www.elsevier.com/locate/jfoodeng
Choosinganappropriatedryingmodelforintermittent
andcontinuousdryingofbananas
R.Baini*,T.A.G.Langrish
DepartmentofChemicalEngineering,TheUniversityofSydney,NSW2006,Australia
Received24September2005;accepted27January2006
Availableonline5April2006
Abstract
Empiricalmodels,thediffusionmodelandtheconceptofacharacteristicdryingcurvehavebeenusedintheliteraturetodescribethedryingofbananas.Theiruseiscomparedhereforcontinuousandintermittentdrying,asinsolarkilns.Experimentswerecarriedoutinapilot-scalekilnwiththewetbulbtemperaturesetat40°Candanairvelocityof1.3msÀ1.Bananasweredriedcontinuouslyatdry-bulbtemperaturesof60and80°Candintermittentlyat60°C.Theaveragemoisturecontentsofthedriedbananaswerecomparedwiththosepredictedbyanempirical(Newton)modelandtheFickiandiffusionmodel.Aleastsquaresprocedurewasusedinfitting,inwhichthestandarderrorforthedifferencebetweenthemoisturecontentsfromtheexperimentaldataandeachofthedryingmodelswasminimized.Thefitteddryingconstantswerefoundtobe0.11hÀ1and0.09hÀ1forcontinuousandintermittentdrying,respectively.Thefittedacti-vationenergyandtheestimateddiffusioncoefficientwere13.4kJ/moland1.09·10À9m2/s,respectively.Thediffusionmodelwasfoundtofitthemoisturecontentsbetterthantheempiricalmodel.Thiswasshownbythestandarderrorsobtained,whichwere50%and88%lowerthanthoseobtainedinthefittingoftheempiricalmodelforcontinuousandintermittentdrying,respectively.Theanalysisofthedryingratesusingtheconceptofacharacteristicdryingcurveshowedthatthisconceptonlygaveareasonablefittingofmoisturecon-tentsforcontinuousdrying.Itwasalsofoundthatthesugarcontentofthebananasatdifferentdegreesofripenesshadaninsignificanteffectonthecharacteristicdryingcurve.Overall,theresultsofthisworksuggestthattheempiricalmodelandthecharacteristicdryingcurveareapplicablefordescribingthedryingkineticsofbananadriedcontinuouslybutnotintermittently.Ontheotherhand,thedif-fusionmodel,whichincludesthevariationofmoisturecontentandtemperaturethroughoutthebananainitssolution,describesthedry-ingkineticsofbananawellforbothcontinuousandintermittentdrying.Thismakesthediffusionmodelsuitableforpredictingtherelaxationprocessesthatoccurwhenthedryingconditionsareinterrupted,suchasinintermittentdrying.Ó2006ElsevierLtd.Allrightsreserved.
Keywords:Bananadryingkinetics;Continuousdrying;Intermittentdrying;Relaxationmoisturecontent
1.Introduction
Theworldproductionofbananasisincreasingalmosteveryyear(FAO,2005).Itreachedapproximately100mil-liontonnesin2004.Thissupplyofbananasandplantainsmakesthemthelargestquantityoffruitproducedintheworld(FAO,2005).However,thehighwatercontentofbananasmakesthemsusceptibletomouldgrowth.India,
Correspondingauthor.
E-mailaddresses:rbaini@chem.eng.usyd.edu.au(R.Baini),timl@chem.eng.usyd.edu.au(T.A.G.Langrish).
0260-8774/$-seefrontmatterÓ2006ElsevierLtd.Allrightsreserved.doi:10.1016/j.jfoodeng.2006.01.068
*theworld’slargestbananaexporter,reportedpostharvestlossesofbananafruitsashighas35–45%,whereasBrazil,theworld’ssecondproducerofbananas,reportedaprod-uctlossofapproximately40%(Cano-Chauca,Ramos,Stringheta,&Pereira,2004).
Preservationofbananasmayreducetheselossesandincreasetheavailabilityofworldfoodsupply.Dryingasamethodoffoodpreservationisonewaytoprocessagri-culturalproductsafterharvesting.Theapplicationofappropriatedryingtechniquesisdesirableinordertopro-ducegood-qualitydriedproducts.Thedryingtechniquehasbeenobservedtohaveanimpactonthedryingkinetics
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Nomenclature
A,Bshrinkageparametersforbanana(–)a,b,c,dparametersoftheDCCmodel(–)awrelativehumidity(%or–)
C,C0,K,K0constantsforGABcorrelation(–)Cpheatcapacity(JkgÀ1KÀ1)Ddiffusioncoefficient(m2sÀ1)DEactivationenergy(K)Drpre-exponentialfactor(m2sÀ1)dpdiameter(m)hFheat-transfercoefficient(WmÀ2KÀ1)Jmassfluxofmoisturefromsurface(kgmÀ2sÀ1)k0constantinEq.(5)(hÀ1)k1,k2,k3,k4constantsinEq.(5)(–)kddryingconstant(hÀ1)kmmass-transfercoefficient(msÀ1)ktcthermalconductivity(WmÀ1KÀ1)Llength(m)MRmoistureratio(–)Ndryingrate(kgmÀ2sÀ1)Nvunhindereddryingrate(kgmÀ2sÀ1)Qnetheatfluxtobanana(WmÀ2)Rgasconstant(8.314JmolÀ1KÀ1)RDRrelativedryingrate(–)rradius(m)ttime(s)Ttemperature(°C)Tgdrybulbtemperature(°C)Twwetbulbtemperature(°C)
u
VVsVwXXcrXeXmX0YGreeksbnkskwqbqairDHcDHk/
velocity(msÀ1)
volumeofthebanana(m3)volumeofthesolid(m3)volumeofthemoisture(m3)
moisturecontent(kgperkgdrybasis)
criticalmoisturecontent(kgperkgdrybasis)equilibriummoisturecontent(kgperkgdrybasis)
monolayermoisturecontent(kgperkgdrybasis)
initialmoisturecontent(kgperkgdrybasis)humidity(kgperkgdryair)
constantinEq.(4)(–)heatofsorption(JkgÀ1)
latentheatofvaporization(JkgÀ1)bananadensity(kgmÀ3)airdensity(kgmÀ3)
heatofcondensationofwatervapor(JmolÀ1)heatofcondensationofwatervapor(JmolÀ1)characteristicmoisturecontent(–)
SubscriptsiatnodepointIGbulkairstreamsatthesurfacecatthecentre
andthequalityofthedriedproducts.WorkbyChua,
Mujumdar,Hawlader,Chou,andHo(2001)showedthatvaryingthedryingtemperaturereducedthecolourdegra-dationofthebananaslicesaswellasthedryingtime.Tem-peraturestepsof5°Cwereappliedintheirwork,inwhichexperimentswererunfora6-hperiod.Twodryingapproacheswereusedandweredescribedasstep-upandstep-downtechniques.Thetemperaturewasincreasedby5°Ceveryhourfrom20to35°Cforthestep-uptechniqueandwasdecreasedby5°Ceveryhourfrom35to20°Cforthestep-downtechnique.Resultsobtainedfromthesetech-niqueswerecomparedwiththoseobtainedfromcontinu-ousdryingat25°C.Theabsolutehumiditywasmaintainedintherangeof0.007–0.010kgperkgdryairforalltests.Theirresultsshowedthattherequiredtimetodrybananastoamoisturecontentofapproximately2.0kgperkgbasisusingthestep-downdryingtechniquewas300%shorterthanthatusingthecontinuousdryingtechnique.Bycontrast,thestep-updryingtechniquerequiredapproximately25%extratimetodrythebananastoamoisturecontentofapproximately2.1kgperkgdrybasiscomparedwiththatrequiredbythecontinuousdry-ingtechnique.Forthestep-downtechnique,thelargehumiditygradientbetweenthebulkairandatthesurfaceofthebananaincreasedthedryingrateduringtheearlystageofdrying(thebananawasmoist).Therapidmoisturelossthatoccurredduringtheearlystageofdryingreducedthemoisturecontentofthedriedbananasignificantly.Therefore,lesstimewasrequiredtodrythebananausingthestep-downtechniquethanthatcomparedwiththestep-uptechnique.
