig.9.Velocityprofilesincrosssectionofthecorrugatedchannel(mock-up‘e’).

0 500 1000 1500 2000 2500

0 5 10 15 20 25

ΔP/bend (Pa)

Flowrate (kg.h^-1)

Mock-up b Mock-up c Mock-up d Mock-up e

Fig.10. Influenceofthegeometryandtheflowrateonthepressuredropofone bend(curvedlength+vorticespropagation).

Inthemock-up‘b’,thestraightlengthisequalto6.94mmbut thecorrugationangleishigherthaninthemock-up‘c’(respectively 75^◦and45^◦),thebendpressuredropisthushigher.

Finally,thehighestpressuredropperbendisobservedinmock- up‘e’.Thisisduetothecurvedlengthwhichisactually2.6times longerthantheoneofmock-up‘d’withthesamestraightlength.

Thisistheinfluenceoftheinternalcurvatureradius.

Finally,itcanbeestablishedthatpressuredropsinacorrugated channelaremadeofthreecomponents:thepressuredropinthe straightlengths,thepressuredropinthecurvedlengthsandthe pressuredropduetothepropagationofthevorticesdownstream eachbend.

Thestraightlengthbetweentwobends seemstobethekey parameter.Itsincreasereducesthenumberofbendsperunitof lengthandthusthetotalpressuredrop.However,whenthestraight lengthislowerthanthepropagationlength,itsincreasepromotes thepropagationofthevorticesandthusthepressuredrop.This increasesthepressure dropper benduntilreachinga maximal valuewhenthestraightlengthisequaltothepropagationone.

Thepressuredropalsodependsonthecorrugationangle.The higherthecorrugationangle,thehigherthecurvedlength,thus promotingthepressuredrops(seeEq.(6)).

Finally,thepressuredropprofilesarestronglydependenton thenumberofbendsandontheirgeometry.Thefollowingsection detailstheirinfluenceontheRTD.

3.1.2. Residencetimedistribution

TheRTDhasbeeninvestigatedforeachgeometry(mock-ups‘a’

to‘e’).Thefollowinggraph(Fig.11)illustratestheexperimental results.Thenumberofequivalentstirredtankreactors(NETR)per unitoflengthhasbeenassessedineachmock-up.

Firstofall,whateverthegeometry,theNETRishighenough (NETR>50 formock-upsfrom 3.4to5mlong)toconsiderthat theflowbehaveslikeaquasi-plugflow.Howevertheinfluenceof thewavygeometryonNETRisnotnegligible.Ashighlightedin thepressuredropstudy,themainparametersarealsothenum- berofbendperunitoflengthandtheratiobetweenthecurved lengthandthestraightone.OnFig.11,theNETRofmock-up‘c’is around2timeshigherthantheNETRofmock-up‘d’.Thebends ofthesegeometries arestrictlythesame (curvatureradiusand corrugationangle).Thus,thedifferenceliesonthestraightlength betweentwobendsandasaconsequenceonthenumberofbends perunitoflengthwhichpromotestheplugflowbehaviour.Actu- ally,theflowinstabilitiesgeneratedineachbendpromotetheradial

0 20 40 60 80 100 120 140

300 500 700 900 1100 1300 1500 1700 1900 2100

NETR / L

Reynolds number

Mock-up a Mock-up b Mock-up c Mock-up d Mock-up e

Fig.11.Numberofequivalentstirredtankreactor(NETR)perunitoflengthvs.

Reynoldsnumberforeachmock-upgeometry.

homogenizationoftheconcentrationgradientswhichreducesaxial dispersion.

Similarly,theincreaseoftheratiobetweenthecurvedlength andthestraightone,leadstoanincreaseoftheNETRperunitof length.TheNETRperunitoflengthinmock-up‘b’andmock-up‘e’

arerespectivelyhigherthaninmock-up‘c’andmock-up‘d’.This isduetotheirratiobetweenthecurvedlengthandthestraight onewhichincreaseswiththeincreaseofthebendangle(mock-up

‘b’comparedtomock-up‘c’)orthecurvatureradius(mock-up‘e’

comparedtomock-up‘d’),theothergeometricalparametersbeing keptconstant.

Theseresultsareconfirmedwiththegraphof Fig.12 which showstheinfluenceofthewavygeometryontheNETRperbend.

OnFig.12,theNETRinthemock-ups‘c’and‘d’areveryclose.

