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Computer-aided modeling of controlled release through surface erosion with and without microencapsulation

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Title:
Computer-aided modeling of controlled release through surface erosion with and without microencapsulation
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Book
Language:
English
Creator:
Wong, Stephanie Tomita
Publisher:
University of South Florida
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Tampa, Fla.
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Subjects / Keywords:
Encapsulation
Release rate
Mathematical modeling
MATLAB
COMSOL
Dissertations, Academic -- Chemical and Biomedical Engineering -- Masters -- USF   ( lcsh )
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bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

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Abstract:
ABSTRACT: Predictive models for diffusion-controlled particle dissolution are important for designing advanced and efficient solid products for controlled release applications. A computer-aided modeling framework was developed to derive the effective dissolution rates of multiple particles as the solid surface material eroded gradually into the surrounding liquid phase. The mathematical models were solved with numerical methods using the computational software MATLAB. Results from the models were imported into COMSOL Script to create three-dimensional plots of the particle size data as a function of time. The release model found for the monodispersed particles was manipulated to incorporate polydisperse solids, as these are found more frequently in chemical processes. The program was further developed to calculate the particle size as a function of time for particles encapsulated for use in controlled release. The parameters, such as radius size, coating material and encapsulation thickness, can be altered in the computer models to aid in the design of particles for different desired applications. Simulations produced conversion profiles and three-dimensional visualizations for the dissolution processes. Experiments for the dissolution of citric acid in water were performed using a reaction microcalorimeter to verify results found from the computer models.
Thesis:
Thesis (M.S.C.H.)--University of South Florida, 2007.
Bibliography:
Includes bibliographical references.
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Mode of access: World Wide Web.
Statement of Responsibility:
Stephanie Tomita Wong.
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Title from PDF of title page.
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Document formatted into pages; contains 155 pages.

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aleph - 001956427
oclc - 247413942
usfldc doi - E14-SFE0002206
usfldc handle - e14.2206
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Computer-aided modeling of controlled release through surface erosion with and without microencapsulation
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ABSTRACT: Predictive models for diffusion-controlled particle dissolution are important for designing advanced and efficient solid products for controlled release applications. A computer-aided modeling framework was developed to derive the effective dissolution rates of multiple particles as the solid surface material eroded gradually into the surrounding liquid phase. The mathematical models were solved with numerical methods using the computational software MATLAB. Results from the models were imported into COMSOL Script to create three-dimensional plots of the particle size data as a function of time. The release model found for the monodispersed particles was manipulated to incorporate polydisperse solids, as these are found more frequently in chemical processes. The program was further developed to calculate the particle size as a function of time for particles encapsulated for use in controlled release. The parameters, such as radius size, coating material and encapsulation thickness, can be altered in the computer models to aid in the design of particles for different desired applications. Simulations produced conversion profiles and three-dimensional visualizations for the dissolution processes. Experiments for the dissolution of citric acid in water were performed using a reaction microcalorimeter to verify results found from the computer models.
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Computer AidedModeling of ControlledRelease throughSurfaceErosion withandwithoutMicroencapsulation by StephanieTomitaWong Athesissubmittedinpartialfulfillment of therequirementsforthedegreeof MasterofScienceinChemicalEngineering DepartmentofChemicalEngineering CollegeofEngineering UniversityofSouthFlorida MajorProfessor:AydinKSunol,Ph.D. JohnT.Wolan,Ph.D. SerminG.Sunol,Ph.D. Date ofApproval: November2,2007 Keywords: encapsulation,releaserate,mathematicalmodeling,MATLAB,COMSOL Copyright2007,StephanieTomitaWong

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Dedication Thisthesisisdedicatedtomyparentswhoselove,guidanceand sacrificehavegivenme theopportunitytolivemydreams. Thisthesisisalsodedicatedin loving memoryofmyGrandmotherMay.Eventhough shewasnotabletoseethecompletionofthisthesis,shealwaysknewIwouldfinish.

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Acknowledg ments Fouryearsago,IreturnedtoschoolunsureofwhetherIhadtheabilitytocompletemy academicgoals.Itwasonlywiththesupportandhelpofaspecialfewthatthisthesiscouldbe completed. MythanksandappreciationtoDr.AydinSunoland Dr.SerminSunolfortheir encouragementandguidancethroughoutthetimeittookformetocompletethisresearchandto writethisthesis.Mycommitteemember,Dr.JohnWolan,hasprovidedyearsofinspiration.His ntificlibertyandkindnesstothosearoundhimhasfilledme withadmiration.ManythanksgotothepastandpresentmembersoftheEnvironmentally FriendlyEngineeringSystemsresearchgroup,whosharedtheirknowledgeandexperiencewith me:Dr.Navee dAslam,RaquelCarvallo,KeyurPatel,Hait a oLi,andWadeMack.Special thanksisgiventoBrandonSmeltzer,whoseconstanthelpthroughouttheseyearshasbeen immeasurable.IalsoappreciatethemanycontributionsfrommembersofChemicalEngineering DepartmentatUSF:Dr.VinayGupta,Dr.Scot tCampbell,Dr.BabuJoseph,Ms.CarlaWebb, Ms.CayPelaez,Mr.EdVanEttenandMr.JamieFargen.Iwouldalsoliketoacknowledgethe useoftheservicesprovidedbyResearchComputingattheUniversityofSouthFlorida. Finally,Iw ouldliketothankmyfamilyandfriendstowhomthisthesisisdedicatedto. Myfatherhas always encouragedandmotivatedmetolearnandhasbeenmysoleinspirationto thinkingcritically.Hehasinstilledwithinmetheimportancemakingmeaningfulc ontributions usingtheseskills.Mylifewouldnotbecompletewithouttheunconditionalsupportandloveof mymother.Herdedicationandkeenperspective area constantinspirationtome.Ialsohave deepappreciationformysister,whoisthemostam azingpersonIknow.Shecontinuestofillmy lifewithcompassionasmyclosestconfidantandwithhappinessasmybestfriend.Iam especiallygratefultoSharon,whoseexceptionalfriendshiphaslastedthroughoutthemost dynamicyearsofmylife.Th isthesisisalsodedicatedtoDat,whoseenduringsupport,patience andbeliefinmehasgivenmethestrengthtobecomewhoIam.

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NotetoReader Theoriginalofthisdocumentcontainscolorthatisnecessaryforunderstandingthedata. The original thesis isonfilewiththeUSFlibraryinTampa,F lorida

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i T ableofContents ListofTables ................................ ................................ ................................ ................................ ... v ListofFigures ................................ ................................ ................................ ................................ vi ListofSymbols ................................ ................................ ................................ .............................. ix ListofMATLAB/COMSOLScriptCodes ................................ ................................ .................... xi Abstract ................................ ................................ ................................ ................................ ......... xii Chapter1: Introduction ................................ ................................ ................................ .................. 1 1.1 Background ................................ ................................ ................................ ........... 1 1.2 Applications ................................ ................................ ................................ .......... 3 1.2.1 Pharamaceuticals ................................ ................................ ................... 3 1.2.2 Food s ................................ ................................ ................................ .... 3 1.2.3 Fertilizers,PesticidesandDetergents ................................ ................... 4 1.3 ReviewofDissolutionModels ................................ ................................ .............. 5 1.3.1 HistoricalBackground ................................ ................................ .......... 6 1.3.2 AdditionalConsiderations ................................ ................................ ..... 8 1.3.3 Computer AidedModeling ................................ ................................ ... 8 1. 4 Limitations ................................ ................................ ................................ .......... 10 1.5 PurposeofStudy ................................ ................................ ................................ 10 Chapter2: Overview ................................ ................................ ................................ .................... 12 2.1 GeneralDissolution ................................ ................................ ............................. 12 2.2 GoverningEquations ................................ ................................ ........................... 13 2. 3 ModelingAssumptions ................................ ................................ ........................ 14

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ii Chapter3: MonodisperseParticleDissolution ................................ ................................ ............ 15 3.1 MATLAB ................................ ................................ ................................ ............ 15 3.2 ProgramDesign ................................ ................................ ................................ .. 15 3.2.1 Start ingtheProgram ................................ ................................ ........... 16 3.2.2 DeterminationofDiffusionCoefficientforaSystem ......................... 17 3.2.3 CalculationoftheNumberofParticlesbasedonMass ...................... 17 3.2.4 RadiusRelationshiptoTime ................................ ............................... 18 3.2.5 RelationshipofRadiustoConcentrationandConversion .................. 19 3.2.6 VisualizationofResults ................................ ................................ ...... 20 3.3 AnExample ................................ ................................ ................................ ......... 20 3.3.1 SimulationSetup ................................ ................................ ................. 20 3.3.2 SomeResults ................................ ................................ ....................... 21 3.4 COMSOLScript ................................ ................................ ................................ .. 26 3.5 Visualization ................................ ................................ ................................ ........ 27 3.6 SummaryofResults ................................ ................................ ............................ 33 Chapter4: PolydisperseParticleDissolution ................................ ................................ .............. 35 4.1 ProgrammingModifications ................................ ................................ ................ 35 4.1.1 CalculationofNumberofParticleswith VaryingSizeDistribution .. 36 4.1.2 CalculationofConstantsforAllSizeDistributions ............................ 38 4.1.3 DeterminationofRadiiforeachGroupatSpecifiedTimes ................ 39 4.1.4 RelatingRadiustoConcentrationandConversionforEachGroup ... 41 4.1.5 VisualizingtheResultsofthePolydisperseModel ............................ 42 4.2 ExamplesusingthePolydisperseModel ................................ ............................. 42 4.2.1 SimulationforTestCases ................................ ................................ ... 42 4.2. 2 DeterminationofExperimentalRadiusSizeDistribution ................... 49 4.2.3 MATLABOutput ................................ ................................ ................ 54

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iii 4.2.4 Resul tsforVariousInitialConcentrations ................................ .......... 57 4.3 VisualizationforMultipleParticles ................................ ................................ .... 59 4.4 Summary ................................ ................................ ................................ ............ 66 Chapter5: Encapsul atedMonodisperseParticles ................................ ................................ ........ 67 5.1 ModelforEncapsulation ................................ ................................ ..................... 69 5.1.1 CalculationofDiffusionCoefficientfortheEncapsulatedLayer ....... 69 5.1.2 Determi ningtheRadiiofEncapsulated Layer ................................ .... 70 5.1. 3 ConcentrationandConversionofEncapsulationMaterial ................. 71 5.1. 4 VisualizationofResultsforEncapsulatedModel ............................... 72 5.2 ExampleforEncapsulatedModel ................................ ................................ ....... 72 5.2.1 SimulationSetupforEncapsulationModel ................................ ........ 72 5.2.2 EffectofEncapsulationThickness ................................ ...................... 74 5.2.3 COMSOLVisualizationforEncapsulatedModel .............................. 76 5.2.4 EncapsulationThicknessEffect ................................ .......................... 84 Chapter6: EncapsulatedPolysdisperseParticleDissolution ................................ ....................... 86 6.1 ProgramBuildUp ................................ ................................ ............................... 86 6.1.1 Cal culatingEquivalentNumberofParticles ................................ ....... 87 6.2 EncapsulatedPolydisperseParticleExample ................................ ...................... 88 6.3 EffectofEncapsulationThickness ................................ ................................ ...... 91 Chapter7: ResultsandDiscussion ................................ ................................ .............................. 99 7.1 ExperimentalValidation ................................ ................................ ...................... 99 7.1.1 ReactionMicrocalorimeterExperiments ................................ ......... 100 7.1.2 AdditionalExperimentalTests ................................ ......................... 105 7.2 InterpretingDiscrepancies ................................ ................................ ................. 111 7.2.1 ModelingAssumptions ................................ ................................ ..... 111 7.2.2 ExperimentalInaccuracies ................................ ................................ 114

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iv 7.3 EncapsulatedModels ................................ ................................ ......................... 115 7.3.1 SourcesofError ................................ ................................ ................ 117 7.4 Com parisonofMethods ................................ ................................ ................... 1 18 Chapter8: Conclusion ................................ ................................ ................................ ............... 119 8.1 Implications ................................ ................................ ................................ ....... 119 8.2 FutureWo rk ................................ ................................ ................................ ...... 121 References ................................ ................................ ................................ ................................ ... 124 Appendices ................................ ................................ ................................ ................................ .. 130 AppendixA: Derivat ionofDissolutionofSolidParticlesinaLiquid ................... 131 AppendixB: MATLABSampleSourceCode ................................ ....................... 137 AppendixC: COMSOLSampleSourceCode ................................ ....................... 142 AppendixD: ExperimentalProcedureusingReactionMicrocalorimeter .............. 144 AppendixE: SampleData ................................ ................................ ...................... 150

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v ListofTables Table3.1: Parametersforcitricacidusedintestcase ................................ ................................ 20 Table3.2: Resultsfromsimulationforvariousradiisizesfor0.10 grams citricacid ................ 21 Table4.1: Sizedistributionspercentsforfivetestcases ................................ ............................. 42 Table4.2: Averagemasspercentofeachradiussize. ................................ ................................ 54 Table4.3: Numberofparticlesineachradiussizedistribution. ................................ ................. 55 Table5.1: Parametersforglucosecoatingmaterial. ................................ ................................ ... 72 Table5.2: Effectofencapsulationthickn essonglucoseamountanddissolutiontime. ............. 84

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vi L ist of F igures Figure1.1 : Schematicillustrationoftheprincipleofsurfaceandbulkerosion[52]. ................. 2 Figure1.2: Commonactiveingredientapplicationv ersus controlledreleaseapplication[6]. ... 5 Figure1.3: Dissolutionofmonodispersepowder[11]. ................................ ............................... 7 Figure3.1: Programalgorithmformonodisperseparticlesmodel. ................................ ........... 16 Figure3.2: Monodispersemodelfor0.10gramscitricacidforvariousradiussizes. ............... 22 Figure3.3: Monodispersemodelfor0.1 0 gramscitricacidwithinitialradiusof0.059cm. ... 23 Figure3.4: Monodispersemodelfor0.10grams citricacid atdifferenttemperatures. ............ 23 Figure3.5: Monodispersemodelforcitricacidwithdifferentinitialamounts. ....................... 25 Figure3.6: Visualizationwithcorrespondingconversionata)t=0min,b)t=2mins, c)t=4mins d)t=6mins andt=8mins ................................ .............................. 28 Figure3.7: Visualizationof3Dparticleata)t=0min,b)t=2mins,c)t=4minsand d)t=6mins ................................ ................................ ................................ ........... 33 Figure4.1: Programalgorithmforpolydisperseparticlesmodel. ................................ ............. 36 Figure4.2: Schematicillustrationofpolydisperseparticledissolution. ................................ ... 37 Figure4.3: Conversionversustimegraphsandradiussizedistributionsfora)Test1, b)Test2,c)Test3,d)Test4 ande)Test5. ................................ ........................... 43 Figure4.4: Polydispersemodelforcitricacidatdifferentinitialamounts. .............................. 48 Figure4.5: Equivalentsphericalradius. ................................ ................................ .................... 49 Figure4.6 : SEMand particlenumberingforcollectionofequivalentareaforsix samplesa) f) ................................ ................................ ................................ ........... 50 Figure4.7: Radiussizedistributionsfor sixsamplesa) f). ................................ ....................... 52 Figure4.8: Averageradiussizedistributionofsixsamples usingequivalentareafromSEM. 54 Figure4.9: Concentrationversustimeforeachradiussizegroup. ................................ ........... 56 Figure4.10: Totalconcentrationchangeofpolydisperseparticles. ................................ ............ 56 Figure4.11: Totalconversionofpolydisperseparticles. ................................ ............................ 57 Figure4.12: Polydisperseconvers ionversustimeforcitricacidsatdifferentinitial concentrations. ................................ ................................ ................................ ........ 58

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vii Figure4.13: Polydisperseparticledissolutionfora)t=0min,b)t=2mins,c)t=6mins, d)t=10minse)t=15minsandf)t=20mins ................................ .................... 60 Figure5.1: Encapsulatedparticlea)fullviewb)sidesectionviewc)frontsectionview ........ 67 Figure5.2: Programalgorithmforencapsulatedmonodisperseparticlesmodel. ..................... 68 Figure5.3: Innerparticleradius,r 0, withencapsulationlayerthickness,h,yieldsradiusof encapsulatedparticler 0 enc ................................ ................................ ....................... 69 Figure5.4: Schematicillustrationforencapsulatedmo nodisperseparticledissolution ........... 70 Fi gure5.5: Concentrationversustimeforencapsulationthicknessof0.0010cm. ................... 73 Figure5.6: Conversionversustimefor encapsulationthicknessof0.0010cm ........................ 74 Figure5.7: Concentrationversustimeforencapsulationthicknessof0.010cm. ..................... 75 Figure5.8: Conversionversustimeforencapsulationthicknessof0.010cm. ......................... 75 Figure5.9: Encapsulatedparticleconversionfora)t=0min,b)t=1mins,c)t=3 mins, d)t=5 mins,e)t=7minsandf)t=9mins. ................................ ......................... 77 Figure5.10: Effectofencapsulationthicknessonthedelayedconversionofcitricacid. .......... 85 Figure6.1: Schematicillustrationforencapsulatedpolydisperseparticles. ............................. 86 Figure6.2: Programalgorithmforencapsulatedpolydisperseparticlesmodel. ....................... 87 Figure6.3: Concentrationchangesforencapsulationandinnerparticleforallsize distributions ................................ ................................ ................................ ............ 88 Figure6.4: Concentrationchangeforvarioussizedistributions. ................................ .............. 89 Figure6.5: Total concentrationversustimeforpolydisperseencapsulatedparticles. .............. 90 Figure6.6: Totalconversionversustime for poly disperseenc apsulatedparticles ................... 90 Figure6.7: Increasedencapsulationlayerthicknessforpolydispersemodel. .......................... 91 Figure6.8: Concentrationchangeversustimeforallsizedistributionswithincreasedcoating thickness. ................................ ................................ ................................ ................ 91 Figure6.9: Encapsulatedp olydisperse particledissolutionfor a) t= 0 mins ,b)t=2mins, c)t=4mins,d)t=6mins,e)t=8mins,f)t=10minsandg)t=12mins ......... 92 Figure7.1: OmniCalSuperCRCreactionmicrocalorimeter ................................ .................... 100 Figure7.2: Reactionmicrocalorimeterheatflowandconversiongraph ................................ 102 Figure7.3: Experimentaldataversusmonodispersemode lforvaryinginitialradii ............... 103 Figure7.4: Experimentaldataver suspolydispersemodelfo r0.10gramsofcitricacid ......... 104 Figure7.5: Comparisonofexperimentaldatafor0.10grams citricacidwithboth models .... 105 Figure7.6: Experimentalresultsford ifferentamountsofcitricacid ................................ .... 106 Figure7.7: Comparisonof experimentaldatafor0.02gramscitricacidwithbothmodels ... 107