Thecolourchangeofthedriedbananaswasestimatedbysubtractingthecolourofthedriedbananasfromthecolourofthefreshbananas.Thenetchangeofcolourwasestimatedasthesquarerootofthesumforthesquaredcolourchange.Thenetchangesofcolourwerefoundtobe40%and23%forthestep-upandstep-downtechniques,respectively,lowerthanthatobtainedinthecontinu-ousdryingtechnique.ItwasalsoreportedbyChua,Mujumdar,andChou(2003),intermittentdryingwasfoundtobebeneficialformaterialsthatdriedprimarilyinthefallingrateperiodbecauseitproducedbetterqualityproducts,comparedwiththosedriedusingconventionalcontinuousdrying,inthesensethatthelevelsofascorbicacidandb-carotenecontentinthedriedfruitsfrominter-mittentdryingwerefoundtobesignificantlyimproved.
332R.Baini,T.A.G.Langrish/JournalofFoodEngineering79(2007)330–343
Theinterruptionsofdryingconditions,suchasinthetemperature-steppingandintermittentdryingtechniques,mayinterruptmoisturelossfrommaterials,whichwillgivedifferentdryingprofilesfordifferentdryingtechniques(Chua,Hawlader,Chou,&Ho,2002).Italsomeansthatselectinganappropriatedryingmodeltoevaluatethedry-ingbehaviorisnecessarytogetgoodpredictionofthedry-ingkinetics.Dryingkineticsofbananascanbemodeledempiricallyusingfirst-orderreactionkinetics,andmecha-nisticallyusingFicks’Lawofdiffusion.Empiricalmodelsincludelumpedparametermodelsthatgenerallypredictonlytheaveragemoisturecontentsofbananasasafunc-tionofdryingtime.Ontheotherhand,onetypeofmech-anisticmodelistheFickiandiffusionone,includingthediffusioncoefficient.Thistypeofmodelpredictsthemois-turecontentthatvarieswithtimeandspacewithinthebananas.Themodeldescribesthemoisturetransportphys-icallywithinthebananas,givingresultsofmorefundamen-talvaluecomparedwithempiricalmodels.However,empiricalmodelsaresimplercomparedwiththediffusionmodeland,becauseofthis,theyaresometimewellacceptedandappliedinthestudyoffruitdrying.Dryingbananasmayalsobeanalyzedusingtheconceptofachar-acteristicdryingcurve.Thisconceptassumesthat,duringhindereddrying,theshapeofthefalling-ratedryingcurveisuniqueforeachmaterial.
Theapplicationofthesemodelsmaybelimitedtocer-taindryingconditions.Thethreedifferenttypesofmodelshavebeenappliedinthiswork.Theirapplicationsindescribingthedryingkineticsofbananas,driedcontinu-ouslyandintermittently,havebeendiscussed.Duringintermittentdrying,theheatsupplyandairflowinsidethekilnareoperatedinacyclicfashion.Suchanoperationmethodchangesthedrybulbtemperatureandthehumidityoftheair.Whentheheatsupplyisinterruptedduringinter-mittentdrying,thedry-bulbtemperaturedecreases,reduc-ingtheexternaldryingrate.However,eveninthisperiodofaninterruptedheatsupply,internalmovementofmoisturemayoccurduetodiffusionorothertransportprocesses.Itcausesarelaxationprocesstooccurintheinternalprofileofthemoisturecontent,anditincreasestheinitialrateof
dryingwhentheheatsupplyisresumed.Thediffusionmodel,whichpredictsthevariationofmoisturecontentswithtimeandspace,takesthisrelaxationofmoisturecon-tentintoaccount.Therefore,thisphenomenonmaybedescribedbetterbythediffusionmodelcomparedwithempiricalmodelsandtheconceptofacharacteristicdryingcurve.
1.1.Empiricalmodel
Thedryingkineticsofbananashasbeenempiricallypre-dictedusingafirstorderapproachdescribedbyEq.(1).Dryingrate,À
dX
¼kdðXÀXeÞdt
ð1Þ
Thismodelhasbeenfoundtofittheaveragemoisturecon-tentofbananasthatweredriedcontinuouslyverywell(Baini&Langrish,2004;Hofsetz&Lopes,2004;Kiranou-dis,Tsami,Maroulis,&Marinos-Kouris,1997;Maskan,2000;Sousa&Marsaioli,2004;Phoungchandang&Woods,2000).
ThismodelwasfirstsuggestedbyLewis(1921).Here,thedryingrateisassumedtobeproportionaltothemois-turecontentdifference.kdisadryingconstant.Theequa-tionmaybeintegratedtogivethefollowingones:MR¼
XÀXe
¼expðÀkdtÞ
X0ÀXe
X¼ðX0ÀXeÞexpðÀkdtÞþXe
ð2Þð3Þ
Theseequationsexpressthefirst-orderkineticsintermsofthemoistureratio(Eq.(2))togivethemoisturecontent,X,(Eq.(3))thatdecreasesexponentiallywithtime.Xeistheequilibriummoisturecontentexpressedonadrybasis.Thereareotherexpressionsforthemoistureratio(MR)thathavebeenusedtodescribethedryingkineticsforfruits.TogrulandPehlivan(2004)usedthemodelsinTable1intheirdryingworkforgrapes,peaches,figsandplums.Theyfoundthatthefollowingmodelswerethebestonestodescribethedryingofthefruitsinquestion;the‘‘approxi-mationofdiffusionmodel’’forapricots(non-pre-treatedorSO2-sulfured)andfigs;themodifiedHendersonandPabis
Table1
Empiricalmodelsforfruitsdrying,TogrulandPehlivan(2004)Modelno.123456789101112
Modelequation
MR=exp(Àkt)MR=exp(Àktn)MR=exp(À(kt)n)MR=aexp(Àkt)MR=aexp(Àkt)+c
MR=aexp(Àk0t)+aexp(Àk1t)MR=1+at+bt2MR=aexp(Àkt)+(1Àa)exp(Àkbt)MR=aexp(Àkt)+(1Àa)exp(Àgt)
MR=aexp(Àkt)+bexp(Àgt)+cexp(Àht)MR=aexp(Àkt)+(1Àa)exp(Àkat)MR=exp(Àk(t/L2)n)
Name
NewtonPage
ModifiedPage
HendersonandPabisLogarithmicTwo-term
WandandSingh
‘‘Approximationofdiffusion’’Verma
ModifiedHendersonandPabisTwotermexponential
ModifiedPageequation–II
References
O’Callaghanetal.(1971)Page(1949)
Overhultsetal.(1973)
HendersonandPabis(1969)Yagcioluetal.(1999)
Sharaf-Eldeenetal.(1980)WangandSingh(1978)Sharaf-Eldeenetal.(1979)Vermaetal.(1985)Karathanos(1999)
Sharaf-Eldeenetal.(1980)DiamanteandMunro(1991)
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modelforapricots(NaHSO3-sulfured),grapesandplums;andtheVermamodelforpeaches.Theseresultsshowedthattheapplicabilityofthebestempiricalmodeldependsonthematerial,andthattheremaybemorethanonemod-elthatisreasonableforeachtypeoffruit.Intheirwork,twodifferentmodelswerefoundtobereasonablemodelsforapricotsthatweretreatedbydifferentchemicals(SO2andNaHSO3wereusedinthiswork).