ThebendsarestrictlyidenticalandsoistheirinfluenceontheRTD measurements.Thusthisis,indeed,thebendswhichgovernthe RTDandthehigherthenumberofbendsperunitoflength,the narrowertheRTD.

ThedecreaseoftheNETR/bendbetweenmock-ups‘b’and‘c’

orbetweenmock-ups‘e’and‘d’maybeduetotwoeffects.The firstoneisthedecreaseoftheratiobetweencurvedlengthand straightlength.Thesecondonecouldbeduetotheappearanceof axialre-circulatingcellsgenerateddownstreameachbendwhen thecurvatureradiusdecreases[26,31].Thisphenomenoncreates recirculatingzonesdownstreameachbendthatcoulddisturbthe plugflowbehaviour.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

300 500 700 900 1100 1300 1500 1700 1900 2100

NETR / Number of bends

Reynolds number

Mock-up b Mock-up c Mock-up d Mock-up e

Fig.12.InfluenceoftheReynoldsnumberandthegeometryontheRTD.

0.01 0.10 1.00 10.00

100 300 500 700 900 1100 1300 1500

Mixing time (s)

Reynolds number

Mock-up 'a' Mock-up 'b' Mock-up 'c' Mock-up 'd' Mock-up 'e'

Fig.13.Mixingtimesvs.Reynoldsnumberaccordingtothecorrugationgeometry.

However,whateverthegeometry,thenumberofbendsperunit oflengthandtheNETRperbendarehighenoughtoassumeaquasi plugflowwhateverthechanneldesign.

3.1.3. Mixingperformances

Mixingperformanceshavebeenassessedbymeasuringthemix- ingtimesversustheReynoldsnumber.ResultsaregiveninFig.13.

Asexpected,thetrendisasfollows:themixingtimesaredecreasing withtheincreasingReynoldsnumber.

Whatever theReynolds number, mixing timesin themock- up‘a’,whichismadeof90mmstraightlengthsand180^◦turning bends,arehigherthaninthecorrugatedgeometries(mock-ups‘b’

to‘e’).Moreover,theinfluenceofthegeometryisrelevantforlower Reynoldsnumber.AboveRe=700,mixingtimestendtobesimilar whateverthecorrugation(exceptformock-up‘a’).

AtaconstantReynoldsnumber,mixinginmock-up‘e’isless efficientthanintheothercorrugatedmock-upswithagaparound Re=150.Thisresult maybedue totheflowstructure which is differentinthemock-up‘e’.

Intheliterature,thecriticalDeannumber,abovewhichtheDean vorticesappear,isestimatedbetween100and250[16].Thesevor- ticespromotethehomogenizationofthetemperature,thevelocity butalsotheconcentrationgradients.Thusitcontributestoimprove themixingperformancesofthecorrugatedflow.

Inourexperiments,thegeometricalparameterwhichisdiffer- entbetweenmock-up‘d’andmock-up‘e’isthecurvatureradius (respectively1.5and4mm).Asaconsequence,whentheReynolds numberequals150,theDeannumberequals160inthemock-up

‘d’and100inthemock-up‘e’.ThusitseemsthatthecriticalDean numberhasbeenreachedinmock-up‘d’(andalsoinmock-ups

‘b’and‘c’)whichisnotthecaseformock-up‘e’.TheFig.14illus- tratesthisresultbyconsideringthemixingtimesversustheDean number.

Theseresultsconfirmtheformerhypothesissinceforaconstant Deannumber themixingperformances areequal whateverthe geometry.Thecurvatureradiuswillbethusanimportantparame- tertobeconsideredformixingapplicationsandparticularlywith viscousfluidsforwhichtheReynoldsnumbersarelow.

3.1.4. Thermalperformances

From temperature measurements, the influence of wavy geometriesonthethermalperformancesisstudied.Inthatway,an intensificationfactorisintroduced.Ittakesintoaccounttheglobal

0.01 0.10 1.00 10.00

0 200 400 600 800 1000 1200 1400 1600 1800

Mixing time (s)

Dean number

Mock-up 'b' Mock-up 'c' Mock-up 'd' Mock-up 'e'

Fig.14.Mixingtimesvs.Deannumberaccordingtothecorrugationgeometry.

heattransfercoefficient,U(Wm⁻²K⁻¹)butalsothegeometrycom- pactness.Itisdefinedas:

U×A

V (Wm⁻³K⁻¹)

withAtheheatexchangearea(m²)andVthevolumeoffluid(m³).

ThegraphofFig.15showstheintensificationfactorversustheflow rateforeachgeometry.

