5118Ind.Eng.Chem.Res.,Vol.46,No.15,2007

Table7.ResultsoftheApplicationoftheMearsCriterionforthePresenceofExternalDiffusiona

runnumber

ksfromcorrelation

resultofMearscriterion

R290.003601.263R300.003610.730R310.003571.457R320.003601.602R330.003610.709R340.003750.842R350.003570.245R360.003580.112R370.003580.112R380.004900.113R390.003560.694R400.004150.326R410.004180.322R420.003850.428R430.003590.538R440.003610.585R450.003580.518R460.003580.621R470.003570.657R480.003580.732R490.003380.727R500.003630.606R510.003890.503R52

0.00512

0.210

a

Otherparametersforthecalculation: B)0.21;σ)0.8;τB)3;andRp)0.045cm.

reportedinthereviewofSeguinetal.12withoutsignificantdifferencesintheresults.

Intheexpression7anaveragediffusioncoefficientforthemixture(embeddedintotheSherwood’snumber)hasbeenevaluatedbymeansoftherelation4.Byvaryingtheconversionfromzerototheoutletexperimentalvalueacompositionprofilealongthereactorhasbeengeneratedand,incorrespondencewitheachcomposition,adiffusioncoefficientofoleicacidDA,Mixhasbeenevaluatedandaveraged.Thediffusioncoef-ficientinthemixturehasthenbeencorrectedbytakingintoaccounttheporosity,tortuosity,andconstrictionfactorrelatedtothewholebedasmadepreviouslyforthesingleparticle(seeeq3):

Deff,B)

DA,Mix Bσ

τ(8)

B

Thisapproximationcanbejustifiedbyconsideringthereductionindiffusivityoccasionedbythenondirectflowpathcharacteristicofthefixedbed,10particularlyinthecaseoflow-porositybeds(inourcaseexperimentallyevaluatedandresultingin B)0.21),probablyoriginatedfromaratherwideparticlesizedistributionandatverylowReynoldsnumbers12inconditionsofcreepingflowregimeofviscousfluid.Theunusuallylowbedporositymeasuredcanprobablybeduetothespatialcollocationofsmallresinparticlesinsidethespringsusedascatalystdiluentresultinginalow-voidagebedstructure.Despitethisbedcharacteristic,thefluidflowwasnothinderedandthereactorhasbeenoperatedforseveralweeks,includingstart-upandshut-down,withoutpressuredropincrease.

Theeffectivediffusivityapproach,relatedtothewholebed,canbefurtherjustifiedbyconsideringtheverythinchannelsspacingavailabletofluidflowacrossthebedandtotherelativelyhighviscosityandlowlinearvelocities.Intheseconditionsthethicknessofstagnantfilmsurroundingtheparticles,throughwhichdiffusionalresistancesareoperative,isprobablyofthesameorderofmagnitudeofthechannelsinwhichthecreepingflowregimeisestablished.InTable7arereportedtheresultsoftheapplicationoftheMearscriterion,forwhichalimitvalueforneglectingtheinterventionoffluid-solidmass-transferlimitationis0.15,togetherwiththevaluesofthekscoefficientasobtainedfromCoeuretcorrelation(eq7).Aswecanobservefromthementionedtable,theinfluenceofthistypeofresistancecannotbeexcludedandmustbeproperlytakenintoaccountinthereactorsimulationmodelforthedescriptionofourexperimentaldata.

ModelConsideringtheExternalDiffusionLimitation.Withtheaimofintroducingthefluid-solidmass-transferresistance,eq5mustbemodifiedasfollows:8

-U

dCi

dz

+JS,iaS)0i)A,M,E,W(9)

whereUisthefluidlinearvelocity,zistheaxialreactorcoordinate,Ciisthebulkconcentrationofcomponenti,aSisthespecificsurfaceareaofcatalystperunitofbedvolume,andJS,iisthemass-transferrate.Thislastvariablerepresentstherateatwhicheachcomponentistransferredfrombulkliquidtothesolidsurfaceor,reversely,fromthecatalystsurfacetothebulkofliquid.AsuitableexpressionforJS,iisthefollowing:

-JS,i)ksi(Ci-CSi)

(10)

Inthisexpressionksiisthemass-transfercoefficientreferencedtoliquid-soliddiffusionaltransportresistanceandCiSistheconcentrationofcomponentievaluatedattheliquid-solidinterface.Accordingtotheseassumptionsrelation10canbesubstitutedintoeq9toyieldanordinarydifferentialequation(ODE)system:

-U

dCi

dz

)ksiaS(Ci-CSi)i)A,M,E,W(11)

Byassumingthatthepseudo-steady-stateconditionisfulfilledatthecatalyst-fluidinterface,theamountofeachcomponenttransferredbydiffusionmustbeequaltotheamountconsumedorformedbychemicalreaction.Inotherwords,thesurfaceaccumulationtermforeachcomponentisassumedasnegligibleandthefollowingexpressionsetcanbewritten:

ksiaS(Ci-CSi)-rFcat)0

i)A,M,E,W(12)

whereFcatisthecatalystconcentrationinthebed,expressedascatalystweightperunitofbedvolume.Theequationsinrelation12mustbesolvedsimultaneouslyateachintegrationstepintimeofODEsystem11togivethevaluesofsurfaceconcentra-tions,forallthecomponents.ThereactionrateexpressionisalsoreportedinTable6but,inthiscase,theconcentrationsaretakenatthecatalyst-liquidinterface.Moreover,fromageneralpointofview,eachcomponentischaracterizedbyitsownmass-transfercoefficientksi.

Themass-transfer-limitedreactormodel,representedbythedifferential-algebraicsystem(eqs11and12),hasbeenappliedtothedescriptionofthethreeexperimentaldatasetsrelatedtotheusedtubularreactors.Themodelhasbeensolvediteratively,andtheexternalmass-transfercoefficienthasbeenconsideredasanadjustableparameterthathasbeenvaried,ineachrun,toreproduceexactlytheexperimentalconversion.Inthisapproachtheksivaluescanbeconsideredasexperimentalvaluesforthemass-transfercoefficientwiththemainassumptionthattheentireeffectoflowerperformances,withrespecttoapurelykineticmodel,isattributedtotheexternalmass-transferresistance.Intherealsituationothereffectsareoperativeandsuperimposed