Otherdryingmodelsbasedontheanalyticalsolutionofthediffusionmodel(Crank,1975)inradialcoordinateshavealsobeenusedtodescribethedryingofbananas.Theana-lyticalsolutionofthediffusionmodelassumesthatthemois-turecontentwithinthebananaisinitiallyequaltotheinitialmoisturecontentandthatthesurfaceofthebananainstantlycomestoequilibriumduringdrying.Theexternalconditions,andhencetheequilibriummoisturecontent,donotchangeduringdrying.ThreemodelsbasedonthesolutionofthediffusionmodelhavebeenusedbyDandam-rongrak,Young,andMason(2002)todescribethedryingofbananas.Intheirwork,bananasweretreatedusingfourdif-ferentpre-treatments(blanching,chilling,freezingandfreezingcombinedwithblanching)priortodrying.Thedry-ingconditionswereatemperatureof50°C,anairvelocityof3msÀ1andrelativehumidityof10–35%.ThemoistureratiowasexpressedinEq.(4),whichcanbesimplifiedtogivethethreeempiricalmodelsinTable2.
MR¼X1422expÀbnDtn¼1bnr2ð4ÞInthiscase,theradiusofthebananaandthediffusioncoef-ficientareconsideredtobeconstant.Thefittingofthe
averagemoisturecontentsusingthemodelswasfoundtobesatisfactory.Thefittedequilibriummoisturecontent(kgperkgdrybasis)wasfoundtobeintherangeof(0.221–0.443)fortheNewtonmodel,(0.072–0.158)forthetwo-termmodeland(À0.053to0.045)forthePagemodel.Basedontheseresults,thetwo-termmodelwaschosenasthebestmodeltorepresentthedryingofbananasthatweredriedtolowermoisturecontents.ThenegativevalueofthefittedequilibriummoisturecontentforthePagemodelwasconsideredtobephysicallyunrealistic.Itisunusualthattheequilibriummoisturecontentshouldde-pendonthesolutionusedforfittingthedryingkinetics,be-causetheequilibriummoisturecontentshouldonlydependonthematerialbeingdried.Theanalyticalsolutionofdif-fusioncoefficientwasalsousedbyNogueiraandPark(1992).Theyfittedtheaveragemoisturecontentsforwhole
Table2
Empiricalmodelsforbananadrying,Dandamrongraketal.(2002)Modelno.ModelInstantaneousNameequation
moisturecontent1MR=Aexp(Àkt)X=Xe+A1exp(Àkt)Simple2MR=Aexp(Àkt)X=Xe+A1exp(Àkt)Two-term+Bexp(Àit)
+A2exp(it)
3
MR=Aexp(ÀktN)
X=Xe+A3exp(ÀktN)
Page
peeledbananasthatweredriedinarangeoftemperaturefrom50to70°Candairvelocityfrom0.5to1.5msÀ1.Theyreportedactivationenergiesforbananathatwerefrom15.5to25.3kJ/mol,whereasthediffusioncoefficientswerefrom2.6·10À10to6.5·10À10m2/s.
Ingeneral,thesimilarityoftheempiricalmodelsgiveninTables1and2istheexponenttermincludedintheequa-tions.Thischaracteristicofexponenttermgivesadecreas-ingexponentialcurvethatfitsthecontinuousdecreaseofmoisturecontentsofthedriedbananas.Thesemodelshavebeenfoundtofittheaveragemoisturecontentsofcontin-uousdryingforfruits,includingbananas,satisfactorilyasillustratedintheliterature,andtheywillnotbefittedagaininthiswork.ThefittingofmoisturecontentsinthisworkwillonlyusetheNewtonmodelasanempiricalapproachtoillustratethecomparisonoffittingbetweentheempiricalanddiffusionmodel,forcontinuousandintermittentdrying.
Thedryingconstantcanbecorrelatedwiththeairdry-ingconditionsandthematerialdimensions,asshowninEq.(5)(Kiranoudisetal.,1997).
kd¼k0dk1k2k3k4
pTuaw
ð5Þ
ThedryingconstanthasunitsoftimeÀ1.Here,dpisthediameterofthedriedmaterial,Tisthedryingtemperature,uistheairvelocityandawistherelativehumidity.
Thedryingconstantsfor35foodmaterialswerecom-piledbyKrokida,Foundoukidis,andMaroulis(2004).Forbananas,thedryingconstantwasfoundtobeintherangebetween0.11and1.9hÀ1.Theexperimentalcondi-tionsforbananaswerelengthsof0.1–0.28m,temperaturesof40–80°C,relativehumidityof3–40%andairflowratesof0.62–4.5m/s.However,SousaandMarsaioli(2004)obtainedhighervaluesofthedryingconstantforbananascomparedwiththosecompiledbyKrokidaetal.(2004).TheydriedCavendishbananasinamicrowaveovenwithairtemperaturesandflowratesthatvaryfrom25to55°Cand0.8to1.8m3/min,respectively.Thefitteddryingconstantshadarangeof0.52–0.69hÀ1.ThedifferencebetweentheresultsofSousaandMarsaioli(2004)andKrokidaetal.(2004)mayreflectsomebiologicalvariabilitybetweenthesamplesinthetwostudies.
Theequilibriummoisturecontent,Xe,issometimestrea-tedaszero(Sousa&Marsaioli,2004)tosimplifythemath-ematicalsolution.Inreality,itisafunctionoftherelativehumidityandthetemperature(Kiranoudisetal.,1997).Oneofthemostpopularfittingequationsfortheequilib-riummoisturecontentsistheGABcorrelation,asdescribedbyXXmCKaw
e¼
½ð1ÀKað6ÞwÞð1ÀKawþC¼CCKawÞ
DHc
0exp
ð7ÞK¼KRTDHk
0exp
RT
ð8Þ
334R.Baini,T.A.G.Langrish/JournalofFoodEngineering79(2007)330–343
Here,awisthewateractivity,Xmisthemonolayermoisturecontent,whileCandKareconstantsthatarerelatedtothedryingtemperature,T(K).DHcandDHkarefunctionsdescribingtheheatofsorptionofwater(monoandmulti-molecularlayers)andtheheatofcondensationofwaterva-por,respectively.Risthegasconstant,takenas8.314JmolÀ1KÀ1,ifDHcandDHkareinJmolÀ1.
Theempiricalmodelsdiscussedonlytheaveragemois-turecontentasafunctionofthedryingtime,t.Theaveragemoisturecontentswithinthedriedmaterialsmaychangewhenthedryingconditionschange.Thevariationofmois-turecontentasfunctionsofbothtimeandspacewithinthedriedmaterialsisnotincludedintheempiricalmodels,whichmaylimittheirapplicationsindescribingtheinter-mittentdryingofbanana.1.2.Characteristicdryingcurve
Theconceptofacharacteristicdryingcurvesuggeststhatthereisauniqueshapeofthefallingorhindereddry-ingcurveforaparticularmaterial.Therefore,acharacter-isticcurveofamoistmaterialmightbedrawnupfromasingleexperimentalrate-dryingcurve.Therelativedryingrateisdefinedas,R¼NDR
N¼fð/Þð9Þ
v
Nvistheunhindereddryingrate.Itmaybetakenasthemaximumrate,ifthereisnoclearunhindereddryingperi-od./isthecharacteristicmoisturecontent,whichisthera-tioofthefreemoisturecontent(XÀXe)tothemoisturecontentdifferencebetweenthecriticalpointandtheequi-libriummoisturecontent./¼
XÀXeXcrÀXð10Þ
e
Xisthevolume-averagedmoisturecontent,Xcristhemois-turecontentatthecriticalpointandXeisthatatequilib-rium.Forproductswithnoclearfirstdryingphase,thecriticalmoisturecontentisusuallyassumedtobeequaltotheinitialmoisturecontent(Jannot,Talla,Nganhou,&Puiggali,2004).Forbananas,thefirstdryingperiodwasnotfoundtobeclear(Baini&Langrish,2004)anditappearedshortlyafterthestartofdryingintheworkofTalla,Jannot,Kapseu,andEtude(2001).