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viii Figure7.8: Comparisonofexperimentaldatafor0.04 grams citricacidwithbothmodels ... 107 Figure7.9: Comparisonofexperimentaldatafor0.06 grams citricacidwithbothmodels ... 108 Figu re7.10: Comparisonofexperimentaldatafor0.08 grams citricacidwithbothmodels ... 108 Figure7.11: Comparisonofexperimentaldatafor0.20 grams citricacidwithbothmodels ... 109 Figure7.12: Comparisonofexperimentaldatafor0.30 grams citrica cidwithbothmodels ... 109 Figure7.13: Comparisonofexperimentaldatafor0.40 grams citricacidwithbothmodels ... 110 Figure7.14: Comparisonofexperimentaldatafor0.50 grams citricacidwithbothmodels ... 110 Figure7.1 5 : SEMphotosofnon sphericalcitricacidparticles. ................................ ............... 111 Figure7.16: Stirredexperimentaldataforcitricacidfordifferentinitialamounts .................. 113 Figure7.17: C omparisonplotforstirreddatausing0.10gramsofcitricaci d ......................... 114 Figure7.1 8 : Experimentalresultsforcitricacidencapsul atedwith0.01and0.10gramsof glucose. ................................ ................................ ................................ ................. 116 Figu re7.1 9 : Monodispersemodelforcitricacidencapsulatedwith0.01and0.10gramsof glucose ................................ ................................ ................................ ........ 116 Figure7. 20 : Polydispersemodelforcitricacidencapsulatedwith0.01and0.10gramsof glucose. ................................ ................................ ................................ ................. 117 Figure8.1: Comprehensiveflowsheetforallfourprogramsdevelopedinthiswork. ............ 120 Figure8.2: Othernon sphericalgeometricshapesofparticles ................................ ............... 122 Figure8.3: Surfaceareachangesbetweena)lowconcentrationandb)highc oncentrationof ................................ ................................ ........................... 123 FigureB.1: MATLABresultingplotforconcentrationformonodispersemodel .................... 140 FigureB.2: MATLABresultingplotforconversionformonodispersemodel ........................ 141 FigureC.1: CommandpromptwindowinCOMSOLScr ipt ................................ .................... 142 FigureC.2: COMSOLScriptvisualizationofsphericalparticle ................................ .............. 143 FigureD.1: WinCRCTurbomicrocalorimeterprogramsetupwindow ................................ ... 145 FigureD.2: Heatflowcurvefromreaction ................................ ................................ ............... 145 FigureD.3: Dynamiccorrectionusingheatcurveoption ................................ ......................... 146 FigureD.4: Applyingtaucorrectiontoreactionheatflowcurve ................................ ............. 147 FigureD.5: Trimmedcorrectedheatflowcurve ................................ ................................ ....... 148 FigureD.6: Integratedheatflowcurveandconversion ................................ ............................ 149

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ix L istofSymbols Symbol Description Unit A Area cm 2 C Concentration g/mol C Constant c(g) Constantforgroup g C o Initial concentration C s Saturationsolubility g/ cm 3 D Diffusioncoefficient cm 2 /sec D 12 Wilke Changdiffusioncoefficient cm 2 /sec H Encapsulationthickness cm H Thicknessofdiffusion layer cm G Group i,j CountervariablesinMATLAB K Dissolution rateconstant ,integrationconstant k(g) Integrationconstantforgroup g k 1 Brunneretal.dissolutionrateconstant k 2 Nieberalletal.square rootlawdissolutionrateconstant k 3 Hixson Crowellcubic rootlawdissolutionrateconstant M Massof particles remaining g M d Massdissolvedattime t g M o Initialmass g M t Dissolvedmaterialattime t g M T Totalmassofsample g M 2 Molecularweightofsolvent g/mol 2 Viscosityofsolventsolution centipoises N Numberofparticles N(g) Numberofparticlesingroup g P s (g) Percentofparticlesingroup g P v (g) Percentbyvolumeofeachsizegroup g Density g/cm 3 R Radiusofparticle cm r o Initialparticle radius cm r o (g) Averageradiussizeforgroup g T Temperature K T Time s W(g) Weightofeachparticledistribution W Particleweight g w 0 Initialparticleweight V Volumeofsphericalparticle cm 3 V(g) Volumeofsingleparticleingroup g

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x V m Dissolutionmediumvolume cm 3 V P Volumeofsingleparticle cm 3 V p (g) Volumepercentofsizedistribution V s (g) Volumeofeachsizedistribution V T Totalvolume cm 3 V 1 Molarvolumesoluteatnormalboilingpoint cm 3 /gmol X Conversion Associationparameterofsolvent Azimuthalcoordinate Polarcoordinate

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xi Listof MATLAB/COMSOL Script C ode s MATLABcode1: Calculationofconstantsand insertionintodissolutionfunction. ................. 19 MATLABcode2: Loopcreatedtosolveforradiusateachspecifiedtime. ............................... 19 COMSOLcode1: Parametricconversionofsphere ................................ ................................ .... 26 MATLABcode3: Calculatingconstantsforvarioussizegroups. ................................ .............. 39 MATLABcode4: Extendedfunctionforvarioussizegroups. ................................ ................... 40 MATLABcode5: Loopcreatedtofindradiusforallsizegroupsatgiventime. ....................... 40 MATLABcode6: Assignmentofmasspercentforeachradiussizegroup. .............................. 55

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xii Computer AidedModeling of ControlledRelease throughSurfaceErosion withandwithoutMicroencapsulation StephanieTomitaWong ABSTRACT Predictivemodelsfordiffusion controlledparticledissolutionareimportantfordesigning advanced andefficientsolidproductsforcontrolledreleaseapplications.Acomputer aided modelingframework wasdevelopedtoderivetheeffectivedissolutionratesofmultipleparticles asthe solid surfacematerialerodedgraduallyintothesurroundingliquid phase. The mathematicalmodelsweresolved with numericalmethodsusingthecomputationalsoftware MATLAB.Resultsfromthe modelswereimportedintoCOMSOL S cripttocreatethree dimensional plotsoftheparticlesizedataasafunctionoftime. Ther eleasemodelfoundforthe monodispersedparticleswasmanipulatedtoincorporatepolydispersesolids,asthesearefound morefrequentlyinchemicalprocesses.Theprogramwasfurtherdevelopedtocalculatethe particlesizeasafunctionoftimeforpar ticlesencapsulatedforuseincontrolledrelease.The parameters,suchasradiussize,coatingmaterialandencapsulationthickness,canbealteredinthe computermodelsto aidinthe design of particle sfordifferent desiredapplication s .Simulations producedconversionprofilesandthree dimensionalvisualization sforthedissolutionprocesses. Experimentsforthedissolutionofcitricacidinwaterwereperformedusinga reaction micro calorimetertoverifyresultsfoundfromthecomputermodels.

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1 Chapter1: Introduction 1.1 Background Controlledreleasetechnologiesareemergingasanovelsolutiontoproducingefficient specialtyproducts for applicationsextendingfarbeyondtheirconventionaluseinmedicine. The worldwidemarketfor controlledreleasematerial sinareassuchasagriculture, foodprocessing and other consumerproduct s is expectedtogrowtoanestimated$787millionby2012 [1] Researchanddevelopmentof controlledrelease systemshasconcentratedonsearchingfor innovationswhichimprovethe efficiencyofproducts whilestayingcosteffective Thedissolutionofsolidparticlesisfrequentlyencounteredint hechemicalprocessing industryandhasbeenstudiedquantitativelyforoveracentury [2,3] Pharmaceuticals,foods, fertilizers,pesticidesanddetergentsareexamplesofformulatedsolidproductswhoseapplication isreliantonthedissolutionbehavio r [4 7] Controlledrelease,theconceptofsustainingor prolongingthereleaseofbeneficialagentsforaspecifiedtimeisanovelapproachtoproduce safeandeffectiveusesofactiveingredients [8,9] Inthe1960 s technologyincontrolledrelease rapidlyevolvedasasolutionfor theapplicationofactiveagentswhichhaddesiredobjectives whileavoidingadversesideeffects [10] Withinthelastdecade,researchinthedissolutionof solidsubstanceshasincreasedwiththeadventofsophisticatedinstrumentationandcomputer access [11] Mathematicalmodelswhichquantitativelydescribethetransportmechanismsof dissolutionhavemanyapplications,includingaidinginthepredictionofreleaserate profiles, understandingtheeffectofimportantformulation/processingparametersandoptimizing advanceddeliverysystems [12] .Theknowledgeof thefundamentalfactorswhichinfluence dissolutionisimportantforboththemanufacturerandadministratorofsoliddosageforms [13] Utilizin gtheessentialmodelsfordissolution,advancedmodelsforcontrolledreleasecanbe developed.Controlledreleaseisusedinvariousfieldstosupplyaneffectiveamountofnecessary materialatadesiredtimeandismostcommonlyattainedusingencapsu latedparticles [14]

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2 Amongthemostpopularclassificationsofcontrolledrelease formulations are erodible devices [15] Thiscategoryisdefinedbythereleaseoftheactiveagentasthecarrieriseroded awaybythesurroundingenvironmentthoughphysicalprocesses [16] Adistinctadvantageof thisdeviceisthato vertime,thedevicedisappears intothesystemsandnoretrievaloftheremains isnecessaryafteractivation.Therearetwomainerosioncases,surfaceandbulkerosion F igure 1 1 illustratestheseprincipalcases .Insurfaceerosion,thesolidsurfacedegradesmuchfaster thantheliquidintrusionintothebulk.Therefore, theresultingdegradationoccursmainlyonthe outerlayers.Incontrast, inbulkerosion,particlesdegradeslowlywhiletheliquiduptakebythe systemoccursrapidly.Erosionoccursthroughouttheparticle,andisnotrestrictedtothesurface This paper focusesonsurfaceerosion,inwhichduringdegradation,thephysicalintegrityofthe particle,suchasdeviceshapeormolecularweight,ismaintained [17] Inthiswork,thevariationsintheapplicationofcontrolledreleasedevicesinthefieldsof pharmaceuticals,foodsandpesticideswillbereviewed.Then,themathematicalmodelsfo r dissolutionwillbesummarizedfollowedbyadiscussionontherecentadvancesusingcomputer aidedmodeling.Aswithanymathematicalmodel,limitationsbasedonthesimplifying assumptionsexist,andwillthenbediscussed.Throughthisexamination, theneedand importanceforthedevelopmentofadvancedmodelsforcontrolledreleasewillbedemonstrated. Surfaceerosion Bulkerosion F igure 1 1 : Schematicillustrationoftheprincipleofsurfaceand bulkerosion [52]

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3 1.2 Applications 1.2.1 Pharmaceuticals Betweenthe1960sandthe1970s,therewasasignificantmovementtoresearchand developmicroencapsulati ontechniques, resultinginnumerouspatentso nmicroencapsulation innovation [18] Microencapsulationinvolvessurroundingtinyparticleswithacoating,suchas sprayingapolymeronafluidizedbedofsoliddrugparticles [19] Utilizationofthesecoated particlesallowdrugreleasewithinthehumanbodytobecontrolledanddistributedoverabroad timeperiod,allowingtheactiveagenttobeabsorbedcontinuou sly [19] Controlledreleaseof drugshaveseveraladvantagesoverconventionaldosageforms,suchasavoidingdrugreleasein specificorgans [19] ,reduceddosefrequency,minimizingadversesideeffects,improved pharmacologicalactivityandprolongingaconstanttherapeuticeffect [20] In traditionaldrug deliverysystems,suchastabletsorintravenousinjections,the entiredoseisadministeredatone time.Thisresults insudde nly elevated,closetotoxic,concentrationsofthedrugintheplasma whichultimatelyleadtoadversereactions [21] Theshortdurationtimesr equirethepatientto inconvenientlyrepeattheadministration,resultinginstrongfluctuatingdruglevelsinthebody. Incontrast,controlledreleaseofferasystematicreleaseoftheappropriatedrugconcentrationfor asustainedtimeperiod.Bypro vidingthedrugonlywhereandwhenitisneeded ,drugdeliveryis morepredictableandefficient [21] Erodibledevicesareespeciallyusefulinpharmaceuticalsas nosurgeryisrequiredtoremovethedevicefromthebodyafterthedrugisdepleted [17] 1.2.2 Foods Encapsulationdevelopedinthefoodindustryasatechniquetoprotectmaterials,suchas foodingredientsorenzymes,frommoisture,heatorotherharshconditions [22] Oneofthemost commonapplicationsisfor theincorporationfunctionalfoods,ingredientswhichexhibit functionalbenefitsbeyondbasicnutriti on.Theseingredientscanbeusedtoimpartnegative propertiesliketaste,suchasbitternessoroxidation,orphysicaltexture,likesedimentationor phaseseparation [23] Flavorsareoneofthemostvaluableingredientsinfoodformulasandare aprimeexampleoftheutilizationofencapsulation .Foodmanufacturesareoftenconcernedwith protectingflavors,suchasaromasubstanceswhichare expensiveandareusuallydelicateand volatile.Coatingtheactiveingredientcanenhancethestabilityandviabilitybyproviding additionalprotectionagainstevaporationoranundesiredreactioninthefood [5] Food manufacturersarealsoutilizingencapsulationinfoodstomaskodorsandtastes,sinceflavor

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4 controlisimportantinfoodqualityandacceptability. Encapsulationcanalsobeusedtorelease activeingredientsoverprolongedperiodsoftime,reducethelossofingredientsthroughcooking processes,andseparatereactiveorincompatiblecomponents [5] However,duetothehighcosts ofspecializedmanufacturingandunavailabilityoffood gradematerials,foodencapsulation techniqueshavebeenlimitedandremainasareaofneededresearch [22] 1.2.3 Fertilizers, P esticidesand D etergents Inagriculture,c ontrolledrelease techniquesareusedtoproduce accurate,reproducible andpredictablerate s ofadministrationoffertilizersandpesticides .Conventionally,fertilizers andpesticidesaredistributedinperiodicintervalswhich create sharprisesintheconcentration levels.Thehighconcentrations maycauseundesirablesideeffectstothetargetsiteofthesystem and/orthesurroundingenvironment.Followingtheinitialpeakoftheactiveagent,the concentration diminishesduetonaturalprocessessuchaseliminationfromthesystem, consumption ordeterioration [16] Inthecaseofpesticides,protectionfrompestsisrequiredfor extendedperiodsoftime.Sustainedreleasecanbeachie vedusingencapsulateddeviceswhich reduceboththeamountofpesticideusedaswellasthenumberoftimesitmustbeappliedtothe crop. Membrane regulateddevicesareusedtoslowlyreleasefertilizers andpesticides through theerosionofthe membrane .Thesedevicescanalsobef ormulatedtoincludecompoundswith lowwatersolubility,creatinglowdissolutionrates,andtheuseofnitrogenouscompoundswhich areactivatedbymicrobialaction [8] Figure 1 2 showstheimprovedapplicationusing controlledreleasecomparedtoconvent ionalmethodofdelivery.Asshown,theconventional applicationinitiallysurpassestoxicconcentrationlevelsandthendropovertimebelowthe minimumeffectivelevel.Incontrast,controlledreleaseprovidessustainedreleasewithinthe desiredconce ntrationrange [6]

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5 Controlledreleasemethodsarealsobeingapplied to manufactur e specializeddetergents. Encapsulationisusedtoprotectandcontrolthedeliveryoffragrances andsofteners toclothesas theyarebeingwashed.Techniquesforencapsulationarecurrentlybeingdevelopedwhichaimat protectingvolatilefragrancemate rialsindryenvironmentsand thenreleasingthemwhen moistureispresent [24] Automaticdishwaterdetergentsarealsobeingencapsulatedtorel ease rinseaids,watersofteners,andshineadditivesatelevatedtemperatures [24] Formulatedmodels forcontrolledreleaseforfertilizers,pes ticidesanddetergentsareimportantforpredictingthe deliveryoftheactiveingredientsaidinthedesignandmarketingoftheseproducts [6 ] 1 .3 ReviewofDissolution Models Dissolutionresearchhasbeendevelopingforaboutacentury,andsincethen,several approacheshavebeenusedtoevaluatethereleaserateanddissolutionbehaviorofsubstances. Mathematicalmodelsfordissolutionhave beendevelopedtoaidindesignofmoresophisticated andeffectivesolidproducts.Theprinciplesoftheactualprocessesoccurringatthemicroscopic levelmustthereforebeunderstoodinordertoachieveaccuratedissolutionmodelswhichcanbe applied incontrolledreleaseresearch. Moresophisticatedmodelingandanalysis is necessary in thestudyof the dissolutionofparticles,becauseunliketablets,thesurfaceareaand/orshape Minimumeffective concentration Toxicconcentration Time Underdosing Overdosing C o n c e n t r a t i o n Figure 1 2 : Common activeingredientapplicationversus controlled releaseapplication [6]

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6 changesasthedissolutionprocessproceeds. Systems w heresurface recedeswithtime,also classifiedas movingboundaries problems ,aresolvedusing numericalmethods [25] 1.3.1 HistoricalBackground Thebasicdiffusion controlledmodelforsolid dissolutionwasdevelopedin189 7by NoyesandWhitney.The equationstatedthat thedissolutionrateisproport ionaltothedifference betweentheinstantaneousconcentration, C attime t, andthesubstancessaturation solubility T he changeinconcentration d C /dt, canbewrittenas ) ( dt dC C C k s (1.1) where k istheintrinsicdissolutionrateconstant.Generally,thisexpressionstatedthattherateat whichthesolidsubstancedissolvedinitsownsolutionisproportionaltothedifferencebetween the concentrationofthesolutionandtheconcentrationofth esaturatedsolution [26] Theequationfordissolutionwasexpandedin1900,whenresearchbyErichBrunnerand StanislausvonTolloczkoshowedt hattherateofdissolutiondependsofthesurfaceareexposed, thestructureofthesurface,thestirringrate,temperatureandarrangementoftheapparatus [27] TheNoyes Whitneywasmodifiedbyallowing k=k 1 A ,where A isthesurfacearea.The model wasexpressed inE q (1.2). ) ( dt dC 1 C C k s (1.2) Thenin1904,BrunnerworkedwithWaltherNersttoincludespecificrelationsbetweenthe constants [28,29] Brunnerequationwasderivedbyletting k 1 =D/(Vh) ,whereDisthediffusioncoefficient,histhe thicknessofthediffusionlayerand V isthedissolutionmediumvo lume.Theequationcanbe writtenintermsofthechangeinconcentration,suchas ) ( dt dC C C Vh DA s (1.3) orintermsofchangeinmassofsolidmaterial, M asshownin Eq (1.4). ) ( dt dM C C A h D s (1.4)

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7 Modelingofdissolutionkineticsofsolid particlesinaliquidisoftendescribedusingtheNerst Brunnerequation ,astheconceptofadiffusionlayerofliquidonthesolidsurfaceallowsthe complexdissolutionprocesstobeanalyzedinatractablefashion [3] In1931,HixsonandCrowell developedanotherdiffusion controlledmodelforsingle sphericalparticledissolutionundersinkconditions.Hixs onandCrowell expressedthesurface area intermsofparticleweight, w usingtheassumption thatthesurfacearea, A ,wasproportional to [30] WhenthisassumptionisappliedtoEq. (1.2),integrationoftheexpressionyields whatis knownasthecubic rootlaw,statedas t k w w 3 3 / 1 3 / 1 0 (1.5) where w 0 istheinitialweightoftheparticleand k 3 isaconstant. Accordingto resultingequation the cubicrootoftheweightoftheparticleislinearwith theslope k 3 ,whichisapropertyofthe solidandhydrodynamiccharacteristicofthereleasesystem HixsonandCrowellderivedthis equationforsingleormonod ispersesphericalparticlesundersinkconditions.Monodisperse particles,which are particleswiththesameinitialradius,undergodissolutionatsamerateas shownin Figure 1 3 Thecubic rootlawisthewidelyacceptedandmostcommonlyused dissolutionmodelbecauseofitssimplicityandgeneralapplicabilitytoawiderangeofparticulate studies [31] Time Zerot(1)t(2)Dissolutionor CriticalTime Zero Zero Zero Figure 1 3 :Dissolutionofmonodispersepowder [11].