First,asobtainedduringthepressuredropandtheRTDstudies, theinfluenceofthenumberofbendsperunitoflengthisrele- vant.Theintensificationfactoralmostreachesaratioof2between geometry‘c’(106bends/m)and‘d’(44bends/m).

UnlikeRTDandpressuredropmeasurements,theinfluenceof thecorrugationangleseemsheretobenegligiblesincethecurves ofgeometries‘c’(=45^◦)and‘b’(=75^◦)aresuperposed.Finally, anincreaseofthecurvatureradiusfrom1.5(geometry‘d’)to4mm (geometry‘e’)leadstoadecreaseoftheintensificationfactor.Two effectsareinvolved;thedecreaseofthenumberofbendsperunit oflength(from44to38bends/m)andthedecreaseoftheDean number(foraconstantflowrate)whichleadstoadecreaseofre- circulatingcellsintensity.ThisphenomenonisillustratedinFig.16.

The effects of both thenumber of bends perunit oflength (mock-up‘c’vs.mock-up‘d’)andthecorrugationangle(mock-up

‘b’vs.mock-up‘c’)areobviouslyunchanged.However,asobserved formixingcharacterizations,theDeannumberisstilltherelevant dimensionlessnumberwhenthecurvatureradiusisconsidered.

Indeed, similar profiles are obtained for both geometries ‘d’

(R_c=1.5mm)and‘e’(R_c=4mm).

0 1000 2000 3000 4000 5000

0 2 4 6 8 10 12 14 16 18

UA/V (kW.m-3.K-1)

Process flow rate (kg.h^-1)

Mock-up 'b' Mock-up 'c' Mock-up 'd' Mock-up 'e'

Fig.15.Intensificationfactorvs.flowrateaccordingtothewavygeometry.

0 5 10 15 20 25 30 35 40 45 50

0 1000 2000 3000 4000 5000 6000 7000 8000

Nusselt number

Dean number

Mock-up 'b' Mock-up 'c' Mock-up 'd' Mock-up 'e'

Fig.16.Nusseltnumbervs.Deannumberaccordingtothecorrugationgeometries.

Theresultsofthethermalcharacterizationscompletethepre- viousexperimental results. The Dean number seemsto bethe relevantparameterandthentheinfluentgeometricalparameter isthenumberofbendsperunitoflength.

3.1.5. Conclusionoftheparametricstudy

Theinfluenceofthegeometricalparametersofawavychannel onheattransfer,pressuredrops,mixingperformancesandresi- dencetimedistributionhasbeeninvestigatedinthefirstpartof thiswork.Themaingoalwastostudyandtounderstandboththe thermo-hydraulicmechanismsandtheflowbehaviourinthecor- rugatedchannelofaplateheatexchanger/reactor.Thetrendsare obviouslydifferentandthemostefficientgeometrywilldependon thechosenperformancecriteria.Actually,themostefficientmock- upsintermsofheatandmasstransfersand/ormomentumoften generatemanyflow instabilities.Asaconsequenceperformance criteriahavetobecomparedtoeachothertofindtheoptimalgeom- etry.Itwilldependonboththeend-usingspecificationsandthe aimedversatilityoftheheatexchanger/reactor.Forinstance,itis interestingtoplotthethermalperformancesvsthepumpingpower asillustratedonFig.17.

Thisgraphshowsthatforafixedpumpingpower,themock-up

‘c’isthemostefficientconcerningtheheattransfercriterion.The mock-up‘b’generatestoohighpressuredropsforanequivalent intensificationfactor(seeFig.15).

Moreover,althoughpressuredropsdecreasebetweenthemock- ups‘c’and‘d’(becauseofthedecreaseofthebendnumberper

0 1000 2000 3000 4000 5000 6000

0.00 0.05 0.10 0.15 0.20 0.25 0.30

UA/V (kW.m-3.K-1)

F.ΔP/L (W.m^-1)

Mock-up 'b' Mock-up 'c' Mock-up 'd' Mock-up 'e'

Fig.17. Intensificationfactorvs.pumpingpowerperunitofcorrugatedchannel length.

0 50 100 150 200 250 300 350 400 450 500

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Number of equivalent stirred tank reactors

Flow velocity (m.s^-1)

mock-up 'e'-2mm square cross section mock-up 'f'-rectangular cross section mock-up 'g'-4mm square cross section Straight duct

Fig.18.InfluenceoftheflowvelocityandthechannelsizeontheRTD.

unitoflengthfrom106to44bends/m),itdoesnotbalancethe decreaseofheattransfercoefficient.SoasillustratedinFig.17,the mock-up‘d’islessefficientthanthemock-up‘c’ifweconsiderthe intensificationfactorvs.thepumpingpower.