Theplotoftherelativedryingrateagainstthecharacter-isticmoisturecontentisnormalizedtopassthroughthepoint(1,1)atthecriticalpointandtopassthroughthepoint(0,0)atequilibrium.Thecriticalpointisthepointoftransitionfromtheunhindered(orthemaximumdryingrateiftheunhinderedperiodisunclear)tothehindereddryingperiod.ThisapproachhasbeenusedbyJannotetal.(2004)todescribethedryingofbanana.Theyobservedthatthereweretwosignificantdryingphases.Thefirstphasewasanexponentialdecreaseofthedryingrateforthecharacteristicmoisturecontentfrom/2to1,asdescribedbyEq.(11).Then,itwasfollowedbyalinear
periodofdryingforthecharacteristicmoisturecontentfrom/1to/2asdescribedbyEq.(12).Thedryingrateasafunctionofthecharacteristicmoisturecontentintherangefrom/1to0wasnotdescribedmathematicallybyJannotetal.(2004),andthereforeitisnotincludedhere.Thevaluesofthecharacteristicmoisturecontent,/1and/2,whichcorrespondtotransitionsbetweenthedryingperiods,weredeterminedgraphicallytobe0.04and0.2,respectively.
fð/Þ¼aexpðb/Þfor/2ð12Þ
TheyalsosuggestedcorrelationsofmoisturecontentasafunctionoftimebysolvingEqs.(9)–(12).Thesecorrela-tionsarepresentedinEqs.(13)and(14).Thefittedparam-etersweregivenasa=0.079,b=2.541,c=0.478,d=À0.004andNv=1.716kgkgÀ1sÀ1.
XðtÞ¼XÀ1abN
bðXlnexpðÀb/Þþv
e0ÀXeÞXÀXt;
0e/2ðXðtÞ¼XXd
cNveþðX0ÀeÞ/þ
cexpÀXðtÀt13Þ
Àd
2Þ;0ÀXec/1ð14Þ
Thisconceptofusingacharacteristicdryingcurveimplies
thattheratioofthedryingraterelativetotheunhinderedrateisindependentoftheexternaldryingconditions.1.3.Diffusionmodel
Fick’slawsofdiffusionhavebeenfrequentlyadoptedintheliteratureindescribingthediffusionofmoisturewithindriedmaterials,includingdriedbananas.Foraxiallysym-metriccylindricalcoordinates,Fick’ssecondlawofdiffu-sionpresentsthemoisturecontent,Xasafunctionoftimeanddistance,r,perpendiculartotheairstreamfromthecoreasgiveninEq.(15).
oXot¼DororroX
orð15ÞThediffusioncoefficient,D,dependsonthetemperaturedistributionthroughoutthebanana,whichisoftenas-sumedtofollowanArrheniusrelationship(Zogzas,Maroulis,&Marinos-Kouris,1996)asshowninEq.(16).Thistemperaturedistributionthroughoutthebananadur-ingthedryinghasnotalwaysbeenclearlyexplainedintheestimationofthediffusioncoefficientinsomepreviousworksofbananadryingthathavealsousedthediffusionmodel(Dandamrongraketal.,2002;Demirel&Turhan,2003;Karim&Hawlader,2004;Nogueira&Park,1992;Phoungchandang&Woods,2000;Queiroz&Nebra,2001).
D¼DDE
rexpÀTþ273ð16Þ
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Here,Dristhepre-exponentialfactorforthediffusioncoefficient,andDEistheactivationenergy.Thisactivationenergyisassociatedwiththeworkrequiredtoremoveonemoleofmoisturefromagivenmaterialandthusdependsonthematerial–moisturebonding.Differenttypesofbind-ingenergiesaregivenbyStrumilloandKudra(1986),oftheorderof5kJ/molforchemicallyboundmoisture(stoichiometric),3kJ/molforphysico-chemicalmoisturebinding(non-stoichiometric)and0.1kJ/molforphysico-mechanicalbinding.Thesefiguressuggestthatwatermol-eculesthatareabsorbedhygroscopicallyontothecellwallatmonolayersitesaremorestronglyboundthanthoseoccupyingsiteswithotherwatermolecules,whicharede-scribedasmultilayers.Thismayalsoexplainthedepen-denceofthediffusioncoefficientonthemoisturecontent.Theevaporationofmoistureatthesurfaceduringdryingreducesthenumberofwatermoleculeswithinthematerial.Therefore,therearefewerwatermoleculesatthesiteswithlowermoisturecontents.Thisphenomenonreducesthediffusioncoefficient(Blakemore,2005).How-ever,shrinkagereducesthepathlengthfordiffusion,whichalsoincreasestheeffectivediffusioncoefficient.Thesetwoeffectsofmoisturecontentandshrinkageonthediffusioncoefficienttendtocanceleachotherout,whichiswhyonlythetemperaturedependenceofdiffusioncoefficientisconsideredinthiswork.Themovementofwatermolecules,whichdependsontheirkineticenergy,explainsthedependenceofthediffusioncoefficientontemperature.
ThetemperaturedistributionofthedriedbananasthatisincludedinEq.(16)canbedescribedbythepartialdif-ferentialequationasgiveninEq.(17).Thebananaproper-ties,whicharethethermalconductivity,kandtheheatcapacity,Cp,arefunctionsofthemoisturecontent.There-fore,thisrelationshiprequiresEqs.(15)and(17)tobesolvedsimultaneously.
oTot¼ktc1oqroT
ð17ÞbCprororAtthebananasurface,themassfluxofmoisturefromthebananas(J)isproportionaltothedifferencebetweenthehumidityofairjustabovethebananas’surface(Ys)andthatinthebulkairstream(YG),asfollows:J¼qairkmðYsÀYGÞ
ð18Þ
Inthisequation,qairrepresentsthedensityofair,whilekmisthemass-transfercoefficient.Whilethehumidityoftheairstream,YG,canbecalculatedfromthedryandwet-bulbtemperatures(TGandTw)inthedryer,Ysisafunc-tionofthemoisturecontent.ThereforeEq.(18)formsaboundaryconditiontoEq.(15).