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8 A semiempirical expressionwasdevelopedbyNiebergall,MilosovichandGovanin1963which relatedtimewiththesquare rootoftheparticleweight [32] Thiseq uationwaswrittenas t k w w 2 2 / 1 2 / 1 0 (1.6) Throughtheirresearch,theyhadfoundthatarelativelyconstantdissolutionconstant, k 2 ,couldbe foundbytakingthesquare rootoftheundissolveddissolutionprofiles.Theyals osuggestedthat thediffusion layer thicknessmayhaveasquare rootdependenceontheparticlesize. In1961,Higuchipublishedamathematicalmodelwhichdescribedthereleaseofasolute inadiffusion controlledsystem.Higuchibasedhi sanalysison thepseudosteady stateofthe kineticreleaseofadrughomogeneouslydispersedinaplanarmatrixintoamediumunderperfect sinkconditions [33] Themodelstates t K M t (1.7) where M t isthedissolvedmaterialatagiventime, t ,and K iskineticconstantfortheHiguchi model,representedasacompositeconstantwithdimensiontime 1/2 TheHiguchimodel agrees well withtheexperimentaldata [31] .However, g iventheassumptionsusedtoderivethismodel,itis recommendedtoonlybeusedfortheinitial 60%ofthereleasecurve s [34] 1.3.2 AdditionalConsiderations Polydispersemedia,characterizedby awidespectrumofparticlesizes,requirethe populationdistributiontobeincorporatedintothemodelfordissolution. Coatedparticlesalsoaddtothecomplexityofthemodel.Giventhediffusioncoefficient, thereleaserateasafunctionofpart iclesizeandcoatingthicknesscanbedetermined. 1.3.3 Computer AidedModeling Theincreasinglycomplicatedmathematicalmodelsfordissolutioncanthereforebestbe solvedwith numericalmethodsimplementedin computationalsoftware.Themanagementof variables,dataimport,calculations,visualizationandfiledevelopmentinvolvedinthe quantitativeanalysiscanbeeasilyhandledwithcomputerprogrammingthemodelsfor dissolution. Computer aidedmodelinghasbeenanattractivetoolinsolvingrel easemodelsduetoits abilityhandlecomplexsystems.AnumericalsolutionwaspresentedbyMaugerandHowardin

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9 1976,inwhichaSystem360 Continuous SystemModelingProgram(CSMP)wasdevelopedto solvetheHiguchi Hiestandequationsforalog normal distribution [35] .TheCSMPwasusedto solvetime variantproblems,wherethelimitsofintegrationwherezerototime t orzeroto infinity.Ho wever aftertheCMSPwasdevelopeditalsocouldbeeasilyappliedtosolve statisticalpopulationswithvariouslimitsdeterminedbythespecificpopulationbeingstudied,for example,aparticle sizepopulation. Withinthelastdecade,computer aided dissolutionresearchhascontinuedtoevolve.In 2004,FrenningdevelopedaseriesofFORTRANroutinestosolvecoupledpartialdifferential equations(PDEs)usedtodescribethedrugreleaseanddissolutionprocesses [36] The FORTRANroutine,providedbyTheNumericalAlgorithmsGroupintheUnitedKingdom, performedspecialdiscretizationusingfinitedifferencestoreducethePDEstoasystemo f ordinarydifferentialequations(ODEs).Thenusingabackwarddifferentiationformulamethod, theresultingODEsystemwassolved.TheworkdonebyStepanekin2004,alsodemonstrated novelcomputer aideddesignmethodologyfordissolutionstudies.St epanekconducteda systematiccomputationalstudywhichrelatedthegranulestructuretodissolutionbehavior [7] The effectsofgranulemicrostr uctureandingredientpropertieswere investigatedanditwas discoveredthatthereleaseratecouldbefine tunedtoadesiredreleaseratebycontrollingthe granuleporosityandbinder solidsratio.In2005,Muro Suneetal.developedpredictivemodels utilizingacomputer aidedmodelingframeworktoanalyzethereleasemodelsofpesticide products.Theresearchhighlightedthebenefitsofincorporatingcontrolledreleasemodelsintoa computerplatform.Thisincludedtheabilitytogenerateandtest variousformulationsofthe product,priortoperformingthefinalstepsexperimentally [6] Researchincomputer aidedreleasemodelingsuggests acontinuedneedforthe developmentofpredictivemodels.Thedevelopmentofcomputerprogramswhichincorporate dissolutionmodelsallowsforextendedanalysisofcomplexsystems,sincethesolutions generatedforreleasemodelscanbemodifiedforvar iousapplications.Intheareaofcontrolled release,computermodelscanoptimizeproductformulationbyidentifyingtheeffectsofthe physicalparametersoftheactiveingredientandpredictingtheresultsofmultiplecoatinglayer alternatives.These advancementsinvokethepossibilityofrunningvirtual( insilico )experiments ratherthanphysicalones.Virtualdissolutionexperimentscanbeperformedquicklyandhavethe unprecedentedadva conditionscanbecompletel ycontrolled [7] Advancednumericalmethodsarepresentingnewanalyticalshort timeapproximationsfor

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10 dissolutionmodels,providinganincreasin glymoredescriptiveanalysisthanavailablefrom modelformulationsdevelopedpreviously [37] 1.4 Limitations Despitethesignificantprogressi nthedevelopmentofdissolutionmodels,discrepancies betweentheoryandexperimentaldataarestillpresent [3] .Researchhasnotdistinguished whethertheproblemsoriginatefromexperimentalfactorsorlimitationsinthemathematical models.Modelsformulatedthroughmathematicalmodelingarealway sbasedonsimplifying assumptions,whichineffect,influencetheaccuracyofthemodel [38] .Forexample,therelease ofadrugofteninvolvest womechanisms,diffusionanddissolution.However,sincethe verificationofamodelinvolvingbothmechanismswouldbecomplex,mathematicalmodelsare oftensimplifiedbymodelingonlydominatingmechanismwhileignoringtheotherless predominantone [39] Polydispersityandencapsulationfurthercomplicatethemodelsfordissolution.Inthe caseofpolydispersesolids,assumptionswhichignore thesizedistributionsoftheparticleslead tosignificanterrorsindissolutioncalculations.Heterogeneoussystems,suchasthosewith multipleparticlesizes,needspecialconsiderationinthedissolutionmodelingprocess [40] .The incorporationofencapsulatedparticlesintodissolutionmodelsalsodemandsspecialized attention.Inthepreparationofthecoatedparticles,itisnecessarytob alancethereleaseratewith anappropriatecoatingthickness.Designinganeffectiveencapsulatedparticlerequires accountingfortheactiveingredientpresent,themechanismofrelease,andthefinalfateofthe combinedingredients [41] 1.5 PurposeofStudy Aspreviously stated ,dissolutionmodelinghasapplicationsinawidevarietyoffields, makinginanimportantareaofresearchfor advancedparticulatetechnology.Historically,there havebeenseveralclassicaldissolutionrateexpressionswhichhavebeenusedtointerpretparticle dissolutionratephenomena,mostcommonlydescribedbytheNoyes Whitney,Nernst Brunner andHixson Cr owellequations.Whileeachofthesemodelsprovidesinsightintothedissolution ratebehavior,limitationsbasedsimplifyingassumptionsarestillpresent.

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11 Inaddition,polydispersesolidsandencapsulatedmaterialssignificantlycomplicatethe exist ingmod els.Forthatreason,computer aidedmodelingappearstobetheonlysuitable platformtoperformsuchvigorouscomputationsfoundintheseadvancedmodels. Thescopeofthisthesisinvolvesdevelopingageneralprogrammingcodetosolve complex time variantdissolutionproblemswhichcanbemodifiedandextendedforvarious applications.The computerprogrammingdevelopmentofthis projectwasstructuredasfollows: 1. DevelopageneralcodeusingtheprogrammingplatformMATLABtosolveth e dissolutionmodelformonodisperseparticles( Chapter3 ). 2. ModifythecodeinStep1formonodisperseparticlestoaccountforpolydisperse particles,wherevaryingparticlesizedistributionsarepresent( Chapter4) 3. ModifytheoriginalcodeinStep1 forthedissolutionmonodisperseparticlesto accountforencapsulatedparticles,inwhichtwodifferentmaterialsarepresent,the activecoreingredientandthecoatingmaterial ( Chapter5) 4. CombinethemodelsdevelopedinSteps2and3toproduceauni fyingprogram accountingforthedissolutionofpolydisperseencapsulatedparticles( Chapter6 ). Theaimofthisworkwastoincorporatethemostdescriptivemathematicalmodelfordissolution intotheinteractiveprogrammingenvironmentofMATLAB.Oncethiswasaccomplished, a secondsoftware,COMSOLMultiphysics Script ,wasutilized toproducevari ousgraphsand3 D visualizationsofthedissolutionprocesses. Theresultsofthecomputermodelswerethen comparedtotheexperimentalresultsfoundforthedissolutionofcitricacidinwaterusinga reaction micro calorimeterinChapter7.Thischapte rcomparesthe theoreticalresultstothose foundexperi mentally,anddiscussesthediscrepanciesfound.Thefinalchapter,Chapter8, examin estheimplicationsofcomputer aidedmodelingforparticledesignanddiscussesthe futureworkpossibleinthis area. Itwashopedthatthesimulationsdevelopedcouldbeusedin facilitatingthedesignofspecializedcontrolledreleasesystems,whosereleaseprofilewouldbe predictableandwhosecompositionparameterscouldbemanipulatedtoachievethemostopti mal design.

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12 Chapter2: Overview 2.1 GeneralDissolution Mathematicalmodelsforthedissolutionofsolidparticlesinvolveaccountingforthe complicatedchangesinthesurfaceareaand/orshape whichoccurduring dissolution [3] Solid particlesinliquidscanbemodeledusing Ner n st Brunner typekinetics [28] : ) ( dt dM C C A h D s (2.1) whereMisthemassofsolidmaterialattimet,kisthedissolutionrateconstant,Aisthearea availableformasstransfer,Disthediffusioncoefficientof thedissolvingmaterial,histhe diffusionboundarylayerthickness ,CistheconcentrationandC s istheconcentrationsolubility. Inaddition,thefollowingconsiderationswereusedtomodelthedissolutionof monodisperseparticles [42] : 1. Thesurfaceareaoftheparticleschangesastheparticledissolves. 2. Dissolutionofalltheparticlesinthesample contributes totheove rall concentration ofthesolute. 3. Thediffusionboundarylayerthickness hasbeenshowntodecreasefor particles belowacertainsize,dependingont hematerialdissolvingandthe dissolution conditions.Dissolutionismodeledmoreaccuratelywhen the bound arylayer thicknessduringparticulatediss olutionisapproximatedbythe particleradius

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13 2.2 GoverningEquations AgeneralmathematicalmodelwasderivedfromNer n st Brunnertypekineticswhich couldbeusedtopredictthetheoreticaltimerequiredfor dissolutionofmonodispersedparticles. Thesurfaceareaofasphericalparticleisgivenby 2 4 r A (2.2) whereristheradiusoftheparticle.Thevolumeofthesphericalparticleis 3 3 4 r V (2.3) Thechangeinvolumecanbe writtenas dr r dr A dV 2 4 (2.4) wheredVisthechangeinvolumeanddrischangeinradius. Themassoftheparticlesisgiven by V N M (2.5) density.Substitutioninto E q. (2.1)yields ) ( 4 4 2 2 C C r N r D dt dr r N dt dV N dt dM s (2.6) Cancelationofliketermsgives ) ( C C r D dt dr s (2.7) TheconcentrationCcanbe derivedfromamassbalancewhichfindsthetotalmassdissolvedata giventime m m m d V r N r N V M M V M C 3 3 4 3 0 3 4 0 (2.8) whereM d isthemassdissolvedatagiventime,M 0 istheinitialmass,Misthemassremaining, V m isthedissolutionmediumvolume,andr 0 isthei nitialpa rticleradius.ReplacingCinEq. (2.7)by(2.8)gives ) ( 3 3 4 3 0 3 4 m s V r N r N C r D dt dr (2.9)

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14 Rearran gement,showninfulldetailinAppendixA, yields dt V N D r c dr r m 3 4 3 3 (2.10) where 3 1 4 3 3 0 N V C r c m s (2.11) IntegrationofdifferentialEq. (2.10),whichdescribestherateofchangeoftheradiuswithrespect totimeleadsto t V N D k c c r c r c r c c r c dr r m 3 4 3 2 tan 3 1 ) ( ln 6 1 1 3 3 3 3 3 (2.12) where c c r c r c r c c k 3 2 tan 3 1 ) ( ln 6 1 0 1 3 0 3 0 3 (2.13) Thederived Eq. (2.12)describestherelationshipoftimeandparticleradiusandwillbeusedin dissolutioncalculations. Rearranging Eq. (2.12)providestherelationshipbetweentimeand radiusasshownin Eq. (2.14). m V N D k c c r c r c r c c t 3 4 3 2 tan 3 1 ) ( ln 6 1 1 3 3 3 (2.14) 2. 3 Modeling Assumptions Aswithallmodels,simplificationsandassumptionsweremadeinthederivationof mathematicalequationsdescribingthedissolutionofsolidparticlesinaliquid.Allparticlesare assumedtobesphericalinshape,whereallsurfaceareaof thesphereisexposedtotheliquid. Theparticlesareconsideredisotropicsphereswherethegeometricshapedoesnotchange.The assumptionisthatthesolutioniswellstirred,however,noconvectionforcesareaccountedforin themodel.Inadditio n,boththesolubilityanddiffusioncoefficientareassumedtoremainas constantsthroughouttheprocess.

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15 Chapter3: MonodisperseParticleDissolution 3.1 MATLAB MATLABwas selected asthe main program mingandcomputationalplatform forsolving themultipleequationsinvolvedin particledissolution.TheinteractiveenvironmentofMATLAB allowedforadvancedalgorithmdevelopment,managementofvariables,importandexportof data,numericcomputation ,dataanalysisandvisualization [43] Inaddition, MATLABha s the abilitytosolve t echnicalcomputingproblems muchfasterthan traditionalprogramming languagessuchasC, C++ orFortran [43] Thesoftwareenvironmentallowedcodes,filesand datatobemanagedwhileperformingadvancedmathematicalfunctionssuchaslinearalgebra, statisticsandnumericalintegration.MATLAB a lso support s thevectorandmatrixoperations neededtoimplementtheprogrammingcodeeffectively. 3.2 ProgramDesign Computationalmodelingofdissolutionwasachievedbydesigningapro graminMATLAB whichsolvedfor theparticleradiusasafunction oftimeasdevelopedinSection2.The propertiesforagivensubstanceweretobeinputtedintotheprogram,whichwouldthenbeused tocalculatevarioustransportphenomenaparameters,thatinturnwouldbeusedtosolvethe governingdissolutionequat ion.Th eprocessisoutlinedinFigure3.1 andwasimplementedin MATLABusingthefollowingalgorithm: 1. Firsttheprogramisstartedusingacallfunctionwhichinvokesthefirstm filetobegin calculations. 2. Basedontemperatureandpropertiesofsolu teandsolvent,thediffusioncoefficientis determined.

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16 3. Thetotalnumberofparticlesis calculatedusingthegiveninitial particleradius,totalgramsand densityofmateri al. 4. Usingthecalculatedvaluesfrom thepreviousprograms,theradiusat incrementedtimesisfoundusing theprocedureoutlinedinSection2. 5. Thecalculatedradius ateachgiven time isconvertedto concentration whichisthenchangedinto conversion. 6. Finally the resultsareplottedas concentrationandconversion versustime graphs.Additional visualizationusing3Dplotsare used toshowtheoverallchangeinparticlesizeovertime. Thefollowingsectionswilldiscusstheimplementationoftheparticledissolutionproblemin MATLAB,revealingtheprogrammingcodenecessaryforthecalculations.Thentheprogram willbeusedtos olvethespecificproblemofcitricacidparticlesdissolvinginwater. 3.2.1 S tart ingtheProgram Sincethisprograminvolved runningmultiplesubroutines,aninitialcallfunctionwascreated underthenameof Start .m .Thepurposeofthism filewas toorganizeandsequencethe followingsub programs: Thefirstprogramis DiffPart.m ,which calculatesthediffusion coefficient ,thentheprogram N umPart .m calculatesthetotalnumberofparticles .Thevalues fromtheseprogramsareimplementedinto Fin dRadius .m which findstheradiusateachgiven time .Next, C oncConv .m convertstheradiustoconcentrationandconversion .Thefinal Start Calculatediffusioncoefficient Calculatethenumberofparticles Findtheradius forgiventime Calculatetotalconcentrationchange andtotalconversion Plotresults Figure 3 1 ::Program algorithm formonodisperse particle s model.

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17 program P rintResults .m plotstheresultsingraphs Oncethem file Start .m iscalled,therestof theprogramsare runautomaticallyandinitiated. Thecompletesourcecodeforthemonodisperse particledissolutionmodelisgiveninAppendixB. 3.2.2 Determinationof D iffusion C oefficient foraSystem Thediffusioncoefficientofthesolidparticleintheliquidsolu tewasestimatedusingthe WilkeandChangcorrelation [45] asshownin Eq. (3.1). 6 0 1 2 2 8 12 2 1 ) ( 10 4 7 V T M D (3.1) M 2 2 isthe viscosityofthesolventsolution(centipoises),V 1 isthemolarvolumeofthesoluteatnormal boilingpoint(cm 3 3.2.3 Calculatio noftheNumberofParticlesbasedonMass Ingeneral,the numbersofparticlesinagivensampleisrarelyknown.Thisisduetothe small,sometimesmicroscopic,sizesoftheparticlesbeingexaminedaswellasthelargesample amountsusedinexperimentswhichdrasticallyincreasethenumberofparticlestoa namount whichisimpracticaltophysicallycount.Toobtainanestimatednumberofparticles,calculations involvingthevolumeoftheparticleandsamplewereused. Theparametersofr 0 ,theinitialradiusoftheparticle,M T ,thetotalmassofsamp thesubstancesdensitymustfirstbedefined.Thetotalvolume,V T ,ofthesamplecanthenbe calculatedas 1 T T M V (3.2) Thevolumeofasingleparticle,V P ,isthencalculatedusingtheinitialradiusandtheassumption thatthe particleissphericalinshape 3 0 3 4 r V p (3.3)

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18 Thenumberofparticles N,inthesampleisthenfoundbydividingthetotalvolumeofthe samplebythevolumeofasingleparticle asgiveninEq.(3.4). P T V V N (3.4) 3.2.4 Radius RelationshiptoTime The expressionrelatingtheradiusoftheparticleatagiventime wasderivedinSection2 andgivenas t V N D k c c r c r c r c c m 3 4 3 2 tan 3 1 ) ( ln 6 1 1 3 3 3 (3.5) Inordertosolvethegivenequation,thevaluefortheconstantcmustfirstbecalculatedusingthe provided r 0 initialparticleradius (cm) V m dissolutionmediumvolume (mL) C s solubilityofthe solute (g/mL) 3 ).T henumberofparticlescalculatedearlierin thissectionisthenusedintheequation 3 1 4 3 3 0 N V C r c m s (3.6) tosolvefortheconstantc. Usingtheconstantc,theconstantofintegration,k,canthenbedeterminedusingthe formula c c r c r c r c c k 3 2 tan 3 1 ) ( ln 6 1 0 1 3 0 3 0 3 (3.7) Oncetheconstantcandtheintegrationconstantkisfound, E q. ( 2.12 ) canbesolvedatvarious times. TheconstantsderivedarethenusedinthemainequationF,asshownin t V N D k c c r c r c r c c F m 3 4 3 2 tan 3 1 ) ( ln 6 1 1 3 3 3 (3.8) MATLAB canthenbeimplementedtosolveatvarioustimesoftforwhenthefunctionFequals zero,tocalculatetheradiusratthegiventime.