However,manyothercriteria,likecompactness,manufactur- ingcosts,...canbeconsidered.In thatcase, therankingof the corrugatedgeometriesmaychange.

The firstpartof this workshows that theinfluenceof each geometricalparameterdepends ontheconsideredperformance criterion. Thatis thereasonwhyanoptimizationregarding the specificationsandthetechnologyuseisrequired.Oncetheopti- mumgeometryisdefinedandtoexpectafutureindustrialization, the performance prediction during the scale-up procedure is required.Thefirststepstowardsthescale-upofcorrugatedchannel aredescribedinthefollowingpartofthispaper.

3.2. Influenceofboththehydraulicdiameterandtheaspectratio Threemock-ups(‘e’,‘f’and‘g’)areconsideredinthissection.The channelsofmock-ups‘e’and‘g’arerespectively2mmand4mm square-crosssection.Themock-up‘f’ismadeofarectangularcross sectionchannelof2mmlargeand4mmin-depth.

3.2.1. Residencetimedistribution

TheRTDhasbeenexperimentallyinvestigatedforthosethree geometries.Thefollowinggraphillustratesthenumberofequiv- alentstirredtankreactors(NETR)versustheflowvelocityineach geometry.TheRTDmeasurementshavealsobeenimplementedin arectangularcross-section(2×4mm²)straightchannel.

For each corrugated geometry, the NETR is high enough (NETR>150)toconsiderthattheflowbehaveslikeaplugflowin thecorrugatedchannelswhateverthecharacteristicsize.Thisisnot trueintherectangularcrosssectionstraightduct,sincetheNETR islessthan50.

During thescale-upprocedure, theresidence timehastobe kept constant to complete the chemistry. Since the developed lengthofthethreemock-upsisthesame,thecomparisonatacon- stantflowvelocityisofinterest.ThegraphofFig.18showsthat thehigherthehydraulicdiameter,thenarrowertheRTD.Thisis actuallytheReynoldsnumberinfluencewhichincreaseswiththe hydraulicdiameter.Thus,iftheflowvelocityisequalto0.3ms⁻¹, theReynoldsnumberisrespectivelyof600,800and1200inthe mock-ups‘e’,‘f’and‘g’.ThefollowinggraphrepresentstheNETR versustheReynoldsnumber.

ThecomparisonbetweenFigs.18and19highlightstheinterest of considering the Reynolds number during the scale-up pro- cess.Howeverthereisstilladifferencebetweeneachgeometry,

0 50 100 150 200 250 300 350 400 450 500

500 700 900 1100 1300 1500 1700 1900 2100 2300

Number of equivalent stirred tank reactors

Reynolds number

mock-up 'e' -2mm square cross section mock-up 'f' -rectangular cross section mock-up 'g' -4mm square cross section

Fig.19.NETRvs.Reynoldsnumber.

especiallyaboveRe=1000.ThiscouldbeduetotheinternalDean numberinfluencewhichtendstoincreasewiththeincreaseofthe channelsize(becauseofboththeincreaseofthehydraulicdiam- eterandthedecreaseoftheinternalcurvatureradius).Therefore, theNETRisplottedversustheDeannumberinthefollowinggraph (Fig.20).

Thecharacteristiccurvesareverycloseeachother,highlighting therelevanceofconsideringtheinternalDeannumber.Actually, thescale-upprocessdoesnotpenalizetheplugflowcharacteris- ticswhichareevenimprovediftheflowvelocityortheReynolds numberarekeptconstant.Theperformancesaremaintainedwith theinternalDeannumberwhichisaninterestingresulttopredict theflowbehaviourduringthescale-upstep.

3.2.2. Mixingperformances

Themixingperformanceshavebeenassessedbymeasuringthe mixingtimesversustheReynoldsnumber.Theresultsaregivenin Fig.21.