Atthebananas’surface,bothconvectiveheattransportandheattransferassociatedwiththeevaporationofmois-ture(thetotalrateofheattransferbeingQ)mustbeconsidered:
Q¼hFðTGÀTsÞÀðkwþksÞJ
ð19Þ
Intheaboveequation,hFistheheat-transfercoefficient,kwisthelatentheatofvaporization,ksistheheatofsorptionandTsisthesurfacetemperature.ThisequationformsaboundaryconditionforEq.(17).Animplicitassumptionhere,whichisusedtospecifytheheatfluxtothebananas,isthatmoistureevaporatesclosetothebananas’surface.Ifweassumethatthesameamountofheatandmasstransportisoccurringatallsurfaces,thenthetemperatureandmoistureprofileswillbesymmetricaboutthecentreofthebananas.Consequently,moistureandheattransportcannotoccuracrossthecentre,afactwhichdefinesanothertwoboundarydXconditions:dr¼0ð20ÞdTc
dr¼0ð21Þc
Ifthemoistureandtemperaturegradientsareapproxi-matedbyfinitedifferences(Wan&Langrish,1995),thenthedryingmodelreducestoa‘stiff’systemofordinarydif-ferentialequationsapplicableatthesurface,withintheba-nanaandatthecentreofthebanana:Surface:
dXs11Dr1sðX
dt
¼1ÀXsÞ
rDrÀJð22ÞsDrsdTs11ktcr1sðX1ÀXs
dt
¼Þ
rDrþQð23ÞsqbCpDrsWithinthebanana:
dXi1Driþ1=2ðXiþ1ÀXiÞriÀ1=2ðXiÀX
dt
¼rDrdTi1kDrÀiÀ1Þ
Drð24Þ
riþ1=2ðTiþ1ÀTiÞriÀ1=2ðTiÀTiÀ1Þdt¼tc
rDrqÀð25Þ
bCpDrDrCentre:
dXc1DrcÀ1=2ðXcÀXdt
¼À
cÀ1Þ
rDrð26ÞcDrcdTc1ktcrcÀ1=2ðTcÀTcÀ1
dt
¼À
Þ
rDrð27Þ
cqbCpDrcThenumericalsolutionappliedheredividesthebanana
into30controlvolumes,whichleadstoasystemof60dif-ferentialequationsdescribingconditionsatthecentre,thesurfaceandwithinthebanana,whichcanbesolvedusinganintegratorforstiffsystems.Gear’smethodwasused,asimplementedinthestandardsubroutineordinarydiffer-entialequations(ODE)fromtheODEPACKlibrary.Thesubscriptsiþ1,iÀ1andcÀ1refertothepositionsoftheedgesofcontrol22volumes,since2theintegrationofthemoisturecontentandthetemperaturewithinthebananainvolvesthecentraldifferencesofthecontrolvolumes.Themoisturecontentandtemperaturedistributionswithinthedriedbananas,asdescribedinEqs.(15)and(17),arefunctionsofspace.Thismeansthatthesecorrela-tionsalsodependontheradiusofthebananas,which
336R.Baini,T.A.G.Langrish/JournalofFoodEngineering79(2007)330–343
changesduringdryingduetotheshrinkageeffect.There-fore,itisnecessarytoincludethiseffect.1.4.Shrinkage
Themoisturelossduringdryingcausestheinternaltis-sueofthebananastocontract.Thisreducesthediameterandthelengthofthedriedbananasandreducesthevolumeofthebananas.Thisshrinkagephenomenonreducesthesurfaceareaforheatandmasstransfer,whichmayaffecttheoveralldryingrate.
Abananaiscomposedofsolidsandmoisture.Theini-tialmoisturecontentofabananaisapproximately60–80%ofthetotalmass.Othercomponentssuchasfibre,starch,sugarandnutrientsmakeuptheremainingpercent-age.Thetotalvolumeofabananaisthevolumeofthemoisture(Vw)plusthevolumeofthesolid(Vs).MayorandSereno(2004)reviewedanumberofshrinkagemodelsforfoodmaterialsduringconvectivedrying.Themodelsreviewedwerethosethathadbeenappliedfordifferentgeometries(cylinder,sphere,ellipsoid,slabandcube)anddifferentreduceddimensions(volume,radius,thickness,width,length,diameterandsurfacearea).Theypresentedfitteddataforcarrot,potatoandapple.Shrinkagewasfoundtoincreaselinearlywiththevolumeofremovedwaterforthesematerials.Thisindicatesthatthevolumechangemaybecorrelatedasalinearfunctionofthemois-turecontentofthedriedbananas,asshowninEq.(28).
Therateofchangeinradiusandlengthofthedriedbana-nasmayinitiallybeassumedtooccuratequalrates,sotheradiuschangemaybeassumedtobeproportionaltothecuberootofthevolumechange.Theradiuschangemaythenbecorrelatedwiththemoisturecontentofthedriedbananas,asshowninEq.(29).TheeffectofradiuschangehasinitiallybeenconsideredinthediffusionmodelasdescribedinEqs.(22)–(27).VX¼AþBV0X0
113rV3X¼¼AþBr0V0X02.Materialsandmethod2.1.Experimentalapparatus
Theexperimentalapparatususedinthisworkwasasteam-heatedkilnintheDepartmentofChemicalEngi-neeringattheUniversityofSydney.AsimplifiedschematicdiagramofthekilnisshowninFig.1.Thekilnisapilot-scaletunneldryerwithworkingsectiondimensionsof0.45mheight,0.9mwidth,0.4mlengthandavolumetriccapacityof0.162m3.Acentrifugalfanwithaconstantmotorspeedof1400rpmprovidesanairvelocityof1.3msÀ1throughtheworkingsection.Ajacketedsteam
ð28Þð29Þ
PI PI VENT TOOUTSIDECONTROLVALVEHEATEXCHANGERBLOWER14 KWBOILERBUTTERFLYVALVESOLENOID VALVE PRVAIR INTAKEPI BUTTERFLYVALVESPARGEKILNBUTTERFLYVALVEBALANCEWATERFEEDTOP UPTANKCOMPUTERDRY AND WET BULB TEMPERATURES BANANAS MASS PUMPFig.1.Schematicdiagramofthepilotscalekiln.R.Baini,T.A.G.Langrish/JournalofFoodEngineering79(2007)330–343337
injectionunitisused,inwhichsteamisinjectedthroughsixinjectionportsovera300mmdiameterductintothetun-nel.Steamisgeneratedbya10kWboiler,withthesteamfinalpressurebeingregulatedto1atm(gauge).Asteam-heatedfinnedheatexchangerisusedtoregulatethedry-bulbtemperature.Theflowratesofthesteamtotheheatexchangerandfromthesteam-injectionsystemarecon-trolledbycontrolvalves,tomaintainthedryandwet-bulbtemperaturestothedesiredsetpoints.
Theaveragevaluesforeachofthedryandwet-bulbtemperatureswerecalculatedbasedonthereadingmea-suredbythefourresistancethermometers(RTDs)thatweremountedinpairsatbothsidesofthekiln.Thetwothermometersusedtomeasurethewetbulbtemperatureswerecoveredbyadampwick.Oneendofeachwickwasimmersedinasmallwaterbathfilledwithdistilledwatertokeepitwet.Aplatformbalancewithacapacityof200kg(±0.005kg)undertheworkingspaceofthekilnmeasuredthemassofthebananasasthedryingprogressed.Outputsignalsfromthecontrolvalves,resistancether-mometersandplatformbalanceweresenttoanIBM-compatiblepersonalcomputerthroughaRS-232interface.Theseprovideddataloggingofthemainprocessvariables,aswellassendingcontrolsignalstothecontrolvalves.ThecontrolsoftwarewaswritteninVisualBasic.Humidityreg-ulationwasalsoprovidedbymanuallyoperatedbutterflyvalves,whichactasventsforthekiln.Forsafetypurposes,asolenoidvalvetoeachunit(heatexchangerandsteaminjection)wasusedtototallyisolatethesteamsupplyifnecessary.
2.2.Experimentalprocedures
Fiveexperimentswerecarriedout,inwhichpeeledbananasweredriedinthekiln.Inthefirstandsecondexperiments,ripebananasweredriedcontinuouslyfor72hatawetbulbtemperatureof40°Candthedrybulbtemperaturesof60and80°C,respectively.Themassofthedriedbananaswasmeasuredbythebalanceunderneaththekilnandwassenttothecomputereverythreeseconds.Anotherexperimentwithcontinuousdryingat60°Cwasrepeatedtogetanothersetofdata.Inthethirdexperiment,ripebananasweredriedintermittentlyfor72hatwetanddry-bulbtemperaturesof40and60°C,respectively.Theboilerandthefanwereswitchedonfor8htoletthedryingprogress.Bothunitswerethenswitchedofffortheremain-ing16hoftheday.Thesameschedulewasusedforthefol-lowing48hofdrying.
Unripeandripebananasweredriedfor72hand24hinthefourthandfifthexperiments,respectively.Themassofthedriedbananasweremeasuredmanuallyforbothexper-iments.Thelengthandthediameterofthedriedbananaswerealsomeasuredinthefifthexperimentusingaverniercaliper.