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19 %CALCULATIONOFCONSTANT,c %Solubilityofthesolute,Cs Cs=1.33;%g/mL %DissolutionMediumVolume,Vm Vm=1;%1mLH2O %Constantwithrespecttotime,c c=nthroot((r0.^3. (3.*Cs*Vm)./(N.*rho*4*pi)),3); %CONSTANTOFINTEGRATION,k k=(1./(6*c)).*log((c.^3 r0.^3.)/(c r0).^3.) (1./(sqrt( 3)*c)).*(atan((2*r0+c)./(sqrt(3)*c))); %FINALFUNCTIONRELATINGTIMEANDRADIUS F=(((1./(6*c)).*log((c.^3 r.^3.)./(c r).^3.) (1./(sqrt(3)*c))*(atan((2*r+c)./(sqrt(3)*c)))) k) (((D12*N*4*pi)*t)/(3*Vm)); M ATLAB code 1 :Calculationof constantsandinsertionintodissolutionfunction %Initialtimetostartloop t=0; %Loopsetfrom1 100 fori=1:100 %Assignvaluefortime Time(i)=t; %Solvefunctionforradiusateachgiventime CalR=fzero(@Function,r0); %Checkthatcalculatedradiusisnotnegative,andifitis,assign %theparticleradiustoequalzero. ifCalR<0 r(i)=0; else r(i)=CalR; end %Updateincrementt t=t+100; M ATLAB code 2 :Loopcreatedtosolveforradiusateachspecifiedtime. 3.2.5 Relationship of Radius to C oncentrationand C onversion Thederivedradiiarethenconvertedtoconcentrationandconversion using C oncConv .m makingthedataeasiertoanalyze. Theinitialmassoftheparticles M 0 ingroupg, canbe calculatedusingtheinitialradiusoftheparticleusingtheequation 3 0 0 3 4 r N M (3.9) Similarly,themassoftheparticleremaining ,M, atagiventi metcanbefoundusingequation 3 3 4 r N M (3.10) Therefore,themassofthematerialdissolved,M d ,iscanbefoundasthedifferenceoftheinitial massandthemassremaining.Thiscanbewrittenas M M M d 0 (3.11)

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20 Toconverttoconcentration,theequationfromSection2relatingconcentrationtomassof materialdissolvedinthedissolutionmediumvolumeisused. Concentration ata given timeis givenas m m m d V r N r N V M M V M C 3 3 4 3 0 3 4 0 (3.12) Theinitialconcentrationisdefinedas m V M C 0 0 (3.13) FinallytheconversionXcanbecalculatedby Eq.(3.14). 0 C C X (3.14) 3.2.6 Visualizationof Results Thefunction P rintResults .m firstconvertsthetimeinsecondstotimeinminutes.Then plotsofconcentrationandconversionversustimearedisplayed. 3.3 An E xample Thefollowingsectionwillusedthedesignedprograminthecasestudyofcitricacid particlesdissolvinginwa teratroomtemperature. 3.3.1 Simulation S etup Forthisteststudy,thesampleofcitricacidisassumedtobemonodispersedparticles, indicatingthattheparticlesareuniforminshapeandsize.Theparametersusedinthesimulation areastabulated inTab le 3.1. Table 3 1 :Parameter sforcitricacid usedintestcase Totalgramsofcitricacid Densityofcitricacid Molecularweightofsolvent Temperature Viscosityofsolution(solvent) Molarvolumeo fsolute Associationparameterofsolvent Solubilityofthesolute Dissolutionmediumvolume (water) M T rho M 2 T 2 V 1 C s V m =0. 1086 g =1.665g/cm 3 =18.015g/mol =298K =0.91centipoises =319.88g/cm 3 mol =2.6 =1.33g/mL = 1mL

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21 3.3.2 Some R esults Thesimulationwascarriedoutforvariousinitialradiisizesranging0.05 0.20cm.The numberofparticlesforeachsize,diffusioncoefficient,constantscandkarelisted inTable3.2. Theconversionofcitricacidforvaryingradiussizesw asplotted inFigure3.2. Table 3 2 :Resultsfromsimulationforvariousradiisizes for0.10 g rams citricacid r 0 (cm) N D 12 (cm 2 /s) C k 0.0 4 243 0.0896 2.2992 0.0 5 125 0.1120 1.8393 0.0 6 72 5. 2086 x10 6 0.1344 1.5328 0.07 45 0.1568 1.3138 0.0 8 30 0.1792 1.1496 0.1 0 16 0.2240 0.9197 Asexpected,theresultsshowthatforlargerparticleradii,thetimeforcompleteconversion (X=1)ismuchlonger.Theslopeisverysteepforthe particlesofsmallradii,indicatingthatfor particlesizes0.01 0.07cm,to taldissolutionoccursinunder 10minutes. Usingtheaverage radiussizeof0.059cm,theconversionversustimewasfoundforthegivenamountof0.1086g CA,asshownin Figure 3 3 Thesecondsimulationwasdonefortheinitialradiussizeof r 0 =0.06 cmfortemperaturesof298K,308Kand318K.Theprogramproducedresults seenin Figure 3 4

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Figure 3 2 : Monodispersemodel for0. 10 g rams citricacid forvariousradiussizes. 2 2

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23 Figure 3 3 : Monodispersemodel for0.1 0 g rams citricacid withinitialradius of 0.059cm. Figure 3 4 : Monodispersemodel for0.10 gramsofcitricacid atdifferenttemperatures.

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24 Thegraphshowsthatincreasedtemperaturesincreasetherateofdissolutionofthe particles,whichagreeswiththermodynamicpredictions. In creasingthetemperatureincreasesthe calculateddiffusioncoefficient.For298Kthediffusioncoefficientis5.2086cm 2 /s,whereasata hightemperatureof318Kthediffusioncoefficientisfoundtobe5.5581cm 2 /s. Thesimulationwasthenrunfortheaverageparticlesizeof0. 0 59cmforvaryinginitial amounts.Aplot ,Fig ure 3.4, wascreatedshowingtheconversionv ersus timeforamountsof 0.02 0.50gofcitricacid in1mLH2Oat298 K.

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2 5 Figure 3 5 : Monodispersemodel for citricacid with different initialamounts

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26 3.4 COMSOLScript COMSOLScriptwaschosenasanadditionalscriptinglanguage foritsadvanced technicalcomputingandvisualizationcapabilities [46] Thepowerfulmodelingcapabilities allowformoreadvancedvisualization ofthecalculateddatafoundfromtheMATLABprogram. AgraphicalfunctionwaswritteninCOMSOLScriptwhichconvertedtheradiusd atafoundin theMATLABintoa3D plotinanewfiguredisplaywindow. Thefirstpartoftheprogram parametricallyconver tsthesphereofradius, r ,centeredattheorigin.Thisisaccomplishedusing thefollowingequations forsphericalcoordinates (3.15) y= sin sin (3.16) (3.17) The parametricconversionand meshgrid areimplementedin COMSOL S criptusingthecodeseenin COMSOLcode1. phi=0:pi/20:pi; theta=0:pi/10:2*pi; [Phi,Theta]=meshgrid(phi,theta); %Nextweusetheparametrizationabove. X= r* sin(Phi).*cos(Theta); Y= r* sin(Phi).*sin(Theta); Z= r* cos(Phi); COMSOLcode1:Parametricconversionofsphere. Theradius,r,foragivetimeisfoundfromtheMATLABdatainthevariablevectorPr. Importingt heinitialradius, r0, time ,t, andparticleradiusforatallgiventimes ,Pr, into COMSOL S cript enablesthecreationof 3D animations showingthe changeinasphericalparticle asafunctionoftime. Asample sourcecodefordevelopingtheanimationinCOMSOL Script is giveninAppendixC.

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27 3.5 Visualization ThedatafromMATLABforthedissolutionofparticleswiththeaverageinitialradiusof r0 =0.0 59 cmwasimportedintoCOMSOLScripttocreate3Dvisu alizationsoftheprocessat varioustimes. The3Dplotsgeneratedareshownwiththecorrespondingconversionversustime graphsinFigure3.6for0.10gramsofcitricacidfrom0to8minutes.

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28 a) Figure 3 6 :Visualizationwithcorrespondingconversionata)t=0mi n,b)t=2mins,c)t=4mins, d)t=6mins and e)t=8mins.

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29 b) Figure3.6(Continued).

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30 c) Figure3.6(Continued )

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31 d) Figure3.6(Continued)

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32 e) Figure3.6(Continued)

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33 a) b) c) d ) Figure 3 7 :Visualizationof3Dparticle ata)t= 0 min,b)t= 2 mins,c)t= 4 minsandd)t= 6 mins. Theparticleiscompletelydissolved atabout t=7 .5 mins. COMSOLScriptenablestheuserto visuallymonitorthedecreasingradiussizeofthedissolvingparticleoveraperiodoftime.All particlesareassumedtobemonodisperse,andsothechangein asingleparticleisassumedto representthechangeofalltheparticlesinthesystem. 3. 6 Summaryof R esults SeveralassumptionsweremadeindevelopingtheMATLABcodeinthissection.First, theparticleswereassumedtobemonodispersed.Theini tialparticleradiusforthemonodisperse particleswasestimatedusingtheaverageinitialradiiofalltheparticles.Inaddition,theparticles wereassumedtobesphericalinshape,asthisgreatlysimplifiedthemathematicalcomputations.

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34 Second,th ediffusionlayerthickness oftheparticleisassumedtoremainconstantthroughoutthe particledissolution.Third,sinkconditionswereconsideredwhenanalyzingtheexperimental datainordertosimplifythemathematicalanalysisofachangingbulkso lutionconcentration. Lastly,parametersusedinthecalculations,suchasthediffusioncoefficientofcitricacidinwater wereassumedtobeconstantthroughouttheprocess. BasedontheBrunner theconstant k ,intheequationlumps theeffectsofexposedsurfacearea,rateofstirring, temperature,structureofthesurfaceandarrangementoftheapparatusontherateofdissolution [34] The resultsofthecomputermodelprovideareasonableestimateforthedissolutionrate foundexperimentally.Calculationsbasedonthetheoreticalmodelassumeidealconditions,such asevenexposedsurfaceareaoftheparticlestothesolv ent.Theseconditionsaremostlikelynot foundinreality,forinstance,particlesareoftenincontactwitheachother,whichreducesthe exposedsurfaceareaoftheparticlemakingtheprocessnon ideal.Theassumptionof monodisperseparticlesalsol imitstheaccuracyofthemodel,sincetheinfluenceofparticlesize distributionsareignored.Thisaspectwillbefurtherexplored inChapter4.

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35 Chapter4: PolydisperseParticleDissolution Inthechemicalprocessingindustry, the dissolutionofmostsolidparticlesinliquids involveawidedistributionofparticlesizes [2] Ithasalsobeenshowedthatmodelsformulated intermsoftheaverageparticlesizecanledtosubstantialerrorsi ncalculatingdissolution behavior [40] .TheprogramdesignfromChapter3modeledacollectionofmonodisperse sphereshavinganaverageinitial particlesize.Thischapterwillmodifytheexistingprogramto accountforthepolydispersityfoundinmostsolidparticlepopulations.Theresultsofthismodel willthenbecomparedtothosepreviouslygenerated. 4.1 Programming M odifications Themethodusedforcalculatingpolydisperseparticlesfollowedthesameprogramflow asdevelopedpreviouslywithmodificationsmadetoaccountforparticlesizedistributions. The samecallfunction Final.m wasusedto initiate thesamesub routinesdevelopedbefore. Modificationsweremadetocategorizetheparticlesintoseveralgroupsbasedontheiraverage particlesizes.Theprogramwasupdatedtoevaluateeachoftheseparticlesizecategories separately,allowingfordi ssolutiondatatobefoundforeachoftheindividualgroups.Thedata wasthencompliedsothattheoveralldissolutionbehaviorresultingfromallsizedistributions couldbeaccountedfor.

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36 Theprogramhasanalgorithm similartothemonodispersemodel developedinChapter3.Afterthe programisstarted,thediffusion coeff icientiscalculated Nextthe numberofparticles in aparticular size groupiscalculated.The nthe radiusis calculated foragiventimeandif additionalsizegroupsexist,theprogram isloopedtorecalculatethenumberof particles inthatgiven sizegroupand theradius isfound forthesametime. Aftertheradi i forallsizesare calculated,theconcentrationand conversionforeachgroupisfound.For agiventime,thetotalconcentrationand totalconversionisfoundbysumming thecontribu tionsfromeach ofthesize groups Thefinalplotsincludethe individualsizegroupconcentration changesaswellasthetotal concentrationandtotalconversionplots. Theprogramstructureisdiagramedin Figure4.1. 4.1.1 Calculationof N umberof P articleswith V arying S ize D istribution Theprogramfile PartDist.m wasmodifiedtoaccountforthesizedistributionchanges. Thiswasachievedbyseparatingtheparticlesintogroupsbasedontheparticleradiussize.Size rangesforthegroupswerede signated,alongwithaweightofparticlesinthatsizerange, indicatingthepercentofthetotalparticlesthatwereinthedesignatedgroup. Groupsare assignedintegervalues,beginningat1,andaredesignatedwiththevariableg. Foreachgroup, Start Calculatediffusion coefficient Forgivensize group calculate numberof particles Findtheradius forgiventime Calculatetotalconcentrationchange andtotalconversion Ifadditional sizegroups exist Plotresults Compute concentrationand conversionforeach sizegroup Figure 4 1 : Programalgorithmforpolydisperseparticle s model.

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37 th epercentsizeandaverageradiusareassignedasfollows : P s (g) isthe percentofparticleswith radiussizerangeingroupg and r 0 (g) isthe averageradiussizeforparticlesingroupg Next,itisnecessarytocalculatethevolumepercentofeachparticlesizedistributionin ordertofindthetotalnumberofparticlesineachgroup. Thepercentagesofparticleswitha givenaverageradiusareconvertedtovolumepercentswhicharethen usedtofindthepercent massineachsizedistribution.First,t hevolumeofasingleparticleineachgroupiscalculated usingtheaverageinitialradius ,r 0 (g), by 3 0 ) ( 3 4 ) ( g r g V (4.1) whereV(g)isthevolumeofaparticleingroupg.The volumepercentofeachsizedistribution V p (g), is givenby Eq.(4.2). ) ( ) ( ) ( g P g V g V s p (4.2) Thetotalvolume ,V T iscalculatedbysummingthevolumepercentsineachsizedistributionas expressedas Eq.(4.3). g i p T i V V 1 ) ( (4.3) Time ZeroCriticalTimet(3)Dissolution Time Zero Zero Zero Size r max r min Figure 4 2 : Schematicillustrationofpolydisperseparticledissolution.

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38 Thepercentby volume,P V (g),ofeach sizegroupisthengivenas Eq.(4.4). T p V V g V g P ) ( ) ( (4.4) Usingthispercentbyvolume,theweightofeachparticlesizedistribution,W(g),canbefound from Eq.(4.5). T V M g P g W ) ( ) ( (4.5) Convertingweight ofeachsizedistribution to volume ,V s (g)is achievedby usingEq.(4.6). 1 ) ( ) ( g W g V s (4.6) Dividingthetotalvolumeineachsizedistributionbythevolumeofasingleparticleinthegiven groupgivesthetotalnumberofparti cles,N(g).Thenumberofparticlesinagivensize distributiongroup isgivenbyEq.(4.7). ) ( ) ( ) ( g V g V g N s (4.7) 4.1.2 Calculationof C onstants forAllSizeDistributions Afterthenumbersofparticlesineachsizedistributionhavebeen calculated,theconstant cisrecalculatedforeachgroupusingtheformula showninEq.(4.8). 3 1 4 ) ( 3 ) ( ) ( 3 0 g N V C g r g c m s (4.8) Theintegrationconstantkisalsocalculatedforeachgroup givenbyEq.(4.9). ) ( 3 ) ( ) ( 2 tan ) ( 3 1 )) ( ) ( ( ) ( ) ( ln ) ( 6 1 ) ( 0 1 3 0 3 0 3 g c g c g r g c g r g c g r g c g c g k (4.9) Thecalculationsoftheconstantsforeachofthesizegroupsareimplemented intheprogramas showninMATLABcode3. Theinitialmass, M 0 (g) iscalculated foreachsizedistribution using Eq.(4.10). 3 0 0 ) ( 3 4 ) ( g r g M (4.10)

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39 Theinitial concentration,C 0 (g),iscalculated foreachsizedistribution isgivenbyEq.(4.11). m V g M g C ) ( ) ( 0 0 (4.11) Thet otalinitialconcentration,C 0 T ,isthesumofallsizedistribution asshowninEq.(4.12) g i OT g C C ) ( 0 (4.12) %CALCULATIONOF CONSTANT,c %Solubilityofthesolute,Cs Cs=1.33;%g/mL %Constantwithrespecttotime,c %Createsloopthelengthofthesizedistribution forg=1:length(Ps) %Checksifthepercentofthesizedistributionisgr eaterthan %zero,andifsocalculatestheconstantc ifPs(g)>0; c(g)=nthroot((r0(g)^3. (3.*Cs*Vm)./(N(g)*rho*4*pi)),3); %Ifthepercentinasizedistributioniszero,cisassignedthe %valueofzero else c(g)=0 end end %CONSTANTOFINTEGRATION,k %Createsloopthelengthofthesizedistribution forg=1:length(Ps) %Checksthatthepercentofthesizedistributionisgreaterthan %zero,andcalculatestheconstantofintegration ifPs(g)>0; k(g)=(1./(6*c(g))).*log((c(g)^3 r0(g).^3.)/(c(g) r0(g)).^3.) (1./(sqrt(3)*c(g))).*(atan((2*r0(g)+c(g))./(sqrt(3)*c(g)))); else %Otherwiseas signstheconstantofintegrationtobeequaltozero k(g)=0; end end %INITIALMASSOFPARTICLESINEACHSIZEDISTRIBUTION,Mo Mo=(r0.^3)*(4/3)*rho.*N*pi; %INITIALCONCENTRATIONOFPARTICLESINEACHSIZEDISTRIBUTION,Co Co=Mo./Vm; M ATLAB code 3 :Calculatingconstantsforvarioussizegroups. 4.1.3 Determinationof R adiiforeach G roup atSpecifiedTimes AloopinMATLABwasgeneratedtoincludeallsize distributionsgroupswhichsolved fortheradiusasafunctionoftime.ThegeneralequationforF(g),which againwassolvedfor

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40 theradius,r,whenitwasequat edtozeroatagiventime,t,isgivenbyEq.(4.13). t V 3 4 N(g) D k(g) c(g) 3 c(g) 2r(g) tan c(g) 3 1 r(g)) (c(g) r(g) c(g) ln 6c(g) 1 F(g) m 1 3 3 3 (4.13) Inthe MATLAB code,itwasnecessarytowritethefunctiongroupsusingthevariablej,sinceg hadpreviouslybeendefinedasalocalvariableandcouldnotbeusedasaglobalvariable. %TIMEASFUNCTIONOFPARTICLERADIUS F=(((1./(6*c(j))).*log((c(j).^3 r.^3.) ./(c(j) r).^3.) (1./(sqrt(3)*c(j)))*(atan((2*r+c(j))./(sqrt(3)*c(j))))) k(j)) (((D12*N(j)*4*pi)*t)/(3*Vm)); M ATLAB code 4 :Extendedfunctionforvarioussizegroups Dueto the increaseddatasize, amatrixwascreatedincludingthefollowing ) ( ) ( g F g t P r (4.14) whereP r (t,g)istheparticleradiusofthesizegroupgattimet.Theparticleradiusforeachgroup isthenusedtofindthemassandconcentrationatagiventime. %Creates looptoincludeallsizedistributions forj=1:length(r0) %Initialtimesettozero t=0; %Checksthatthepercentinsizedistributionisgreaterthanzero ifN(j)>0 fori=1:100 %Assignsvaluefortime Time(i)=t; %Calculatestheradiusateachtimegiven CalR(j,i)=fzero(@Function2,r0(j)); %Checksthatthevalueofradius check=isnan(CalR(j,i)); %Iftheradiusvaluecannotbefoundassignszerofort hemass %andconcentration ifCalR(j,i)<0|check==1 Pr(j,i)=0; M(j,i)=0; C(j,i)=0; else %Iftheradiusisfounditassignedastheparticleradiusandthe %massandcon centrationarecalculated Pr(j,i)=CalR(j,i); M(j,i)=(Pr(j,i)^3)*(4/3)*rho*N(j)*pi; C(j,i)=M(j,i)/Vm; end %Updatesthevalueoft t=t+10; end M ATLAB code 5 :Loopcreatedtofindradiusforallsizegroupsatgiventime.