Themixingtimesarequicklydecreasingwiththeincreaseof theReynoldsnumber.Themock-up‘g’ismoreeffectiveintermsof masstransferthantheotherones,thiscouldbeduetotheDean numberasobserved duringtheRTDexperiments. Ontheother hand,themixingtimesofmock-ups‘e’and‘f’arequiteclose.The internalcurvatureradiusesofthesetwogeometriesareactually similar(3mm).AsaconsequencewhentheReynoldsnumberis

0 50 100 150 200 250 300 350 400 450 500

500 700 900 1100 1300 1500 1700 1900 2100 2300 2500

Number of equivalent stirred tank reactors

Internal Dean number

mock-up 'e' - 2mm square cross section mock-up 'f' -rectangular cross section mock-up 'g' -4mm square cross section

Fig.20.NETRvs.internalDeannumber.

0 0 1 10 100

0 200 400 600 800 1000 1200 1400 1600

Mixing time (s)

Reynolds number

mock-up 'e' -2mm square cross section mock-up 'f' -rectangular cross section 'mock-up 'g' -4mm square cross section

Fig.21.Mixingtimesvs.Reynoldsnumberaccordingtothechanneldesign.

equal,theratiobetweentheDeannumbersofbothmock-ups‘e’

and‘f’isequalto:

De^′e^′

De^′f^′

= Re^′e^′

Re^′f^′

×

s

dh′f′

=1×

r

2.67 =0.9 (11)

Thisratioiscloseto1andthatcouldbethereasonwhythecharac- teristiccurvesareveryclose.Ontheotherhand,theDeannumber ratiobetweenmock-ups‘e’and‘g’isequalto:

De^′e^′

De^′g^′

= Re^′e^′

Re^′g^′

×

s

dh′g′/Rc′g′

=1×

r

4/2=0.6 (12)

Deannumbersinmock-ups‘e’and‘g’arequitedifferentandthe flowmightnothavethesamebehaviour.Thisiswhatisobserved whenplottingthemixingtimesversustheDeannumberinFig.22.

Asobserved duringthe RTDexperiments,the Dean number isstill therelevant parametertopredicttheperformancesdur- ingthescale-upprocedurefrom2to4mm.However,whatever theconsideredscale-upparameter(velocity,Reynoldsnumberor Deannumber),themixingperformancesdonotdecreaseduringthe scale-upprocessandareevenimproved.Asaconclusion,bothRTD andmasstransferarenotlimitingprocessesduringthescale-up herefrom2to4mm.

0 0 1 10 100

0 200 400 600 800 1000 1200

Mixing time (s)

'Internal' Dean number

mock-up 'e'-2mm square cross section mock-up 'f'-rectangular cross section mock-up 'g'-4mm square cross section

Fig.22.Mixingtimesvs.Deannumberaccordingtothechannelcharacteristicsize.

0.01 0.1

1 10

10 100 1000 10000

Darcy coefficient

Reynolds number

mock-up 'e'-2mm square cross section mock-up 'f'-rectangular cross section mock-up 'g'-4mm square cross section rectangular straight duct

Fig.23. DarcycoefficientvsReynoldsnumber,experimentswithwaterandglycerol (67w%)at25^◦C.

3.2.3. Pressuredrops

TheDarcycoefficient,whichischaracteristicofthesizeandof thegeometryofthechannel,hasbeenplottedagainsttheReynolds numberinFig.23.

Whateverthegeometry,thegraphisdividedintotwozones apartRe=200. Belowthis valuetheDarcy coefficientsareclose tothevaluetypicalofastraightchannel.Inthisregime(viscous fluidsapplicationsforinstance),theeffectofthebendsonthepres- suredropsisquitenegligible.However,itcanbenoticedthatthe Darcycoefficientsofthetwosquare crosssectionmock-upsare eachothercloserthanwiththerectangularcrosssectionmock- up.Asaconsequence,whentheaspectratioiskeptconstant,the DarcycoefficientcanbecorrelatedtotheReynoldsnumberinthe verylaminarregime(Re<200):

=21.3·Re^−0.78 (13)

Boththerectangularcorrugatedchanneland therectangular straight oneleadtohigher pressure dropsin thelow Reynolds regime(Re<200).Sincethecorrugationshavefeweffectsonthe pressuredrop,thedifferencebetweensquareandrectangularcross sectionsmaybeduetotheaspectratioeffectwhichhasbeenstud- iedbyShah&London[32]andisillustratedinFig.24.

Shah&London[32]showedthatthefrictionfactorincreases whentheaspectratiodecreases.ThisiswhatisobservedinFig.23 belowRe=200. BeyondRe=200, thechannel characteristicsize beginstoinfluencetheDarcycoefficients.ThismaybetheDean numberinfluenceasillustratedonthegraphofFig.25.