Thesugarcontentsofthebananasatdifferentdegreesofripenessweredeterminedusingthephenol-sulfuricacidmethod(Masuko,Minami,Iwasaki,&Majima,2005).In
thiswork,aweighedsectionofbananawasblendedwith100mlofwater.Thiswastoallowthesugarinthebananatodissolveinthewater.Thewaterwasthenseparatedfromthesolidusingacentrifuge.1mlofthesupernatantwasdilutedwith100mlofwater,andthissolutionwasthenusedasthesampleintheanalysis.Theanalysisinvolvedtheadditionof1mlofphenoland5mlofconcentratedsulfuricacidto1mlofsample.Theadditionofphenolandsulfuricacidtosugarsgaveanorange-yellowcolour,whichcouldbedetectedbyaspectrophotometerthatwassettodetectabsorptionat485nm.Thesugarcontentwasthencalculatedbasedonthestandardcurvedevelopedusingaglucosestandardsolutionbetween0and300lgglucose/ml.Thesugarcontentdeterminedinthisanalysiswastheamountofsugarthathadbeendissolvedinthewaterexpressedasaglucoseequivalent,neglectingthesugarthathadbeenleftundissolvedinthesolidpartofthebanana.
Forthepurposeofestimatingtheinitialandfinalmois-turecontents,bananasamplesfromeachexperimentweredriedinanovenat105°Cfor24±2h.Bananasampleswerepreparedfrom(1)thefreshbananasand(2)thedriedbananasthathadbeendriedinthekiln.Thesampleswerepreparedbycuttingapproximately1cmlengthsofdifferentpartsofthebananas,whichwerethenputinsmallcontain-ers.Theemptycontainerswereweighedbeforetheywereused.Themassesofthecontainerswiththesamplesinthemweremeasuredbeforeandafterdrying.
Aleastsquaresprocedurewasusedinthefittingofpre-dictionsofthedryingmodelstothemeasuredmoisturecontents.ThisprocedureminimizesthesumofsquaresofthedifferencebetweenthepredictedandexperimentalmoisturecontentsasillustratedsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiinEq.(30).
Standarderror,se¼Pi½ðXExpÀXPredÞ2
n¼1nÀ1ð30Þ
Thedryingconstantwasthefittedparameterfortheempir-icalmodel,whereasthepre-exponentialfactor,theactiva-tionenergyandtheheat-transfercoefficientwerefittedforthediffusionmodel.3.Resultsanddiscussion
3.1.Initialandfinalmoisturecontents
TheinitialandfinalmoisturecontentsaregiveninTable3.Thefinalmoisturecontentsappearedtobemoreuniformamongthebananas.Thecoefficientofvariationforthefinalmoisturecontentswasfoundtobelowerthanthatfortheinitialmoisturecontents.Therefore,thefinalvaluesweretakenasthebasisforallmoisturecontents.Amassbalanceforthetotalmoisturelossofthebananaswasusedtocheckthequalityofthedata.Therewasan11%differ-encebetweenthemoisturelossestimatedfromthediffer-encebetweentheinitialandfinalmoisturecontentsfromovendryingandfromtheamountofmoistureloss
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Table3
InitialandfinalmoisturecontentsofbananasforthefirstexperimentExperiments
SamplenumberContinuousdrying
Averagemoisturecontent(kgperkgdrybasis)
Standarddeviationofmoisturecontent(kgperkgdrybasis)Coefficientvariationofmoisturecontent(dimensionless)Intermittentdrying
Averagemoisturecontent(kgperkgdrybasis)
Standarddeviationofmoisturecontent(kgperkgdrybasis)Coefficientvariationofmoisturecontent(dimensionless)
Initialmoisturecontent,X0(kgperkgdrybasis)S1S23.8413.4653.6170.1730:173
¼0:0483:623.7463.9373.5250.3210:32
¼0:0913:53
S33.496
S43.664
Finalmoisturecontent,Xf(kgperkgdrybasis)S1S20.1950.2050.2000.0080:008
¼0:0390:200.2580.2610.2600.0030:003
¼0:0130:26
S30.191
S40.207
3.3173.1020.2640.257
V/V0 (dimensionless)recordedinthekiln.Thereasonforthisdiscrepancymaybethesignificantvariabilityininitialmoisturecontents,whichmakesaccuratesamplingofinitialmoisturecontentsverydifficult.Hencethemoisturecontentsherehadbeenbasedonthefinaloven-dryvalues,whichhadlessvariationcomparedwiththeinitialvalues,andtherecordedmoisturelossinthekilnfromtheplatformbalance.3.2.Shrinkageanalysis
Thevolumeofthedriedbananachangedlinearlywiththemoisturecontent.Thislinearfunctionofvolumewasnotaffectedbythetemperature(Fig.2)butwasaffectedbythedegreeofripeness(Fig.3).TheconstantsAandB,discussedinSection1,werefoundtobe0.65and0.35,respectively.Thus,theshrinkagemaybedescribedbythefollowingcorrelation:VX¼0:6þ0:4V0X0
ð31Þ
1.00.80.60.40.20.00.0UnripeRipefitted line for ripefitted line for unripe0.20.40.6X/X0 (dimensionless)0.81.0Fig.3.Thevolumeasafunctionofthemoisturecontentofthedriedbananasfordifferentdegreeofripeness.0.0160.0140.012TheradiuschangeofthedriedbananaswascorrelatedusingEq.(29)andwascomparedwiththeexperimentaldataasshowninFig.4.Thefittingisactuallybetterwhenthevolumechangeiscorrelatedasthesquareoftheradiuschange(shownasthesolidline)comparedwiththecubed
1.00.80.60.40.20.00.060 °C80 °Cfitted line0.20.40.6X/X0 (dimensionless)0.81.0Radius (m)0.0100.0080.0060.0040.0020.0000.000.50Radius squaredRadius cubed1.001.502.002.503.00Moisture content (kg per kg dry basis)3.504.00V/V0 (dimensionless)Fig.4.Theradiusasafunctionofthemoisturecontentofthedriedbananas.oftheradiuschange(shownasthedottedline)assuggestedintheintroduction.Thisindicatesthatthechangeinlengthisinsignificantcomparedwiththechangeinradiusduringthedryingofthebananas.Thisisconsistentwiththeshrinkagebeingmoresignificantintheradialdirectioncomparedwiththatinthelongitudinaldirection.There-fore,thechangeofradiusmayberepresentedbestbythefollowingcorrelation:r¼r0
VV0
12¼
X0:6þ0:4X0
12ð32Þ
Fig.2.Thevolumeasafunctionofthemoisturecontentofthedriedbananasfordifferenttemperatures.R.Baini,T.A.G.Langrish/JournalofFoodEngineering79(2007)330–343339
Thisequationfitstheactualdatabetterthanthecuberootrelationship,asshowninFig.4.3.3.Moisturecontentfittinganalysis
Theaveragemoisturecontentofthedriedbananasisplottedasafunctionoftime,asshowninFigs.5and6forcontinuousandintermittentdryingexperimentsat60°C,respectively.Themoisturecontentdecreaseswithtimeuntilitreachestheequilibriumstate,asusualduringdrying.Boththeempiricalanddiffusionmodelspredictthemoisturecontentsverywell.Forintermittentdrying,therelaxationperiodwaspredictedbythediffusionmodelbutnotbytheempiricalmodel.Thediffusionmodeltookintoaccountthevariationofmoisturecontentandtemper-aturewithtimeandspace,meaningthismodelincludedtheeffectofthedry-bulbtemperaturechangesduringthenightperiod(relaxationperiod).Thechangesinthedry-bulbtemperatureaffectedtheheatandmass-transferratesatthebananasurface,whichalsoaffectedthemoisturecon-tentwithinthebanana.Thedecreaseintheheatandmass-transferrates,whichalsodecreasedthemoistureloss
3.00)sExperimental dataisabPredicted by empirical model 2.50yrPredicted by diffusion modeld gk 2.00rep gk(1.50 tnetno1.00c erutsi0.50oM0.0001020304050607080Time (hr)Fig.5.Fittingofthemoisturecontentsofthedriedforcontinuousdrying.2.50)sExperimental dataisaPredicted by empirical modelb y2.00Predicted by diffusion modelrd gk rep1.50 gk( tnet1.00noc erut0.50sioM0.0001020304050607080Time (hr)Fig.6.Fittingofthemoisturecontentsofthedriedbananasforintermittentdrying.atthebananasurfaceduringthenightperiod,wasincludedintheestimationoftheaveragemoisturecontent.Thishelpedthediffusionmodeltofittherelaxationperiodbet-terthantheempiricalmodel(Fig.6).Thediffusionmodelgavestandarderrorsof0.04kgperkgdrybasisand0.09kgperkgdrybasisforcontinuousandintermittentdrying,respectively.Thesestandarderrorswere50%and88%lowerthanthoseobtainedinthefittingoftheempir-icalmodel,whichwere0.08kgperkgdrybasisand0.77kgperkgdrybasisforcontinuousandintermittentdrying,respectively.