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41 %Whenthepercentinsizedistributionislessthanzero else %Createsloopwhichassignszeroforthevaluesoftime,particle %radi us,massandconcentration fori=1:100 Time(i)=t; Pr(j,i)=0; M(j,i)=0; C(j,i)=0; t=t+10; end end end MATLABcode5(Continued). 4.1.4 RelatingRadiusto C oncentrationand C onversionfor E ach G roup Themassofsizegroup gattimetisgivenas M(t,g)as ) ( )) ( ( 3 4 ) ( 3 g N g t P g t M r (4.15) The concentration,C(t,g),for asizegroup g atagiventimeisfoundby m V g t M g t C ) ( ) ( (4.16) whilet hetotalconcentrationofthe solution,C T (t),iscalculates bysummingallsizedistributions ateachgiventime inEq.(4.17). g i T g t C t C ) ( ) ( (4.17) Theconcentrationofdissolved material,C d (t),atagiventime isfoundby ) ( ) ( 0 t C C t C T d (4.18) andt hetotalconversionisthenfoundby Eq.(4.19). 0 ) ( ) ( C t C t X d (4.19)

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42 4.1.5 Visualizing theResultsofthePolydisperseModel Thefunction PrintResults mfirstconvertsthetimeinsecondstotimeinminutes.Then plotsofconcentrationand conversionversustimearedisplayed. Thefirstgraphdisplaysthe concentrationversustimeresultsforeachsizedistribution. 4.2 E xample susingthePolydisperseModel Thefollowingsectionwilluse thedesignedprogramin several casestud ies ofcitricacid particlesdissolvinginwateratroomtemperature. 4.2.1 Simulation for T est C ases First,theMATLABcodewastestedtoseetheresultsofchangingtheparticlesize distribution.Theparametersusedforcitricacid( Table 3 1 )wereusedinthetestruns.Allcases usedaninitialamountofcitricacidof0.25g. Fivetestcaseswererun,eachwithdifferentsize distributionshapes ,asshownin Table 4 1 Table 4 1 :Sizedistributionspercentsforfivetestcases RadiusSize(cm) Test1 Test2 Test3 Test4 Test5 >0.04 16.67% 2.15% 34% 2.15% 34% 0.05 16.67% 13.6% 34% 2.15% 13.6% 0.06 16.67% 34% 13.6% 13.6% 2.15% 0.07 16.67% 34% 13.6% 13.6% 2.15% 0.08 16.67% 13.6% 2.15% 34% 13.6% <0.09 16.67% 2.15% 2.15% 34% 34% Test1hadanequalpercentsofallradiussizes,test2hadastandardnormaldistribution,test3 hadahighpercentofradiiinthelowrange,test4hadahighpercentofradiiinthehighrange andtest5hadaU shapeddistribution.Theperce ntsweregraphedtoillustratethesize distributionandtheresultsoftheconversion versus timegraphsfromMATLABwereplotted

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43 a) Figure 4 3 : Conversionversustimegraphs andradiussizedistributions fora)Test1,b)Test2,c)Test3,d)Test 4ande)Test5.

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44 b) Figure4. 3 (Continued)

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45 c ) Figure4. 3 (Continued )

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46 d) Figure4. 3 (Continued)

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47 e ) Figure4. 3 (Continued )

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4 8 Figure 4 4 : Polydispersemodelforcitricacidatdifferentinitialamounts.

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49 4. 2. 2 Determin ationofE xperimental R adius S ize D istribution Aspart oftheparticlesizeanalysis,particlesareestimatedtobesphericalinshapesothat anequivalentsphericaldiameter(ESD)canbederived [47 49]. Thecitricacidobta inedfrom Fis herChemicalshadcrystalgeometriesthatwerenotwelldefined,therefo re,itwasconvenient to estimatethestructureofthecitricacidusedintheexperiments intermsofequivalentspherical shape.Forthis analysis,theequivalentradiuswasdefinedastheradiusofaspherewithequal surfacearea.Themeasurementswe remadeusingascanningelectronmicroscope(HitachiS 800)toimagethesurfaceoftheparticlesofcitricacid. Imagesofs ixsamplesof<30particles were taken atamagnificationof13X. Shape Actual S hape Equivalentspherical shape Surfaceareaof shape Equivalentsphericalradiususingequivalenceby surfacearea Undefined geometry CrystalStructure S A r=(S A 1/2 Figure 4 5 : Equivalentsphericalradius

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50 a) b) c) Figure 4 6 : SEMandparticlenumberingforcollectionofequivalentareaforsixsamplesa) f).

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51 d) e) f) Figure4. 6 (Continued )

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52 a) RadiusSize Percent >0.04 11.4% 0.05 31.4% 0.06 34.3% 0.07 17.1% 0.08 5.7% <0.09 0% b) RadiusSize Percent >0.04 9.8% 0.05 31.7% 0.06 31.7% 0.07 22.0% 0.08 4.9% <0.09 0% c) RadiusSize Percent >0.04 4.8% 0.05 35.7% 0.06 40.5% 0.07 9.5% 0.08 4.8% <0.09 4.8 % Figure 4 7 :Radiussizedistributionsforsixsamplesa) f).

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53 d) RadiusSize Percent >0.04 3.0% 0.05 21.2% 0.06 35.4% 0.07 24.2% 0.08 12.1% <0.09 3.0% e) RadiusSize Percent >0.04 5.7% 0.05 42.9% 0.06 25.7% 0.07 14.3% 0.08 8.6% <0.09 2.9% f) RadiusSize Percent >0.04 11.1% 0.05 30.6% 0.06 27.8% 0.07 22.2% 0.08 5.6% <0.09 2.8% Figure4. 7 (Continued ).

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54 RadiusSize > 0.04 0.05 0.06 0.07 0.08 <0.09 Percent 7.7% 32.4% 32.9% 18.0% 6.8% 2.3% Figure 4 8 :AverageradiussizedistributionofsixsamplesusingequivalentareafromSEM. 4. 2 .3 MATLABOutput Sincetheaverageareaisnotausualmeasurementthatwouldbeknown,thiswas convertedtotheequivalentmasspercent.Thiswouldallowtheusertoadjusttheradiussize distributioninanexperimentalsetupbymeasuringthedesiredmassofcitricaci dfromagiven radiussize.Theequivalentmasspercentofthedistributionfoundin Figure 4 8 isgivenbelow. Table 4 2 :Averagemasspercentofeachradiussize. RadiusSize <0.04 0.05 0.06 0.07 0.08 <0.09 MassPercent 2.1% 17.8% 30.9% 26.9% 15.1% 7.2% Byrepresentingthedistributionasamasspercent,theusersimplyhastomultiplythe masspercentbythetotalinitialamountofcitricacidtofindouthowmuchcitricacidinagiven sizedistributionmustbeused. Themasspercentisinputtedinto MATLABinthe NumPart.m sectionoftheprogram andtheaverageradiusforeachsizedistributionisassigned.Theprogramisdesignedtoallowas manysizegroupsasnecessary.Forthisexamplesixsizegroupsaredefined.

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55 %PARTICLEDISTRIBUTION PERCENTANDAVERAGEPARTICLERADIUS %Percentofparticleswithradiussizelessthan400um(micrometers) Ps(1)=0.0214; r0(1)=0.04;%cm %Percentofparticleswithradiusbetweenthesizeof401 500um(micrometers) Ps(2)=0.1 768; r0(2)=0.05;%cm %Percentofparticleswithradiusbetweenthesizeof501 600um(micrometers) Ps(3)=0.3098; r0(3)=0.059;%cm %Percentofparticleswithradiusbetweenthesizeof601 700um(micrometers) Ps(4)=0.2696; r0(4)=0.07;%cm %Percentofparticleswithradiusbetweenthesizeof701 800um(micrometers) Ps(5)=0.1508; r0(5)=0.08;%cm %Percentofparticleswithradiusbetweenthesizeof801 1000um(mi crometers) Ps(6)=0.0716; r0(6)=0.1;%cm M ATLAB code 6 :Assignmentofmasspercentforeachradiussizegroup. Theprogramgeneratedforpolydisperseparticleswasmorecomplicatedbecauseit involved multipledatasetsforthedifferentsizegroups.Therefore,threegraphswereproduced tovisualizethechanges.Thefirstgraph Figure 4 9 showed theconcentrationchangeineach sizegroupasfunctionoftime. Thenumberofparticlesforeachgroupwascalculatedusingthe averagemasspercentsfoundinthesamplesofcitricacid andisshownin Table 4 3 Table 4 3 :Numberofparticlesineachradiussizedistribution. InitialRadius,r 0 Numberofparticles,N 5 401 22 501 23 601 12 701 5 1

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56 Figure 4 9 : Concentrationversustime foreachradiussizegroup Thesecondgraphrepresented thetotalconcentrationchange.Thiswascalculatedbysumming theconcentrationchangesfromeachoftheradiussizegroups. Figure 4 10 :Totalconcentrationchange ofpolydisperseparticles

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57 Thenbydividingthetotalconcentrationforagiventimebythetotalinitialconcentration, theconversion(X)oftheentiresystemasafunctionoftimewasfound. Figure 4 11 :Totalconversionofpo lydisperseparticles. The slopeofthe totalconversiongraph usingthepolydispersewasnoticeablylesssteep tha n the conversiong raphsproducedusing themonodispersemodel. For radiussizesoflessthan0.1 cm t hetimefordissolutiontocomplete wa salso longerthanfor themonodisperseparticlemodel. 4. 2 .4 Resultsfor V arious I nitial C oncentrations Theaverageradiussizedistributionwasappliedto calculatetheconversionfor initial concentrationsvaryingfrom0.2 0.5gCA. Thepercentofeachradiussizefoundin Figure 4 8 wasbasedontheequivalentareafoundfromtheSEM. Theprogramforpolydisperseparticleswasrunforinitialconcentrationsof0.02 0.50 gramsofcitricacid,thesameasdonepreviouslyforthemonodispersemodel.Thepolydisperse modelresultedinasmootherandmoreprolongedconversion. Forexample,completeconversion of0.10gramsofcitricacidinthemonodispersemodeloccurredat7.5minutes,whileforthe polydispersemodel,dissolutionwascompletein21minutes.Thepolydispersemodelresultedin anincreasefordissolutiontim eforallconcentrations,asseenin Figure 4 12

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5 8 Figure 4 12 : Polydispersec onversionv ersu stime forcitricacidsatdifferent initi alconcentrations.

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59 4.3 Visualizationfor M ultiple P articles TheCOMSOLscriptcodedevelopedinChapter3wasadju stedtosimultaneouslyoutput 3 Dgraphsforparticlesofsixdifferentinitialradiussizes.Forthetestcaseof0.10gramsofcitric acidandthesizedistributionfoundthroughSEMtechniques,thegraphsrepresentingthechange ofradiusforapolydisperseparticlesystemproduceda reshownin Figures4.1 3 1 8 Thegraphs showthechangesforthesixsizegroupsfromtheinitialtimeof0minutesto20minutes.Itcan beseenfromthepolydisperseresultsthattheradiussizehasasignificanteffectonthetime requiredfordissol ution.Whilethemonodispersemodelfor0.10gramsusinganaverageradius of0.059cmtookonly8minutesforcompletedissolution,thepolydispersemodelusingthe averagesizedistributionhasalongerdissolutiontimeofalmost20minutes.

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60 Figure 4 13 :Polydisperseparticledissolutionfor a)t =0min,b)t=2mins,c)t=6mins d)t=10minse)t=15 mins andf)t=20mins a)

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61 Figure4.13(Continued). b)

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62 Figure4.13 (Continued). c)

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63 Figure4.13(Continued). d)

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64 Figure4.13(Continued). e)

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65 Figure4.13(Continued). f)

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66 4. 4 Summary ByincorporatingtheradiussizedistributionintotheMATLABprogram,anincreasingly comprehensive codewasgenerated.Foreverysizegroupcreated,itwasnecessarytore runthe initialcode,producedinChapter3,usingtheradius sizeofthespecifiedgroup .Inaddition,the resultsfromallradiussizegroupsneededtobecombinedtode scribeth eoveralleffect.The polydisperseparticles model provided morerealisticresultsfordissolution,sincemostindustrial chemicalsinvolveparticlesizedistributions. Theprogram producedinthissectionalsohasadditionalbenefitsinoptimizationan d productdesign.Bymanipulatingthepercentagesofeachoftheradiussizedistributions,therate ofdissolutioncanbechangedsignificantly,evenwhentotalinitialweightremainsconstant.This allowsthemanufacturertodeterminethebestradiuss izetouseforagivenapplicationwhile keepingtheamountofresourcesusedtoaminimum.Whena desired dissolutionprofileis given theprogramcanbeusedtoback calculatewhatsizedistributionswillproducethedesiredeffect.

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67 Chapter5: EncapsulatedMonodisperseParticles Controlledreleasethroughencapsulationhasbeenstudiedforuseinpharmaceuticals [17] ,chemicals [2] ,cosmetics,foods [5] andpesticides [6] Thee ncapsulat ionof citricacidhas benefitsinmanufacturing,storageanduse.Theencapsulationcanincreasetheprocessingspeed oftheparticlesandallowsformoredurabilitybyhardeningtheoutershell.Theadditionallayer alsoprotectsthecore citricacid material fromundergoingundesiredreactionsduringstorage. Finall y,theencapsulationimprovesthefunctionalityofthecitricacidbycontrollingoftheacid releaseallowingforthecreationproductswhichcanbetailordesiredtomeetthemanufacturers needs Forsimplificationpurposes,theencapsulatedparticles inthissectionare assumedtobe monodispersed. Thecorematerialconsiderediscitricacidandisdesignatedastheredmaterial in Figure 5 1 .Glucosewaschosenasasuitablecoatingmaterialandisrepresentedusingthe colorgreen.Sincethisapplicationconsideredthedissolutioninwater,glucosewasan appropriatecoatingmaterialbeca useitwasawatersolublecarbohydrate whichcouldbe manufacturedusingspraydriedglucosesyrup.Glucoseisalsoanapprovedsweeteningfood additive,makingitasafechoiceforuseapplicationsinthefoodindustry.Processedcitricacid hasfuncti onalbenefitsinpharmaceuticals,foodapplications,healthcareanddetergents. a) b) c) Figure 5 1 :Encapsulatedparticlea)fullviewb)sidesectionviewc)frontsectionview

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68 Themathematicalmodelusedto describethedissolutionofencapsulated monodisperseparticlesissimilartheone developedinChapter3.Sinceallthe particlesareassumedtobe initiallyth e same sizeandhavethesameencapsulation thickness,thedissolutionofalltheparticles occursatthesametime,asseenin Figure 5.4 Thee ncapsulationlayer complicates themodelbyrequiringtwosetsofchemical propertyparameters,oneforthe e ncapsulationlayerandthesecondforthe innerparticle.Calculationsfortheeroding encapsulationlayerareperformedfirstuntil allofthecoatingisdissolved.Thenthe innerparticlecalculationsareperformedin amannerresemblingthemethoddev eloped formonodisperseparticledissolution The concentrationandconversion mustbecalculatedtwice,onceforthe encapsulationmaterialandthesecondfor theinnerparticlematerial.The concentrationandconversionsforboth materialswillbeto taledandplottedasa functionoftime.Theprogramalgorithmis showninFigure5.2. Calculatediffusioncoefficientfor encapsulationmaterial Findradiusof encapsulatedparticle forgiventime Ifencapsulated radius < particle radius Findtheinnerparticle radiusforgiventime Ifencapsulated radius > particle radius Plotresults Calculatethenumberofparticles Calculatediffusioncoefficientfor innermostparticle Start Calculateconcentrationchangeand conversionforinnerparticleand encapsulation Calculatetotalconcentrationchange andtotalconversion Figure 5 2 :Programalgorithmforencapsulated monodisperseparticlesmodel.

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69 5.1 Model for E ncapsulation In thissection,theprogramdesignedinChapter3wasextendedtoincludecalculations foranerodiblelayerencapsulatedaroundtheoriginalsphericalparticle.Similarassumptionson thederivationonthedissolutionkineticsweremadeforthelayerofen capsulation,howeverthe programwasdesignedtoincludeseparatecalculation s basedontheparametersofthematerial usedinthecoatinglayer.Theprogramalsohadtobedesignedtocheckforwhenthecoating layercompletelydissolved,andtobeginca lculationsatthattimefortheinnerparticledissolution. Figure 5 3 :Innerparticleradius,r 0, withencapsulationlayerthickness, h ,yieldsradiusofencapsulatedparticle r 0 enc 5.1.1 Calculationof D iffusion C oefficientforthe E ncapsulated L ayer Thethicknessoftheencapsulationlayerisgivenas h .Theradiusoftheencapsulated particle,whichincludestheinnerparticleandthecoatinglayer,isdefinedas h r r enc 0 0 (5.1) where r0 enc istheinitialradiusoftheencapsulationandparticleand r0 istheinitialradiusofthe particle.FollowingtheprocedureoutlinedinSection3.2,thediffusioncoefficientusingthe Wilke Changmethodwasfoundf orthecoatingmaterial, D 12enc by 6 0 1 2 2 8 12 2 1 ) ( 10 4 7 enc enc enc enc V T M D (5.2) r 0 r 0 enc h

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70 Time Zerot(1)t(2) t(3)t(4)Dissolutionor CriticalTime Zero Zero Zero where M 2 isthemolecularweightofthesolvent(g/mol), T isthetemperature(Kelvins), 2 enc is theviscosityofthesolventsolution(centipoises), V 1 enc isthemolarvolumeofthesoluteat normalboilingpoint(cm 3 /gmol)and enc istheassociationparameter ofthesolventforthe encapsulationmaterialspecified. 5.1.2 Determiningthe R adii ofEncapsulatedLayer Theadditionoftheencapsulatedlayerincreasesthecomplexityoftheproblemby creatingtwoseparateradiiwhichmustbesolvedinasequentialorder. Thedissolutionofthe outercoatingiscalculatedfirst.Sincethecoatingmaterialdiffersfromtheparticlebeing encapsulated,differentconstantsmustbecalculatedforuseintheequationtosolvefortheradius atagiventime.Theconstantcfor theencapsulatedmaterial,expressedas c enc ,isgivenby 3 1 4 3 3 0 enc m enc s enc enc N V C r c (5.3) where C s_enc isthesolubilityofthesolute(g/mL)and enc isthedensityoftheencapsulation material. Usingtheconstant c enc ,theconstantofintegration, k enc fortheencapsulationmaterial canthenbedeterminedusingtheformula givenin Eq. (5.4). Figure 5 4 :Schematicillustrationf or encapsulatedmonodisperseparticledissolution.