Consideringthesquarecrosssectioncorrugatedchannels,the Darcycoefficientcurvesofbothgeometries‘e’and‘g’areveryclose eachotheraboveRe=200.InthisregimetheDeannumberisagain therelevantdimensionlessnumbergoverningtheflow.Itallows thepredictionofpressuredropsduringthescale-upprocedureand thefollowingcorrelationcanbeusedforcorrugatedmillichannels:

=3.68×De_i^−0.38 (14)

Theinfluence of theaspectratio issignificant whateverthe flowregime.BelowRe=200,therectangularcrosssectiontendsto promotetheDarcycoefficientwhereasbeyondRe=200theDarcy coefficientintherectangularcrosssectionchannelislowerthanin thesquarecrosssectionchannel.Theinfluenceofthisparameter shouldbefurtherstudied.

3.2.4. Thermalperformances

The graph on Fig. 26 plots the Nusselt number versus the Reynoldsnumberobtainedineachcorrugatedgeometry.

Thisgraphhighlightstworesults.Thefirstoneistheimprove- ment of the thermal performances with the increase of the

Fig.24.Frictionfactor(equivalenttotheDarcycoefficient)vsaspectratioina laminarregime[31].

hydraulicdiameter(from2to4mm).ThismaybeduetotheDean numberinfluence.Thesecondresultistheinfluenceoftheaspect ratio.Thecrosssectionelongationfrom1to0.5seemstoimprove thethermalperformances.Thisisnotthetendencyobservedby Shah&London[32]inastraightrectangularduct(case4onFig.27).

However,unlikeShah&London[32],thepresentworkconcerns corrugatedchannelandtheaspectratioseemstohaveaninfluence ontheradialrecirculationloops.Indeed,inarectangularcrosssec- tionchannel,inadditiontothecounterrotatingvorticesclosetothe wall,additionalvorticesappearinthecrosssectionalongtheouter wall(Fellouahetal.,[16]).ThisisillustratedintheFig.28where velocityprofileshavebeenplottedupstreamabendinmock-ups

‘e’,‘f’and‘g’(Re=400).

GivenaReynoldsnumber,theadditionalvorticesobservedin therectangularcrosssectionpromotethehomogenizationoftem- peraturegradient.Thebetterheatperformancesobservedinthe mock-up ‘f’ maybe due tothis phenomenon. However further developmentsare required,particularlytodissociatetheaspect

0.01 0.1 1 10

10 100 1000 10000

Darcy coefficient

Internal Dean number

mock-up 'e' - 2mm square cross section mock-up 'f' - rectangular cross section mock-up 'g' - 4mm square cross section

Fig.25.Darcycoefficientvs.Deannumber,experimentswithwaterandglycerol (67w%)at25^◦C.

0 5 10 15 20 25 30

0 1000 2000 3000 4000 5000 6000

Nusselt number

Reynolds number

mock-up 'e' - 2mm square cross section mock-up 'f' - rectangular cross section mock-up 'g' - 4mm square cross section

Fig.26.NusseltnumbervsReynoldsnumberaccordingtothecorrugationgeometries.

Fig.27.Nusseltnumber(constantwalltemperature)vsaspectratioinalaminar regimeandstraightduct.Mock-up‘f’configurationiscomparedwithcase4with

˛*=2b/2a=0,5(Shah&London,1978).

ratioinfluencefromtheoneoftheratiobetweencurvatureradius andhydraulicdiameter.Thesebothparametersareindeedvarying inthemock-up‘f’comparedtomock-ups‘e’and‘g’.

TheNusseltnumbervstheDeannumberhasbeenplottedfor thetwosquarecrosssectiongeometries(seeFig.29).

GivenaconstantDeannumbertheheattransferperformances areveryclosewhatever thechannel hydraulicdiameter.Unlike straight tubecorrelations ofNusseltnumber vs Reynolds num- ber,theheattransferperformancesofacorrugatedchannelseem todependontheDeannumber.Thisnoticeableresultshowsthat aslongastheDeannumberiskeptconstant,thescale-upproce- dure(herefrom2to4mmhydraulicdiameter)hasnoeffectonthe Nusseltnumber.

AsaconsequenceaNusseltnumbercorrelationhasbeenestab- lished:

Nu=0.45×De^0.43_i ×Pr^0.33 (15)

Thiscorrelationisthefirstbrickforthepredictionofheattransfer performances during the scale-up process of corrugated milli- channels.

Fig.28.Velocityprofilesatthebendoutletinmock-ups‘e’,‘f’and‘g’(Re=400).