ThesurfacetemperaturespredictedbythediffusionmodelforbothcontinuousandintermittentdryingareshowninFig.7.Forcontinuousdrying,itincreasesfromapproximately50–60°C(dry-bulbtemperature).Forinter-mittentdrying,thesurfacetemperaturechangedaccordingtothechangesofthekilntemperature.Thekilntempera-tureincreasedfromapproximately50°Ctoavalueclosetothedry-bulbtemperatureduringtheday(theboilerwason)andfluctuatedbetween25°Cand35°Cduringthenight(theboilerwasoff).Theeffectofthischangeoftemperatureistakenintoaccountintheapproximationofdiffusioncoefficient,whichfollowstheArrheniusrelationship.
Thesurfacemoisturecontentispredictedbythediffu-sionmodeltodropfromtheinitialmoisturecontenttoitsequilibriummoisturecontentforcontinuousdrying,asshowninFig.8.Thedatapresentedinthisfigureareforexperimentsat60°C.Forintermittentdrying,thesurfacemoisturecontentispredictedtodecreaseduringthedayandincreaseduringthenight.Duringtheday(heatingper-iod),theheatandmass-transferratesatthesurfacecontrolthedryingandcausetherapiddropinthesurfacemoisturecontent.Theincreaseinthesurfacemoisturecontentdur-ingthenightisconnectedwiththedryingbeingcontrolledbyinternaldiffusion,inwhichthemoisturemovesfromtheinnerwetregionofthebananastothedrysurfaceregion.Sincetheboilerandthefanwereoff,thehumidityoftheairstreamabovethesurfaceofthedriedbananasincreased.Hence,themassandheat-transferratesfrom
70)C°60( eruta50repme40t ecafr30us det20ciderP10Continuous dryingIntermittent drying001020304050607080Time (hr)Fig.7.Thepredictedsurfacetemperaturefordryingexperimentat60°C.340
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3.00 Continuous dryingtnetn2.50Intermittent dryingoc )esriustasb2.00i oymrd ecg1.50ak frrueps dg1.00ek(tcider0.50P0.0001020304050607080Time (hr)Fig.8.Thepredictedsurfacemoisturecontentfordryingexperimentat60°C. 0.90tnetn0.80Intermittent dryingoContinuous dryingc e0.70r)ustissioa0.60mb ymrd0.50u igrkb 0.40irlieup qgek0.30 d(etc0.20ider0.10P0.0001020304050607080Time (hr)Fig.9.EquilibriummoisturecontentspredictedusingtheGABcorrela-tionsforexperimentsat60°C.thesurfacewerereduced.Thisencouragedthemoisturetoaccumulateonthesurface,asshownbytheincreasingsur-facemoisturecontentatnight.Thisphenomenonisnotpredictedwellbytheanalyticalsolutionofthepartialdif-ferentialequationofFick’ssecondlaw(Eq.(4)),asopposedtoournumericalsolution.Theanalyticalsolutionassumesthatthesurfacemoisturecontentreachesitsequi-libriumvaluequickly(Ma&Rudolph,2001).However,ournumericalsolutiondoesallowforthisphenomenonandsoismoreaccuratephysically.
TheequilibriummoisturecontentpredictedbytheGABcorrelationsisshowninFig.9.Thepredictedfinalequilib-riummoisturecontentsarefoundtobelowerthantheexperimentalequilibriummoisturecontents;thepredictedvaluesare0.039kgperkgdrybasisand0.120kgperkgdrybasisandtheexperimentalvaluesare0.200kgperkgdrybasisand0.260kgperkgdrybasisforcontinuousandintermittentdryingexperimentsat60°C,respectively.3.4.Analysisoffitteddryingparameters
Thedryingconstants,fittedinthefittingoftheempiricalmodelwere0.11hÀ1and0.09hÀ1,forcontinuousand
intermittentdrying,respectively.ThefitteddryingconstantforcontinuousdryingherewaswithintherangegivenbyKrokidaetal.(2004),wherecontinuousdryingwasalsoused.Theirreporteddryingconstantswereintherangebetween0.11and1.9hÀ1.
Forthediffusionmodel,theactivationenergyhasbeenfittedsimultaneouslytotwodatasetsofmoisturecontentsforthecontinuousdryingexperimentsat60and80°C.Thesimulationhasbeenrununtiltheglobalminimumofthetotalstandarderrorwasobtained.Thisproceduregaveapre-exponentialfactorandanactivationenergyof1.36·10À7m2/sand1610K(13.4kJ/mol),respectively.Thus,thediffusioncoefficientasafunctionoftemperatureforbananacanbeexpressedasshowninEq.(33).Theesti-mateddiffusioncoefficientusingthisequationforatemper-atureof60°Cwas1.09·10À9m2D¼1:36Â10À7exp/s.
À1610
Tþ273ð33Þ
TheactivationenergyobtainedherewaslowerthanthatgivenbyNogueiraandPark(1992).Theirreportedactiva-tionenergyforbananawasfrom15.5to25.3kJ/mol.Zog-zasetal.(1996)reportedarangeofactivationenergiesfrom44to110kJ/molforapple,39.8kJ/molforavocado,from12.7to28.7kJ/molforcarrotandfrom16.3to108kJ/molforpotato.Theestimateddiffusioncoefficientherewasattheupperendofthosereportedinthelitera-ture,closetotheuppervalueof1.59·10À9m2/sreportedbyQueirozandNebra(2001).Therangeofdiffusioncoef-ficientsgivenbyKarimandHawlader(2004)wasfrom6.61·10À11to2.41·10À10m2/s;NogueiraandPark(1992)from2.6·10À10to6.5·10À10m2/s;QueirozandNebra(2001)from6.98·10À10to1.59·10À9m2/s;Pho-ungchandangandWoods(2000)from4·10À10to7·10À10m2/s;Dandamrongraketal.(2002)from4.6·10À10to13.0·10À10m2/s.Thedifferenceintheesti-mateddiffusioncoefficientsobtainedheremaybeduetothedifferentapproachestakentosolvethedryingmodel.Inthiswork,thetemperaturedistributionthroughoutthebananawasconsideredintheheattransferrelationshipde-scribedbyEq.(17).Bananaproperties,suchasthethermalconductivityandtheheatcapacity,whichchangewiththemoisturecontent,werealsoincluded.Thistemperaturedis-tributionwasthenincludedintheestimationofthediffu-sioncoefficientbasedontheArrheniusrelationshipgiveninEq.(15).TherangesofdiffusioncoefficientsforotherfruitsreportedbyZogzasetal.(1996)werefrom2.2·10À12to6.4·10À9m2/sforappleandfrom1.1·10À10to1.8·10À9m2/sforavocado.