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71 enc enc enc enc enc enc enc enc enc enc c c r c r c r c c k 3 2 tan 3 1 ) ( ln 6 1 0 1 3 0 3 0 3 (5.4) The constant c enc andtheintegrationconstant k enc fortheencapsulationmaterialare theninserted intothemainequation, F enc ,whichrelatestheradiusoftheencapsulatedparticle, r enc ,toeach giventime as t V 3 4 N D k c 3 c 2r tan c 3 1 ) r (c r c ln 6c 1 F m enc enc enc enc enc 1 enc 3 enc enc 3 enc 3 enc enc enc (5.5) Theradiusfortheencapsulatedparticle, r enc ,iscalculatedusing Eq. (4.24)untilthe coatingiscompletelydissolved,when r enc =0.Whenthisoccurs,theprogramrecordsthetimeat which r enc =0,andbeginssolvingtheinnerparticledissolutionus ingthisasthestartingtime. The procedureforsolvingtheinnerparticledissolutionusingthesameprogramdevelopedin Chapter3 5.1. 3 C oncentrationand C onversionof E ncapsulation M aterial Themassoftheencapsulationmaterial, M enc atagiventimetis 0 3 ) ( 3 4 ) ( M N t r t M enc enc enc (5.6) where r enc (t) istheradiusoftheencapsulatedmaterialattimetand M 0 istheinitialmassofthe particleasderivedinEq. (3.9). Themassoftheencapsulatedmaterialcanbeconvertedto concentrationby m enc enc V t M t C ) ( ) ( (5.7) where C enc (t) istheconcentrationoftheencapsulationmaterialattimet. Theinitialconcentration ofcoatingmaterial iscalculatedbasedontheinitialencapsulatedparticleradius using m enc enc enc V N r C 3 0 0 3 4 (5.8) Theconcentrationofdissolvedencapsulationmaterial C denc isfoundby

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72 ) ( ) ( 0 0 t C C t C enc enc denc (5.9) Theamount dissolvedisusedtofindtheconversionoftheencapsulatedmaterialbyapplyingthe definitionofconversionas giveninEq.(5.10). enc denc enc C t C t X 0 ) ( ) ( (5.10) 5.1. 4 Visualizationof Results forEncapsulatedModel Thefunction PrintResults. mfirst convertsthetimeinsecondstotimeinminutes.Then plotsofconcentrationandconversionversustimearedisplayed.Theencapsulationmaterialand innerparticlearerepresentedasdifferentcolorsandplottedonthesamegraph. 5.2 ExampleforEnca psulatedModel Thefollowingsectionwillusedthedesignedprograminthecasestudyofcitricacid particlesencapsulatedwith glucose dissolvinginwateratroomtemperature. Thepropertiesfor citricacidwereassumedtobethesameasgivenin Table 3 1 andthecoatingmaterialofglucose propertieswerelistedin Table 5 1 Thethickne ssoftheencapsulationlayermustalsobe specifiedatthistime,andisgivenby h .Forthisexample,athicknessof0. 0 01cmofglucose encapsulationisconsidered. Table 5 1 :Parametersforglucosecoa tingmaterial. Densityofglucose Molecularweightofsolvent Temperature Viscosityofsolution(solvent) Molarvolumeofsolute Associationparameterofsolvent Solubilityofthesolute Dissolutionmediumvolume(water) Encapsulationlayerthickness M 2 T 2 V 1 C s V m h =1.54g/cm 3 =18.015g/mol =298K =0.91centipoises =319.88g/cm 3 mol =2.6 =0.91g/mL =1mL =0.0 0 1cm 5.2.1 Simulation S etu p forEncapsulationModel Theparametersforglucosewereinputtedintothe MATLAB programinthe DiffusionEnc.m and Function2.m sections. Thediffusioncoefficientforthe glucose

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73 encapsulationlayer isfoundtobe5.67x10 6 cm 2 /sec,comparedtothediffusioncoefficientof citricacidinwaterof5.21x10 6 cm 2 /sec.Itisimportanttonotethat thediffusioncoefficient calculatedinChapter3is usedinthissection only as anestimateforcitricacid,sincethesolvent propertieswill differduetotheglucosedissolvedinthesolution.Sincetestshadnotbedoneto determinethenewternary systemproperties,thevaluescalculatedinthissectionprovidea reasonableestimateofthediffusioncharacteristics.Itwasnotwithinthescopeofthisprojectto experimentallyfindthechangesinthediffusionrateofcitricacidinaglucosesolut ion,however, resultsfromthistypeofinvestigationcouldsignificantlyimprovetheaccuracyofthemodel. Aftertheprogramwasrun,tworesultinggraphswereproduced.Thefirstwasshowed thechangeoftheencapsulationandinnerparticleconcentrat ion, shownin Figure 5 5 ,whilethe secondgraphmonitoredtheconversionofthetwomaterials,asshownin Figure 5 6 .The encapsulationthicknessisverythininthisexample,soithaslittleeffectonthereleaserate of citricacid .Withinthefirstminute,theglucosecoatingiscompletelyd issolved,exposingthe innerparticl e.Oncethewaterisincontactwiththecitricacid,dissolutionoccursinafashion similartotheonedescribedinChapter4. Figure 5 5 :Concentrationversus timeforencapsulationthicknessof0.0010cm.

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74 Figure 5 6 :Conversionversustimeforencapsulationthicknessof0.0010cm 5.2.2 Effectof E ncapsulation T hickness Asimilar simulation wascompleted fora n encapsulationthicknessof0.01cm,a10% increase fromthethicknesstestedpreviously TheresultsfromthisrunareshowninFigures5.7 and5.8. Theresultsshowedthatt heconcentrationoftheencapsulatingmaterialdrop ped from 0.06gramso fglucoseto0gramsinabout 4 minutesandtheconcentrationofcitricacidatthis timeremain ed constant.Afteralloftheglucoseisdissolved,thecitricacidbeginsdissolution untilthereisnolongeranycitricacidispresent,whichoccursataro und 11 minutes.Compared totheunencapsulatedparticlesmodeledinChapter3,whichhadatotaldissolutiontimeof around 7minutes,the anencapsulatedparticlewillnotcompletedissolutionforanaddition4 5 minutes.Thischaracteristiccouldbeusefulindesigningfunctionalparticleswheretheactivityof theacidneedstobedelayed.Forfoodapplications,postponementofacidrel easecanprolongthe flavorenhancementofaproduct.Inpharmaceuticals,theactiveingredientmaywanttobe protecteduntilitreachesatargetsite,suchasaspecificorgan,beforeitisreleased.Thesame controlisalsodesiredindetergents,sinc ecertaincleaningagentsshouldnotbereleaseduntila specifictimeinthewashingcycle.

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75 Figure 5 7 : Concentrationversustimeforencapsulationthicknessof0.010cm. Figure 5 8 : Conversion versustimeforencapsulationthicknessof0.010cm.

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76 5.2.3 COMSOLVisualization forEncapsulatedModel COMSOLwasu sedto visualizethechange s inradius ofthe particleasdissolution proceeded.DatacalculatedfromtheMATLABprogramwasim portedintoCOMSOLScriptand 3 Dgraphofthesphericalparticlewasgenerated.Thecodewaswrittentodisplaythe encapsulationmaterialingreenandintheinnercoreparticleasred.

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77 a) Figure 5 9 :Encapsulatedparticleconversionfora)t=0min,b)t=1mins,c)t=3mins,d)t=5mins,e)t=7 minsandf)t=9mins.

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78 b) Figure5.9( Continued )

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79 c) Figure5.9( Continued )

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80 d ) Figure5. 9 ( Continued )

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81 e ) Figure5. 9 ( Continued )

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82 e ) Figure5. 9( Continued )

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83 f ) Figure5. 9( Continued )

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84 5.2. 4 Encapsulation T hickness E ffect Thethicknessoftheencapsulationcanbemodifiedto fitadesiredreleaseprofile.To studytheeffectoftheglucosethicknessonthedelayofcitricacidrelease,fivethicknessbetween 0.0025 0.02cmweretested.Theamountofglucoserequiredforeachtestwasalsocalculated andcouldbeusedindete rminingthematerialdemandsforaspecificproject,perhapsaidingthe decisionofthemosteconomicaldesign. Table 5 2 :Effectofencapsulationthicknessonglucoseamountanddissolutiontime. h (cm) Glucose(g) Delayin Dissolution Time ofC itricAcid (mins) 0.0025 0.0133 1 .5 0.0050 0.0278 2 .5 0.0100 0.0602 4 .5 0.0150 0.0977 6 .5 0.0200 0.1407 9 .0 Aspredicted,theincreaseinthicknessoftheencapsulationlayercansignificantlydelay the delaysthereleaseof citricacidby 1 .5minutes,whileathickerlayerof0.0200cm , delaysthereleaseby almosttwicethatamount givingatotaldissolution time of 9 minutes. Theamountofglucose requiredforthis increasedthicknessisamount0.13grams.Thecostofthisadditionalmaterial couldbeusedtodeterminewhetherthefunctionaldesignisalsoaneconomicallyfavorable choice.

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85 Figure 5 10 :Effectofencapsulationthicknessonthedelayedconversionofcitricacid.

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86 Chapter6: EncapsulatedPolysdisperseParticleDissolution Asdiscussedpreviously,substantialerrorscanoccurwhenthedissolutionof particlesis assumedtobemonodispersed.The programdevelopedinChapter5fortheencapsulated particlescanbefurtherimprovedwhen multisized encapsulated particles areconsidered.This wasaccomplishedbycombingthetechniquedevelopedinChapte r4forpolydispersedparticles withthemodelingmethodforencapsulatedparticledissolutionfoundinChapter5. 6.1 ProgramBuildUp Themethodsforsolvingmultisized particlesandencapsulatedparticleswerecombined inthisdesign.Theprogramwouldfirstbeinitializedandthediffusioncoefficientswouldbe solvedfortheencapsulationandtheinnerparticlematerial,follo wingthecodedevelopedfrom Eq (3.4)and Eq. (5.2).Next,thenumberofparticlesineachsizegroupwouldbedetermined basedonthegivenpercentsineachsizedistribution,si milartothosefoundinEqs.(4.1 4.7) The radiusfortheencapsulatedlayerwascalculateduntilallofthecoat inghaderoded.This calculationwas Time Zero CriticalTim et(3) DissolutionTime Zero Size r max r min Zero Zero Figure 6 1 : Schematicillustrationforencapsulatedpolydisperseparticles.

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87 wasrepeateduntilallofthesize groups.Whentheencapsulation layercompletelyeroded,the programwouldbegincalculations fortheinnerparticle. Again,this wasrepeatedforallofthesize groupspresent.Finally,thetotal concentrationandconversionforthe innerparticleandencapsulationwas calculatedforallsizegroups.The concentrationandconversiondata forencapsulationlayerand inner particlewereplotted.Theprogram algorithmforthismodelisshownin Figure6.2. 6.1.1 Calculating E quivalent N umberof P articles Therateofdiffusionis dependentontheconcentration gradientbetweenthesoluteand solvent.Atechniquesimilartothe oneusedinChapter4,for polydisperseparticles,was developedtocalculatethe equivalentnumberofparticlesi n eachsizedistributionforaspecified concentration.Thisequivalent numberofparticleswasusedto calculatetherateofdissolution basedonthetotalconcentrationgradientofthesystem,andthenwasadjustedtomatchtheactual sizedistribution. Findradiusof encapsulatedparticle forgiventime Ifencapsulated radius < particle radius Findtheinnerparticle radiusforgiventime Ifencapsulated radius > particle radius Plotresults Calculatediffusioncoefficientfor innermostparticle Start Calculatediffusioncoefficientfor encapsulationmaterial Forgivensizegroup calculate numberofparticles Ifadditional sizegroups exist Calculateconcentrationand conversionforinnerparticleand encapsulationforallsizegroups Calculatetotalconcentrationchange andtotalconversion Figure 6 2 :Programalgorithmforencapsulatedpolydisperse particlesmodel.

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88 6.2 EncapsulatedPolydisperseParticle E xample Forcomparisonpurposes,asimulationforpolydispersecitricacidencapsulatedwith glucosewastested.Theparametersforcitricacidandglucosewerethesameasprovidedin Table 3 1 and Table 5 1 ,respectively.Thesizedistributionfoundexp erimentallyfromSEM techniques, Figure 4 8 ,wasusedinthistestrun.Inaddition,theencapsulationlayerthickness wassetto0.0010cm,thesameasthetestruncompletedinChapter5.Thefirstgraph, Figure 6 3 ,producedbythisprogramshowstheconcentrationchangesforeachofthesizegroups.The dashedlinerepresentstheencapsulationlayerandthesolidlinerepresentstheinnerparticle.The encapsulationresultscanbeviewed betterbyzoominginontheareabetween0 30seconds, shownin Figure 6 4 .Sincetheencapsulationlayerissmallwithrespecttothetotalsizeof the particle,ithasvirtuallynoeffectondelayingthereleaseofcitricacid. Figure 6 3 :Concentrationchangesforencapsulationandinnerparticleforallsizedistributions.

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89 Figure 6 4 : C oncentrationchangefor various sizedistributions. Themaindifferencewiththepolydisperseencapsulatedmodelcanbeseeninthetimefor citricacidtocompletedissolution.Inthemodeldevelopedin Chapter5,totaldissolutionofcitric acidoccurredinabout8minutes.Forthepolydispersemodel,thecitricacidtooktwicethe amountoftime,about 16minutestocomplete.Aswouldbeexpectedbasedontheresultsfound inChapter4,thedissolutio ncurveforthepolydispersemodelwaslesssteepandhadamuch smoothercurve.

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90 Figure 6 5 : T otalconcentrationversustimefor polydisperse encapsulated particles Figure 6 6 :Totalconversionversustime for polydisperse encapsulatedparticles.

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91 6. 3 EffectofEncapsulationThickness Forcomparisonpurposes,anincreasedencapsulationlayerthicknesswastestedtoseeits effectonthe releaserateofthecitricacid.Theglucoselayerwasincreasedfrom0.0010cmto 0.01c m.Thegraphsareshownin Figure6.7andFigure6.8 Figure 6 7 :Increasedencapsulationlayerthickness forp olydispersemodel Figure 6 8 : Concentrationchangeversustime forallsizedistributions withincreasedcoating thickness.

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92 Figure 6 9 : Encapsulated p olydisperse particledissolutionfor a) t= 0 mins ,b)t=2mins,c)t=4mins,d)t=6 mins,e)t=8mins,f)t=10minsandg)t=12mins. a)

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93 Figure6.9 (Continued). b)

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94 Figure6.9 (Continued). c)

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95 Figure 6.9 (Continued). d)

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96 Figure6.9 (Continued). e)

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97 Figure6.9 (Continued). f)

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98 Figure6.9(Continued) g)

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99 Chapter7: ResultsandDiscussion Theprevious chapters outlinedthedevelopmentofacomprehensivecomputercodetoaid in estimating thedissolutionratesofsubstancesforcontrolledrelease applications .Whilethe longtermgoalsofsuchaprogramwouldbetoeliminateunnecessaryexperimentaltesting,th e initialvalidationofthemodelsmustbecompletedbycomparingthemodelresultstothosefound inexperiments.Thischapterisdedicatedtoexaminingtheresultsfoundexperimentallywiththe simulatedresultsfoundfromChapters3 6. Dissolution testsofcitricacidinwaterwereperformedexperimentallytovalidatethe resultsfoundinthesimulation. An OmniCalTechnologiesSuperCRC20 305 2.4 reaction microcalorimeterwas used wasusedtomeasuretheconversionversustime.Theprocessfor experimentallymeasuringt heconversionwiththeuseofareactionmicrocalorimeter isoutlined inthefollowingsection. 7.1 ExperimentalValidation Reactionheatflowcalorimeteryhas beenusedinmanytechnicalfieldsfromdrug discoverytoprocessdevelopment [50 55 ] .Microcalorimetersareusedtomonitorthechangein heatwhichaccompaniesthechemicalandphysicalprocessesastheyundergomechanismssuch asadsorption,dilution, dissolution,mixingorchemicalreactions.Microcalorimetersmeasurethe heatquantityofachemicalprocessalongwithitstimederivative,heatflow.Heatflowisalso proportionaltothereactionrateandcanbeusedtocalculateaotherreactionkine ticparameters. Theheatflow ,q, measuredduringanexperimentis proportional tothereactionrate.r. rxn istheheatofthereactionandVisthevolume. rxn V (11.1)

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100 Theintegrationoftheobservedheatflowversustimecurvesyieldstheheatofreaction.The fractionalconversioncanalobeobtainedbyusingthecalculatedfractionalareaunderthe temporalheatflowcurve. Thefollowingequationcanbeusedtocal culatethefractional conversion f t dt q dt q X 0 0 (11.2) Thenumeratorisfoundbycalculatingtheareundertheheatflowcurvetoanytimepointtand thedenominatoristhetotalareaundertheheatflowcurve.Theheatevolvedandthecalorimetric responsecanbefoundbymonitoringthereactionprogressandmeasuringtheenthalpicchanges ofthechemicalreactions. 7.1.1 ReactionMicrocalorimeterExperiments Experimentswereconductedina OmniCalTechnologiesSuperCRC20 305 2.4 reaction microca lorimeter ,asshowninFigure7.1 [51] .Thereactionmicrocalorimeterusesadifferential scanningcalorimeter(DSC)techniquetomeasuretheheatreleasedorconsumedinasample vesselcomparedtoanemptyreferencevessel. Figure 7 1 :OmniCalSuperCRCreactionmicrocalorimeter.

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101 Reactionvesselswere16 mL screw fitthread glassvials fitwith Teflon linedscrew caps.Thedesiredamountofanhydrouscitricacid, obtainedfromFisher Chemicals was m easuredintotheglassvialandwassealed.Boththevesselcontainingthecitricacidandthe emptyreferencevesselwereplacedinthemicrocalorimeterfor35minutes,allowingthesystem toreachthermalequilibrium.Simultaneously,twoliquidsamples of1mLofwaterwere measuredintotwo5mLsyringesandplacedintothesamplebarrelsinthecalorimeterandwere allowedtothermallyequilibrate. The microcalorimeter unitisturnedonandthesamplesandsyringebarrelswereallowed tocometother malequilibriumwiththecalorimeterheatsinktemperature The micro calorimeter operateson linewiththeprogramsoftwareWinCRC Turbo,whichisexecutedafterthesamples havebeeninsertedintotheunitandtheunitisturnedon.WinCRCTurbowasuse dforbothdata acquisitionandconversioncalculations.Samplesweremonitoreduntilasmoothbaselineinthe heatflowwasmaintained. Thereactionwasinitiatedbyinjectingthewaterintothevesselswhile thetemperatureofthemicrocalorimeterwas heldatroomtemperature,approximately25C.The internalthermalcontrollerinthemicrocalorimeterheldthetemperatureconstantthroughoutthe processensuringthattheexperimentwasrununderisothermalconditions.T hedetectionand collectionofd atawasmaintaineduntilnochangeswereseenintheheatflowforatleast30 minutes .Aheatflowcurvewasproducedbymeasuringtheheatflowfromthereactionvesselin incrementsof3samplespersecond,asseeninFigure 7.2 .Thedatacollected wouldthenneedto becalibratedtoaccountforthedelaybetweentheinstantaneousheatflowevolvedandthetime thethermopilesensorisabletodetecttheheatflow. Theprocessofcalibrationinvolvedpassingaknownquantityofheat,producedby pass ingaknowncurrentthrougharesistor,intothesamplechamberofthecalorimeter.The WinCRCsoftwarethenrecordstheheatresponsecurve.Thiscurveisthentransformedintoa squarewaveusingthesoftware,whichallowsfortheresponsetimeofthe microcalorimetertobe calculated.Usingthecalculatedresponsetime,thesoftwarecanthenreadjusttheheatflowcurve datafromtheexperimentalreactiontoaccountforthedelaysfromthesensors.Thenewgraphis savedasataucorrectedgraph.Th etaucorrectedgraphisusedtocalculatethefractional conversionusingthe E q (7.2).

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102 Figure 7 2 :Reactionmicrocalorimeterheatflowandconversion graph. Thefirstexperimentforcitricacidwasconductedatthesameconditionsaslistedin Table 3 1 .Theresultsofthesimulationareshownin Figure 7 3 .Theexperimentalresultsdonot seemtofitanyofthemodelsformonodisperseparticles with one givenaverageradius .Instead, theexper imentaldataseemtobeinbetweentheresultsfoundfortheaverageradiussizesof range0.05to0.10cm.Duringtheinitialminutesofdissolution,theexperimentcloselyfollows themodelfortheradiussizeof0.05and0.06cm.However,astimeprog resses,thetimefortotal dissolutionisdelayedandthecurveappearstomatchtheresultsforlargeparticleradiussizes, greaterthan0.08cm.Thissuggeststhattheparticlesusedintheexperimentincludearangeof particlesizes,whichisindeed thecase,aswasdiscoveredusingtheSEM. ConversionCurve AreaunderHeat FlowCurve

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1 0 3 Figure 7 3 :Experimentaldataversus monodisperse modelforvaryinginitialradii

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104 Afterestablishingthattheparticlesincludedadistributionofradiussizes,the experimentalresultswerecomparedtothemodeldevelopedforpolydisperseparticles.The distributionfoundthroughSEM, Figure 4 8 ,wasinputtedintotheprogramforpolydisperse particlesandtheresultsweregraphedalongwiththeexperimentaldatain Figure 7 4 Forthis concentration,0.10gcitricacidin1mLwater,theresultsfromthepolydispersemodelprovideda goodestimateofthedissolutiontime. Boththeexperimentandpolydispersemodelhadan appr oximatedissolutiontimeof15minutes.Thepolydispersemodelappearedtobeabetterfit fortheexperimentaldataprovided.Foramoredetailedexamination,theexperimentaldatawas plottedagainstbothmodelsin Figure 7 5 Thenoticeabledifferencebetweenthemodelsisthe finaldissolutiontime.Thepolydispersemodelistwiceaslongaspredictedwiththe monodispersemodelandappearstoha vethesamecurveshapeasfoundexperimentally. Figure 7 4 : Experimentaldataversus polydispersemodel for0.10gramsofcitricacid

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105 Figure 7 5 :Comparisonofexperimentaldata for0.10 g rams citricacid withbothmodels. 7.1. 2 Additional E xperimenta lT ests Followingthe samplerun of0.01gramsofcitricacidin1mLofwater,several additional testsweredoneforconcentrationsbelowand abovethisamount.Theresultsshowedthelimits ontheaccuracyofthemodelsforconcentrationsintheextremeranges,fromdilutetomore concentratedsolutions.Thepossibleexplanationsforthediscrepanciesofthemodelsare investigatedlaterint hischapter. Atotalofnineexperimentswereconductedforcitricacidweightsof0.02,0.04,0.08, 0.10,0.20,0.30,0.40and0.50gramsin1mLofwateratapproximately298Kelvin.The experimental results wereplottedin Figure 7 6 .Ingeneral,increasingintheamountofcitricacid addedincreasedthetotaldissolutiontimeoftheparticles.