Theactivationenergyfoundhereof13.4kJ/molrepre-sentstheenergyrequiredtobreakthebanana–moisturebondingofamolewatermolecule,andtobringthemole-culetothesurface,wherethemoleculewillfinallyevapo-rate.Thisvalueishigherthanthematerial–moisturebindingenergiesgivenbyStrumilloandKudra(1986),dis-cussedinSection1.Theactivationenergyfoundheremayalsobeassessedinthebroadercontextofactivation
R.Baini,T.A.G.Langrish/JournalofFoodEngineering79(2007)330–343
Table4
Fittedparametersanddiffusioncoefficientsforcontinuousdryingat60°CandintermittentdryingDryingtechniques
InitialDr(m2/s)
InitialDE(K)
FittedDr(m2/s)
FittedDE(K)
Diffusioncoefficientforthedryingat60°C(m2/s)1.23·10À91.58·10À9341
Standarderrorofmoisturecontents(kgperkgdrybasis)0.0420.090
Continuousdryingat60°CIntermittentdryingat60°C
1.36·10À71.36·10À716101610
1.37·10À71.47·10À715701510
energiesforotherprocesses,asquotedbyDebenedetti(2003).Hesuggestedthathydrogenbondstrengthswereoftheorderof20kJ/mol,similartotheactivationenergyfoundhere,whileregulardispersiveinteractionswereoftheorderof1kJ/molandcovalentbondswerearound400kJ/mol.Theheatoffusionforicewas6kJ/mol,whiletheheatofvaporizationofwaterwas45kJ/mol,andtheheatofsublimationofwateratthetriplepointwas51kJ/mol.Inthecontextofthesenumbers,anactivationenergyof13.4kJ/molsuggeststhatsurfacediffusionofwatermoleculesalongcellularspacesinsidethebananaistheactualdiffusionalprocessthatisoccurring.
Thefittedpairofthepre-exponentialfactorandtheacti-vationenergygiveninEq.(33)wereusedastheinitialguessinthere-fittingofparametersforarepeatedcontin-uousdryingexperimentat60°Candanintermittentdryingexperimentalsoat60°Cduringthe8-hdaytimeheatingperiod.Thefittedparametersobtainedhereareonlyveryslightlydifferent,aspresentedinTable4.Theyareapprox-imately2%and8%higherthanthosegivenbyEq.(33),fortherepeateddatafromcontinuousdryingandintermittentdrying,respectively.Thefittedheat-transfercoefficientwasfoundtobe90WmÀ2KÀ1.Theheat-transfercoefficientindicatesthattheinitialstageofdryingwascontrolledbytheexternaldryingconditions,whileinternaldiffusionwasthecontrollingfactorinthelaterstagesofdrying.3.5.Analysisofbananadryingbehaviorusingtheconceptofacharacteristicdryingcurve
Theconceptofacharacteristicdryingcurveapproachnormalizesthedryingrate(Nv)relativetotheunhinderedrate(Nv).Forbanana,thereisnoclearunhindereddryingperiod,sotheinitialdryingrateissetequaltotheunhin-deredrate.AnormalizeddryingratewasdevelopedusingthefunctionssuggestedbyJannotetal.(2004),describedinEqs.(11)and(12),fortherangeofcharacteristicmois-turecontentfrom0.04to0.2andfrom0.2to1,respec-tively.Thiswasthencomparedwiththeexperimentaldata,asshowninFig.10.ThenormalizeddryingratesforcontinuousdryingmeasuredherearesimilartothefunctionssuggestedbyJannotetal.(2004),alsoforcontin-uousdrying.Numericaldifferentiationofourmoisturecontentsasfunctionsoftimecausesthescatterinthisfig-ure.Similarresultswereobtainedforbananasatdifferentdegreeofripeness,asshowninFig.11.Thismayindicatethattheeffectofthesugarcontentonthediffusionofinter-nalmoistureisinsignificant.Forintermittentdrying,therelativedryingrateisfoundtobescatteredbetweenÀ0.4
1.00s)s0.90elno0.80isne0.70mid(0.60 eta0.50r gin0.40yrd0.30e vita0.20leR0.100.000.000.200.400.600.801.00Characteristic moisture content (dimensionless)Fig.10.Therelativedryingrateasafunctionofthecharacteristicmoisturecontentforcontinuousdryingexperimentat60°C.)s1.00selnoisn0.80emid( 0.60etar gn0.40iyrd evi0.20Ripe-experimenttaPredictedleRUnripe-experiment0.000.000.200.400.600.801.00Characteristic moisture content (dimensionless)Fig.11.Therelativedryingrateasafunctionofthecharacteristicmoisturecontentforcontinuousdrying(at60°C)ofbananasatdifferentripeness.and1,asshowninFig.12.Thenegativevaluescorrespondtotheincreasesinmoisturecontentoccurredovernight.Forintermittentdrying,thenormalizeddryingratesmea-suredherearesignificantlydifferenttothosemeasuredbyJannotetal.(2004)forcontinuousdrying.
Thesimilarshapeofthenormalizeddryingcurvesforcontinuousdrying,asshowninFigs.10and11,mayindi-catethatthecharacteristicdryingcurveconceptappliesforthecontinuousdryingofbanana.However,thisconceptisnotapplicableforintermittentdrying,whichshowsdiffer-entshapeofnormalizeddryingcurveaspresentedinFig.12,comparedwiththatinFigs.10and11forcontin-uousdrying.Forcontinuousdrying,itmaybepossibletodescribetherelativedryingratebyanexponentially
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1.00)ssel0.80noisn0.60emid(0.40 etar g0.20niyrd0.00 evi0.600.801.00ta-0.200.000.200.40leR-0.40Characteristic moisture content (dimensionless)Fig.12.Therelativedryingrateasafunctionofthecharacteristicmoisturecontentforintermittentdryingexperimentat60°C.Thenegativerelativedryingratescorrespondtomoistureabsorptionduringtheperiodsofnoheating.decreasingdryingperiodthatisfollowedbyalineardecreasingperiod(refertoEqs.(11)and(12)),asshowninFig.10and11.However,suchadescriptionislesslikelyforintermittentdrying,whichshowsscattereddryingratesinFig.12.Thisdifferencebetweencontinuousandinter-mittentdryingsuggeststhatacharacteristicdryingcurvemaynotbeanappropriateapproachtomodelingtheinter-mittentdryingofbananas.4.Conclusions
Thekineticsofdriedbananaswasassessedusingempir-icalanddiffusionmodels,togetherwiththeconceptofacharacteristicdryingcurve.Theempiricalmodelwasfoundtosuccessfullypredictthemoisturecontentsofthedriedbananasforcontinuousdryingbutnottherelaxationofmoisturecontentsforintermittentdryingverywell.Similarresultswereobtainedwhentheexperimentaldatawereanalyzedusingtheconceptofacharacteristicdryingcurve.Theanalysisshowedthattheapplicationofthisconceptmayonlybeappropriateforcontinuousdrying,butnotforintermittentdrying.Ontheotherhand,thediffusionmodelpredictedthemoisturecontentsofbothcontinuousandintermittentdryingwell.Thismodeltakesintoaccountthedistributionofmoisturecontentandtemperaturewithinthedriedbananas.Suchacharacteristicofthediffu-sionmodelmakesitbetterintermsofphysicallydescribingthedryingofbananas,particularlyduringtherelaxationperiod,whichwasnotpredictedwellbytheempiricalmodelandtheconceptofacharacteristicdryingcurve.Theseapproachesmaybereasonableonesformodelingcontinuousdryingbutnotintermittentdrying.References
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