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106 Figure 7 6 :Experimentalresultsfor differentamountsofcitricacid Next,graphscomparingeachoftheexperimentalconcentrationswiththecorrespondingresults foundfromthemonodisperseandpolydispersemodel s wereplotted.Thesegraphwereproduced inasimilarmannertotheoneproducedforthefirstsampleof0.10gramsofcitricacid. The valuesfortheexperimentaldataforsixsampledatasetsforconcentrationsof0.2 0.20gramsof citricacidaregive ninAppendixE.Thedifferentconcentrationswerealsotestedusingthe monodispersemodelandpolydispersemodel.Theresidualsbetweentheexperimentaldataand thetwomodelsshowedthatthemonodispersemodelwasmoreaccurateforlowconcentrations whilethepolydispersemodelmoreaccuratelydescribedthedissolutionbehaviorathigher concentrations.

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107 Figure 7 7 : Comparisonofexperimentaldatafor0. 02 g rams citricacidwithbothmodels. Figure 7 8 :Comparisonofexperimentaldatafor0. 04 grams citricacidwithbothmodels.

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108 Figure 7 9 :Comparisonofexperimentaldatafor0.0 6 grams citricacidwithbothmodels. Figure 7 10 : Comparisonofexperimentaldatafor0. 08 grams citricacidwithbothmodels.

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109 Figure 7 11 :Comparison ofexperimentaldatafor0.20 grams citricacidwithbothmodels. Figure 7 12 :Comparisonofexperimentaldatafor0.30 grams citricacidwithbothmodels.

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110 Figure 7 13 : Comparisonofexperimentaldatafor0.40 grams citricacidwithbothmodels. Figure 7 14 :Comparisonofexperimentaldatafor0.50 grams citricacidwithbothmodels.

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111 7.2 Interpreting D iscrepancies Forconcentrationsofcitricacidbetween0.06 0.10grams,themodelsfordissolution provideagoodestimateforreleaseratebehavior.Forconcentrationsbelow0.06grams,the experimentaldatashowsafasterdissolutiontime thanpredicted.Forconcentrationsabove0.10 grams,theexperimentaldatashowsamuchlongerdissolutiontimethanthemodelspredict.This sectionwilllookat thetwo maincausesforthesedisc repancies:modelingassumptions and experimentalinaccur acies. 7.2.1 Modeling A ssumptions N umerousassumptions were madeintheanalyticalderivationoftheoriginalmodelfor calculating thedissolutiontimeofmonodisper separticles,as listedinChapter2.Manyofthe assumptionsmadeconflictwiththeac tualexperimentalsetup. Forinstance,theparticleshape wasassumedtobeperfectlyspherical.SEMresultsshowthattheactualcitricacidparticlesare notsphericalandarenotuniforminshape ,ascanbeseenin Figure 7 15 Particleshapehasa significanteffectonthereleaseratecharacteristics,sincetherateofdissolutionisdependenton thesurfaceareaofsoluteexposedtothesolven t.AsdiscussedinChapter4,thepolydispersesize distributionwasapproximatedusingtheequivalentsphericalareaoftheparticles.Whilea sphericalshapeprovidesagoodinitialestimate,itdoesnotaccuratelydescribethedissolutionof particles withcylindrical,cubicorrectangulargeometries. Figure 7 15 :SEMphotosofnon sphericalcitricacidparticles. Furtherassumptionsreguardingthesizedistributionalsohaveaneffectonthemodeling results.ThemasspercentofeachofthesizedistributionsisinputtedintotheMATLABprogram

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112 andisusedtobackcalculatethenumberofparticlesineachsizegro up.Theassumptionisthat everycitricacidsampletestedhasthesamesizedistribution.However,itwasseenintheSEM results,in Figure 4 6 ,thatthesizedist ributionvariedforeachsample. Forsmallsamplesizes, suchas0.02grams,thereareonlyafewparticlesandinaccuraciesinpredictingthenumberof particleswithagivenradiussizedramaticallychangetheresultsproduced. Themodelformonodisper separticlesutilizedtheconcentrationgradientbetweenthe solubilityofthesolute, C s ,andtheconcentrationatagiventime, C ,asthedrivingforcefor dissolution.Theassumptionthattheconcentrationof C aroundtheparticleisrepresentativeof theconcentrationoftheentiresurroundingsolution furtherimpliesthatthesystemiswellstirred. However,theadditionaleffectsofstirringarenotincorporatedintothemodel.Thisincludes consideringtheagitationastirrerwouldhaveonthesys temandadditionalconvectivemass transfereffectsresultingfromthemovingliquid.Tomonitortheeffectsofstirring,fivesamples ofcitricacidweretestedexperimentally.Thedissolutionoftheconcentrationsof0.02,0.06, 0.10,0.30and0.50gra msin1mL ofwater weretestedwiththeadd itionofaanelectro magnetic stirbar.Thestir reddissolutionexperimentscompletedmuchfasterthantheunstirredtests and canbeseenin Figure 7 16 Theresultsforthestirreddissolutionexperimentof0.10gramsof citricacidwereplottedalongwiththenon stirredexperimentalresults,monodisperseand polydispersemodelin Figure 7 17 .Thedataforthestirredexperimentshowedasteeperslope buthadafinaldissolutiontimesimilartothemonodispersemodel. Themodelsappear edtobe a combination oftheresultsofstirredandunstirredexperiments ,whichisalsosuggestedby research[52] Finally,theparametersforthechemicalsubstancesareassumedtobeconstantwhich canleadtoerroneouscalculations.Viscosityandtemperatureareassumedtobeconstant,which causesthediffusioncoefficienttobecalculatedasaconstantaswell.Inactuality,the temperatureofthesystemchangesbecausethereactionofcitricac idandwaterisendothermic. Thedecreaseintemperaturewouldhaveaneffectontherateofthediffusion,howeverthatwas notconsideredinthesemodels.Anotherinfluentialparameter, C s ,thesolubilityofthesolute, wasconsideredtobeconstantth roughoutthedissolutionprocess.Althoughthiswashelpfulin simplifyingthemathematicsinvolved,itdoesnotfairlyrepresentthesystem,sincesolubilityis neverconstantandis highlydependenton temperature [53 54] Simplificationsmadeinthe d erivationofthedissolutionmodelexplainsomeofthediscrepanciesseenbetweenexperimental dataandtheresultsmodelpredicted

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1 1 3 Figure 7 16 : Stirredexperimental datafor citricacid fordifferent initialamounts.

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114 Figure 7 17 :Comparisonplotforstirreddata using 0.10g rams ofcitricacid. 7.2.2 Experimental I naccuracies Theexperimentalprocedureanddataanalysisshouldalsobeconsideredwhilerelating theresultstotheaccuracyofthemodels.First,thecalorimetricsetupusedwasan OmniCal TechnologiesSuperCRC20 305 2.4 reactionmicrocalorimeter .Thereactionto okplaceona microscale,involvinglessthan5mLofsolvent.Thesmallsamplesizemadeexperimental measurementsdifficult,especiallyfordilutesolutions.Atlowconcentrations,theamountof citricacidpresentwasminuscule,andthechangeinhea tflowmeasuredwasverysmall.With suchlowmeasurements,thelimitationsoftheequipmentcouldhaveaffectedtheresultsfound. Whiletheexperimentwasconductedusingtheisothermalsettinginthecalorimeter,the initialtemperaturesofeachof theexperimentshadslightvariations.Allexperimentswere conductedatroomtemperature,butroomtemperaturevaried dependingonthetemperatureofthe daytheexperimentwasrun .Thetaucorrection,whichisusedtoestablishthecorrectedtimeof th ecalorimeter,requiredraisingthetemperatureofthesystemforanextendedtime,aperiodof about30minutes.Followingthetaucorrectionprocedure,thesystemcouldneverreturnexactly totheinitialtemperature,italwaysremainedafewpercentage shigherthanthestarting temperature.ItwasshowninChapter3,in Figure 3 4 ,thattemperaturecanhasaslightaffecton thedissolutiontime.T herefore,thevariationsintheinitialtemperaturesofeachofthe experimentalrunsshouldbeconsidered.

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115 Reactionmicrocalorimeters ha ve beenused in dissolutiontesting byprovidingtheheat flowproducedbyareaction withrespecttotime .Theintegrationoftheheatcurveallowsforthe determinationoftheconversionasafunctionoftime.Thetotalconversionrepresentsthe integrationofthecompleteheatcurvegeneratedbythereaction.Errorsfromthiscalculationcan befoundat highconcentrationsnearthesolubilitylimit.Evenininstanceswheretheparticles havenotcompletelydissolved,suchasincasesnearthesolubilitylimit,theintegrationofthe completecurvewouldsuggest100%conversion.ThisimpliesthattheDSC results alwaysneed tobevalidatedat highconcentrationstoensurethatcompletedissolutionishasindeedtaken place. 7.3 Encapsulated M odels Atthetimethisprojectwascompleted,glucoseencapsulatedcitricacidparticleswerenot availablefor testing.Forthatreason,itwasnotpossibletotesttheencapsulatedparticles experimentallytothesimulatedmodels.Itwas,however,possibletoestimatetheeffectsofthe glucoseencapsulationlayerbytestingglucoseconcentrationinwaterandth enusingtheglucose watersolutiontodissolvecitricacid.Thetwostepprocessprovidedaninitialestimatetothe dissolutionbehavioranencapsulatedparticlemighthave. Forthemonodispersemodelanencapsulationthicknessof0.001cmwasused,wh ichis approximately0.01gramsofglucose.Forthepolydispersemodel,anencapsulationthicknessof 0.016cmwasused,around0.10gramsofglucose.Thedashedlinesrepresenttheencapsulated layerwhilethesolidlinesaretheinnerparticle.Theam ountsof0.01gramsand0.10gramsof glucoseare graphedalongwiththeinnerparticledata .Thepolydispersemodelassumedthesame radiussizedistributionasfoundinChapter4.Allparticleswereassumedtohaveequal encapsulationlayerthicknesses

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116 Figure 7 18 : Experimentalresultsforc itricacid encapsulatedwith0.01and0.10gramsofglucose. Figure 7 19 :Monodispersemodelfor citricacidencapsulatedwith0.01and0.10gramsofglucose.

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117 Figure 7 20 :Polydispersemodelforcitricacidencapsulatedwith0.01and0.10gramsofglucose. 7.3.1 Sourcesof E rror Comparisonsofth eexperimentaldata,monodisperseencapsulatedmodeland polydisperseencapsulatedmodelshow similardiscrepanciestotheonesoutlinedinSection7.1. Italsoappearsthatfor0.10gramsofglucose,thedissolutionbehaviorismuchdifferentthan expect ed. Oneexplanationforthediscrepanciesisthatboththemonodisperseandpolydisperse modelassumetheglucoseisathinlayersurroundingeachparticle.The reaction micro calorimetryexperimentsconductedwith0.01and0.10gramsofsolidparticlegl ucose, whichhascompletelydifferentsurfaceareacharacteristics.Theparticlesizedistributionofthe glucosewasnottakenintoaccountandthereforedoesnotrepresentthesizeofanencapsulated particleasassumed. Asmentionedpreviously,thepa rametersassociatedwiththeglucose watersolutionand citricacidparticles werenotknownfortheexperimentalconditions.Oncefurther experimentationisdonetoestablishthatthe diffusioncoefficientscalculatedareappropriatefor thesystem,more confidencecanbegiventothemodelsdeveloped.

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118 7. 4 Comparisonof M ethods Themodelsdevelopedinthisworkwereprimarilybased chemicalengineeringtransport phenomenaprinciples.Themaindifferencebetweenthemodelsdevelopedinthisworkwith the dissolutionmodelsreviewedinChapter1isthatchemicalpropertiesalonecanbeusedto determinethedissolutionbehaviorofthesystem.Fortheclassical Nernst Brunnerequation,Eq (1.2),andHixon C rowellcuberootequation,Eq. (1.5), severalconstants k 1 and k 3 areneededto evaluatethemodel.Theseconstantsaregenerallydeterminedbybestfittingexperimentaldata. ) ( dt dC 1 C C k s (1.2) t k w w 3 3 / 1 3 / 1 0 (1.5) Anotherdifferenceinthemodelusedisthatitincorporatestheradiusoftheparticle inthe dissolutionmodel ,rather thanonly totalconcentratio n. Thiscanbeexceedinglyusefulinthe designofparticleswithradiussizedistributions,suchasinpo lydispersesystems,and cannotbe examinedwithmodelswhichconsiderconcentrationorparticleweightalone.

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119 Chapter8: Conclusion Inthiswork,theanalyticalderivationforthedissolutionofsolidparticlesinliquid solventswasformulatedinmathematicalterms.Thestructureoftheprogramconsistedof developingasimplecorestructurewhichallowedfortheparticleradiussize tobecalculatedasa functionoftime.Followingthedesign,additionallevelsofcomplexitywereaddedtothe computationalmodelincludingaccountingforpolydispersityofparticlesandencapsulation. The comprehensivecomputerprogramfromthefour programsdevelopedisshowninFigure8.1. A parametricstudyoftheeffectsofparticlesize,concentration,andcompositionwasperformedto beutilized inthe developmentof speciallydesignparticlesforcontrolledrelease. Theinitialtimedelayin thereleaseoftheactivecoreingredientwasobservedwhenthe particleswereencapsulatedwithawater solublecompound.Thedissolutionratewasfoundtobe dependentontheparticlesize.Parametersassociatedwiththecompositionmainlyaffectedthe calculationofthediffusioncoefficient,whileconcentrationdeterminedthenumberofparticles andalsoimpactedthereleaseratecharacteristics.Accountingforpolydispersityproducedmore naturalcurves,whichmoreaccurate ly resembledthedatafoun d experimentally. 8.1 Implications Themethodology for computer aideddesign of specializedparticleshasbeendeveloped in thiswork. Theprogramcanbeutilizedinseveraldesignapproaches.First,thesimulationcan beusedtopredic ttheresultsofagivenparticle.Theprogramcanbemodifiedtoseetheaffectof variousparameterchangesonthegivenparticles,suchasparticlesizeorconcentration. Additionally,encapsulationoptionscanbeinvestigatedifadelayinthereleas erateis desired. Thesecondappro a ch,whichistheinverseofthefirstapproach, involvesstartingwithadesired releaserateprofileandmanipulatingtheparametersandparticlecharacteristicstodeterminethe typeofparticlewhichcanproducethe desiredeffect.

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1 2 0 Figure 8 1 :Comprehensiveflowsheetforallfourprogramsdevelopedinthiswork.

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121 Sinceallthecomputationsarecarriedoutnumerically,thevirtualexperimentscanbe carriedoutnumeroustimesbeforetheappropriateparticleformulationisdiscovered.Avariety ofparticleoptionswhichmeetaspecified dissolutiontimecanbecompi led,andthesescan then be comparedtodeterminethemostfeasiblealternative basedonmaterialandmanufacturing factors. 8.2 Future W ork Whilethemainframeworkforthecomputerprogramwas developed potential improvementstothecodeexist.Severaloptionscanbeaddedwhichexpandversatilityof programandwouldultimatelyincreasetheaccuracyofthecomputations. Intheanalyticalderivationofthetimerequiredfordissolution,thediffusionb oundary layerthickness, h ,wasapproximatedbytheparticleradius.Thisisapplicableforlarge particles, wherethediffusionboundarylayerthicknessdecreaseswithdecreasingparticleradius.Ifthe diffusionboundarylayerisconstant,thenthetim eforparticlestodissolvecanbemodeledusing thederivationfromSertsou [42] in Eq (8.1), 4 3 3 2 tan 3 2 tan 3 1 ) ( ) ( ) ( ) ( ln 6 1 0 1 1 2 3 3 3 0 3 3 3 3 0 2 N D V h c c r c c r c r c r c r c r c c t m (8.1) where r istheconstantdiffusionlayerthickness.Thisequationisusedtodescribethetransition ofaconstantdiffusionboundarylayertoa boundarylayer whichchangeswiththeradius,as derivedinChapter2.Basedonthedissolvingmaterialanddissoluti onconditions,bothofthe expressionsmayberequiredtofindthebestfittoexperimentaldata. Theprogramcodecanbefurtherimprovedbyadjustingthederivationstoinclude descriptionsofnon spherical shapes,non uniformcoating ,and multiplelay eredcoating Some possiblenon sphericalgeometricshapesforparticlesareshowninFigure8.2. Whilethe assumptionofperfectlysphericalparticleswasmadein the model susedinthiswork incorporatingthederivationadditionalshapescandramatica llyinfluencethedissolution behavior.Forinstance,thederivationof controlleddrugreleasefor cylindr icalbinarysystem similarto wasderivedbyMarentetteandGrosserin [56] .RecentresearchconductedbyAnsari andStepanek [57] focusedonthedissolutionofgranules,whichareagglomerationsofprimary

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122 solidparticles .Incorporationofincreasinglyadvancedgeometriescanexpandtheapplicationsof theprogram. Figure 8 2 : Othern on sphericalgeometricshapes ofparticles Furtherimprovementstothemodelcanbemadeby account ingforcontactofthe particleswithotherparticlesandthesystemcontainer.Themo recontactaparticlehaswithits surroundings,thelesssurfaceareaisexposedwhichaffectsthedissolutiontime ,asshownin Figure8.3 .AswasseeninChapter7,themodelsformonodisperseandpolydisperseparticles fallbetweentheexperimentalre sultsforunstirredandstirredsolutions.Iftheunstirredcaseis considered,thenmodificationsonthemodelmustbemadetoaccountforconcentrationofthe stagnantfilm surroundingtheparticle, whichdonotrepresentthetotalconcentrationofthe system.Ifthestirredcaseisconsidered,thenthemodelmustbeadjustedtoaccountforthe agitationthestirringproduces,suchasanincreaseinconvectivemasstransfer. Theincorporationofadditionalequationstomodelthechemicalparametersofthesystem couldalsoimprovetheresults.Manyoftheparameterswhichwereassumedtobeconstant changethroughoutthedissolutionprocessandshouldbemoreaccuratelydescr ibedusinga function.Forinstance,temperatureisassumedtobeconstantthroughouttheprocess,butthe reactionisendothermic,meaningthetemperaturewilldecreaseasthereactionproceeds. Incorporatingthechangeintemperaturewouldaffectthec alculationofthediffusioncoefficient.

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123 Thesolubilityofthesoluteisalsotakentobeconstant,butisactuallyafunctionoftemperature, andshouldbewrittenasanequationratherthaninputtedasasinglevalue. a) b) Figure 8 3 :Surfaceareachangesbetweena)lowconcentrationandb)highconcentrationofsolidparticles. Furtheradvancementstotheexistingframeworkofthecodecouldpotentiallybe developedtosimulatedissolution testingofthereleaseofsoliddosageformsforpharmaceuticals. AccordingtotheU.S.DepartmentofHealthandHumanServicesFoodandDrugAdministration [58] ,dissolutiontestsareusuallyconductedindissolutionmediumsof500,900,or1000mL. TheaqueousmediumusedusuallyhasapH range between 1.2to6.8,withapH of6.8 usedto simulateintestinalfluid.Thecomputermodeldevelopedsh ouldbetestedundertheseconditions to determine iftheresultsmatchthosefoundexperimentally. The program should then be adjusted to matchthemethodsapprovedbytheU.S.Pharmacopeia(USP)forsolid dosageforms testing.Theimprovedprogramcouldbea valuabletool in drugdevelopmentanddesign ofthe c ontrolledrelease of pharmaceuticalagents. Currentresearchsuggeststhatcontrolled release dosageformscansignificantlyenhanceclinicalefficacywhiledec reasingtreatmentcosts, providingadditionaleconomicvalueoverimmediate releasedosageforms [59] .Theinitial challengeofcontrolledreleaseisthedeterminationoftheformulationforadesired release profiles .Computer aideddesign ,basedonthe fundamentalunderstandingofthedissolution process, couldreducethetrialanderrorassociatedwith experimental prototypetesting Inthe future,c omputersimulat ionscouldbe usedto optimizedrugdeliverysystems byaidinginthe designofspeciali zedcontrolledreleaseformulations.

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124 References [1] BusinessCommunicationsCompany,Inc 2004.RP ,[Online].Available:BBC Research, http://www.bccresearch.com/RepTemplate.cfm?reportID=370 &RepDet=HLT&cat=mst &target=repdetail.cfm [AccessedSep.9,2007]. [2] AI ChEJournal, vol.35,no.4 ,1989,pp.658 661. [3] ControlledDissolutionof AmericanChemicalSocietyandAmerican PharmaceuticalAssociation, vol.88,no.7 ,1999, pp.731 738. [4] AIChEJournal ,vol.18 ,no.2, 1972,pp.446 449. [5] release InternationalJournalofFoodScienceandTechnology, vol.46 2006,pp.1 21. [6] N.Muro basedcomputer aideddesign ComputersandChemicalEngineering ,vol.30 2005,pp.28 41. [7] Chemi cal EngineeringResearchandDesign, vol.82 ,no.11,2004,pp.1458 1466. [8] ControlledReleaseofBiologicallyActiveAgents NewYork:PlenumPress,1974,pp. 15 72. [9] K.Das, Con trolled ReleaseTechnology:BioengineeringAspects ,NewYork:John Wiley&Sons,Inc.,1983. [10] ControlledRelease PolymericFormulations ,WashingtonD.C.:AmericanChemicalSociety,1976.

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125 [11] J.T.Carte nsen, AdvancedPharmaceuticalSolids ,vol.110,NewYork:MarcelDekker, Inc.,2001. [12] Internation alJournalofPharmaceutics, vol.314 ,2006,pp.101 119. [13] ReleasefromSolidsII:TheoreticalandExperiemtnalStudyofInfluencesofBasesand BuffersonRatesofDissolution JournaloftheAmerican PharmaceuticalAssociation, vol.47,no.5,1957,pp .376 383. [14] JournalofControlledRelease, vol.92,2003, pp 361 368. [15] ControlledReleasePolymericFormulations, WashingtonD.C.:AmericanChemical Society,1976. [16] ControlledRelease PolymericFormulations, WashingtonD.C.:AmericanChemicalSociety,1976,pp.1 14. [17] TreatiseonControlledDrugDelivery, NewYork:MarcelDekker,Inc.,1992,pp.199 224. [18] ControlledReleaseSystems:FabricationTechnology 1988,pp.1 16. [19] drug ChemicalEngineeringScience ,vol.58 2003, pp. 1337 1351. [20] M.Z.Zhang,C.Li LingandC. Journalof PharmaceuticalSciences, vol.92,no. 10,2003,pp.2040 2056. [21] U. d a AdvancesinPolymerScience, vol.157 ,2002,pp.67 112.

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126 [22] InternationalJournalofFoodSciencesandNutrition ,vol.50 1999,pp.213 224. [23] En capsulationofwater Journalof Microencapsulation ,vol.23 ,no.7,2006,pp.729 740. [24] ChemicalandEngineeringNews ,vol.85 no.5,2007,pp.21 23. [25] G.L.Flynn,S.H.YalkowskyandT.J.Roseman,MassTransportPhenomenaand Models:TheoreticalConceptsReview. JournalofPharmaceuticalSciences, vol.63, no.4,1974,pp.479 510. [26] A.A. JournaloftheAmericanChemicalSociety ,vol.19 ,1897,pp.930 934. [27] Zeitschriftf urPhysikalischeChemie, vol.35 ,1900,pp.283 290. [28] Zeitschriftfur PhysikalischeChemie, vol. 43 ,1904,pp.56 102. [29] ZeitschriftfurPhysikalischeChemie, vol.47 ,1904,pp.52 55. [30] Industrial&EngineeringChemistryResearch, vol.23 ,1931,pp.923 931. [31] ControlledDissolutionof JournalofPharmaceutical Sciences, vol.91 no. 2,2002,pp.534 542. [32] Journalof PharmaceuticalSciences, vol. 52 ,1963,pp.236 241. [33] ntmentbasescontainingdrugsin JournalofPharmaceuticalSciences, vol.50, 1961,pp.874 875.

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127 [34] Int ernationalJournalof Pharmaceutics, vol.321 ,2006,pp.1 11. [35] JournalofPharmaceuticalSciences, vol.65 ,no.7,1976, pp.1042 1045. [36] oreticalanalysisofthereleaseofslowlydissolvingdrugsfrom JournalofControlledRelease, vol.95 ,2004,pp.109 117. [37] effectsofafinite JournalofControlledRelease, vol.92 ,2003,pp. 331 339. [38] F.C.HoppensteadtandC.S.Peskin, ModelingandSimulationinMedicineandtheLife Sciences. NewYork:Springer,2002. [39] ocaineloadedbiodegradable JournalofControlledRelease vol.60 1999,pp.169 177. [40] AIChEJournal vol.33 ,no.1,1987,pp. 54 63. [41] A.Kondo, MicrocapsuleProcessingandTechnoolgy, NewYork:MarcelDekker,Inc., 1979. [42] JournalofPharmac euticalSciences, vol.93 ,no. 8,2004,pp.1941 1944. [43] ,1994 2007,[Online].Available: TheMathWorksWebsite,http://www.mathworks.com[AccessedJuly2007]. [44] W.J.PalmIII, Introductionto MATLAB 7forEngineers, NewYork:McCrawHill, 2005. [45] R.C.ReidandT.K.Sherwood, ThePropertiesofGasesandLiquids, NewYork: McGraw HillBookCompany,1958. [46] COMSOLScriptUser'sGuide,1994 2006.

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128 [47] eequivalentspherical ProceedingsoftheRoyalSocietyLondonA, vol.419 ,1988,pp.137 149. [48] forencapsulationofall trans Journalof ControlledRelease, vol.103, 2005,pp.369 380. [49] W.B.Zimmerman, ProcessModellingandS imulationwithFiniteElementMethods, London:WorldScientificPublishingCo.,2004. [50] JournalofAppliedPolymer Science, vo l.49 ,1993,pp.1653 1658. [51] Omnical Inc.ReactionCalorimeterInstrucments, 2007 Active.[Online].Available: OmnicalTechnologies, http://www.omnicaltech.com [AccessedSep. 9,2007]. [52] Between TheoryandExperimentalDataforDissolutionfromtheRotatingDiskUnderStirred JournalofPharmaceuticalSciences, vol.57, no.9,1968, pp.1629 1631. [53] AdvancedDrugDeliveryReviews, vol.41 ,2001,pp.229 247. [54] Analysis ofDatawithFirst OrderKineticsandwiththeDiffusion JournalofPharmaceuticalSciences, vol.57,no. 2,1968,pp.274 277. [55] Jour nalofControlledRelease, vol.92 ,2003,pp.361 368. [56] JournalofPharmaceuticalSciences ,vol.81,no.4,1992,pp. 318 320 [57] AIChEJournal ,vol.52,no.11,2006,pp. 3762 3774. [58] U.S.DepartmentofHealthandHumanServicesFoodandDrugAdministrationCenter forDrugEv GuidanceforIndustry:DissolutionTestingof ImmediateReleaseSolidOralDosageForms

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129 [59] releasedosage JournalofControlledRelease ,vol. 48,1997,pp. 237 242.

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130 Appendices

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131 A ppendix A :Derivation of D issolution of S olid P articlesina L iquid ThefollowingmathematicalmodelwasdevelopedbySertsou [42] forthedissolutionof solidparticlesinaliquid.Thepaperomittedseveralmathematicalstepsforderivingtheneeded equations,whichareshowninthissection.Theequationswereimper ativetodevelopingthe MAT LA Bcodeusedthroughoutthisthesis. Dissolutionofsolidparticlesinaliquidasdescribedb yNernst Brunnertypekinetics: ) ( C C A h D s dt dM ( A.1 ) whereM=massofsolidmaterialattimet,k=dissolutionrate constant,A=areaavailablefor masstransfer,D=diffusioncoefficientofthedissolvingmaterial,andh=diff usionboundary layerthickness. Forasphericalparticle,thesurfaceareais 2 4 r A ( A.2 ) Thevolumeis 3 / 4 3 r V ( A.3 ) Thereforethechangeinvolume dr r dr A dV 2 4 ( A.4 ) Substitutionof(4)into(1)andsubstitutingrforhyields ) ( 4 4 2 2 C C r N r D dt dr r N dt dV N dt dM s ( A.5 ) wherethetotalmassoftheparticlesis V N M ,where =density,N=numberofparticles ofradiusr.Cancelingliketermsgives: ) ( C C r D dt dr s ( A.6 ) Themassdissolvedatanytime,M d ,isgivenas M M M d 0 ( A.7 ) whereM 0 istheinitialmassoftheparticles,andM=massoftheparticlesremaining.

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132 AppendixA : (Continued) AmassbalanceofM d dividedbythedissolutionmedium,V m, givestheconcentration,C m m m d V r N r N V M M V M C 3 3 4 3 0 3 4 0 ( A.8 ) wherer 0 =initialparticleradius.Su bstituting(8) into(6)yields ) ( 3 3 4 3 0 3 4 m s V r N r N C r D dt dr ( A.9 ) Expanding m m s V r N V r r N r C D dt dr 2 3 4 3 0 3 4 ( A.10 ) m m s V r N V r r N r C D dt dr 2 3 4 3 0 3 4 ( A.11 ) m m s V r r N r N V C D dt dr ) 4 4 3 ( 3 1 3 3 0 ( A.12 ) Multiplybyr m m s V r N r N V C D dt dr r ) 4 4 3 ( 3 1 3 3 0 ( A.13 ) Multiplybydt dt V r N r N V C D dr r m m s ) 4 4 3 ( 3 1 3 3 0 ( A.14 ) dt V N D N r N r N V C dr r m m s 3 4 4 4 4 3 3 3 0 ( A.15 )

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133 AppendixA : (Continued) dt V N D N r N r N V C dr r m m s 3 4 4 4 4 3 3 3 0 ( A.16 ) dt V N D r r N V C dr r m m s 3 4 4 3 3 3 0 ( A.17 ) dt V N D r N V C r dr r m m s 3 4 4 3 3 3 0 ( A.18 ) dt V N D r N V C r dr r m m s 3 4 4 3 3 3 3 1 3 0 ( A.19 ) Let 3 1 3 0 4 3 N V C r c m s ( A.20 ) Then dt V N D r c dr r m 3 4 3 3 ( A.21 ) rearranginggives dt V N D r c dr r m 3 4 3 3 ( A.22 ) where 3 1 4 3 3 0 N V C r c m s ( A.23 )

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134 AppendixA : (Continued) cisacons tantwithrespecttotime.Eq. ( A.21 )describestherateof radiuswithrespecttotime.Integrationofthelefthandsideyields k c c c r c c c r r c c r r c dr r 3 ) 2 ( 3 1 tan 3 3 1 ) log( 6 1 ) log( 3 1 1 2 2 3 3 ( A.24 ) wherek=integrationconstant. Simplifying k c c c r c c r r c r r c dr r 3 ) 2 ( 3 1 tan 3 2 ) log( ) log( 2 6 1 1 2 2 3 3 ( A.25 ) Expandandrearrange k c c r c c c r r c c r r c dr r 3 3 3 3 2 tan 3 3 1 ) log( 6 1 ) log( 3 1 1 2 2 3 3 ( A.26 ) Forinitialconditionswherer=r 0 att=0 3 3 3 4 r c dr r t d V N D m ( A.27 ) k c c r c c c r r c c r t V N D m 3 3 3 3 2 tan 3 3 1 ) log( 6 1 ) log( 3 1 3 4 1 2 2 ( A.28 )

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135 AppendixA : (Continued) Thensincer=r 0 att=0 c c r c c c r r c c r k 3 3 3 3 2 tan 3 3 1 ) log( 6 1 ) log( 3 1 0 1 2 0 2 0 0 ( A.29 ) Tosolvefortime, c c r c c c r r c c r c c r c c c r r c c r N D V t m 3 3 3 3 2 tan 3 3 1 ) log( 6 1 ) log( 3 1 3 3 3 3 2 tan 3 3 1 ) log( 6 1 ) log( 3 1 4 3 0 1 2 0 2 0 0 1 2 2 ( A.3 0 ) t V N D k c c r c r c r c c r c dr r m 3 4 3 2 tan 3 1 ) ( ln 6 1 1 3 3 3 3 3 ( A.3 1 ) wherek=constantofintegration.Applyingtheinitialconditionsofr=r 0 att=0 c c r c r c r c c k 3 2 tan 3 1 ) ( ln 6 1 0 1 3 0 3 0 3 ( A.3 2 ) Forthedissolutionoflargeparticlesuntilthetimewhendiffusionboundarylayerthickness beginsto decreasewithdecreasingparticleradius,thefollowingdifferentialequationis applicable dt V h N D r c dr m 3 4 3 3 ( A.3 3 )

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136 AppendixA : (Continued) whereh=anapproximateconstantdiffusionboundarylayerthickness.Integrationof(14)yields t V h N D k c c r c r c r c c r c dr m 3 4 3 2 tan 3 1 ) ( ln 6 1 1 1 2 3 3 3 2 3 3 ( A.3 4 ) wherek 1 =theconstantofintegration.Applyinginitialconditionsofr=r 0 att=0 c c r c r c r c c k 3 2 tan 3 1 ) ( ln 6 1 0 1 2 3 0 3 0 3 2 1 ( A.3 5 ) Forcaseswhenthediffusionboundarylayerthicknessisdecreasingandapproximatedbythe particleradius, 4 3 3 1 tan 3 2 tan 3 1 ) ( ) ( ln 6 1 1 0 1 3 0 3 3 0 2 N D V c c r c r c r c c t m ( A.3 6 ) thiscorrespondstofindingtherootof Eq. (A.32). Forcaseswhenthediffusionboundarylayer thicknessisconstant,thetimeforparticlestodissolvecorrespondsto 4 3 3 2 tan 3 2 tan 3 1 ) ( ) ( ) ( ) ( ln 6 1 0 1 1 2 3 3 3 0 3 3 3 3 0 2 N D V h c c r c c r c r c r c r c r c c t m ( A.3 7 )

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137 AppendixB: MATLABSampleSourceCode The followingistheMATLABsourcecodeforthemonodisperseparticlemodel.The function Start.m initiatestheprogramtorunfromthecommandwindow.Theresultingplotsseen aftertheprogramhascompletedcalculationsissh ownattheendofthissection. 1 %MONDISPERSEPARTICLEDISSOLUTION 2 %START.m2007 3 %STARTcallsthethreesubrountinesinthisprogram 4 %1)NUMPARTcalculatesthetotalnumberofparticles 5 %2)DIFFPARTfindsthediff usioncoefficient 6 %3)FINDRADIUScalculatestheradiusateachgiventime 7 %4)CONCCONVcalculatestheconcentrationandconversion 8 %basedontheradius 9 %5)PRINTRESULTScreatestable includingtime,conversion 10 %andconcentration 11 %Plotsgraphofconversionandconcentrationv ersu stime 12 13 globalCsD12Nr0rhotVm 14 15 %NUMPARTcalculatesthetotalnumberofparticles 16 NumPart 17 %DIFFPARTfindsthediffusioncoefficient 18 D iffPart 19 %FINDRADIUScalculatestheradiusateachgiventime 20 FindRadius 21 %CONCCONVcalculatestheconcentrationandconversionbasedontheradius 22 ConcConv 23 %PRINTRESULTScreatestableincludingtime,conversionandconcentration 24 PrintResults 1 %MONDISPER SEPARTICLEDISSOLUTION 2 %NUMPART.m2007 3 %NUMPARTcalculatesthenumberofparticles 4 5 %Givenparameters 6 %InitialRadiusofOneParticle 7 r0=[0. 06 ];%cm 8 %TotalGramsofCitricAcid,Mt 9 Mt=[0.10 86 ];%grams 10 %Densityofparticle,rho 11 rho=1.665;%g/cm^3 12 13 %Calculations 14 %TotalVolumeofCitricAcid,Volume_CA 15 Vt=Mt*(1/rho);%cm^3 16 %VolumeofOneParticle,Particle_Volume 17 Vp=(4/3)*pi*r0.^3;%cm^3 18 %Numberofparticles,N 19 N=Vt./Vp;%particlesofCitricAcid

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138 Appendix B: (Continued) 1 %MONDISPERSEPARTICLEDISSOLUTION 2 %DIFFPART.m2007 3 %DIFFPARTcalculatesthediffusioncoefficientfortheparticle 4 %insolution 5 6 globalCsNr0rhotVm 7 8 %ESTIMATIONOFDIFFUSIONCOEFFIECENT 9 %(Properties ofGasesandLiquidsReid,R.C.andSherwoodT.K) 10 %WilkeandChang 11 %GivenParameters 12 %Molecularweightofsolvent,M2 13 M2=18.015;%grams 14 %Temperature,T 15 Temp=318.15;%Kelvin 16 %Viscosityofsolution(solvent),v2 17 v2=0.91 ;%centipoises 18 %Molecularweightofsolute,M1 19 M1=192.12;%grams 20 %Molalvolumeofthesoluteatitsnormalboilingpoint,V1 21 V1=rho*M1;%cm^3/gmole 22 %Associationparameterofsolvent,phi 23 phi=2.6; %recommendedbyWilkeandChangforwater 24 %Calculationofdiffusioncoefficients 25 %Mutualdiffusionofsolute1insolvent2atverylowsolute 26 %concentration(cm^2/sec),D12 27 D12=7.4e 8*(((phi*M2)^(1/2)*Temp)/(v2*V1^0.6)); 1 %MONDISPERSEPARTICLE DISSOLUTION 2 %FINDRADIUS.m2007 3 %FINDRADIUScreatesaloopbetween1 10000secondsinincrements 4 %of100andcalculatestheradiusaseachtime.Theradius 5 %iscalculatedusingthefzerofunctionfortheequationin 6 % FinalDiss. 7 8 globalCsD12Nr0rhotVm 9 10 %Initialtimetostartloop 11 t=0; 12 %Loopsetfrom1 100 13 fori=1:100 14 %Assignvaluefortime 15 Time(i)=t; 16 %Solvefunctionforradiusateachgiventime 17 CalR=fzero(@Function,r0); 18 19 %Checkthatcalculatedradiu sisnotnegative,andifitis,assign 20 %theparticleradiustoequalzero. 21 ifCalR<0 22 r(i)=0; 23 else 24 r(i)=CalR; 25 end 26 %Updateincrementt 27 t=t+100; end

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139 AppendixB:(Continued) 1 %MONDISPERSEPARTICLEDISSOLUTION 2 %FUNCTION.m2007 3 % FUNCTIONcalculatestheconstant,c,andtheintegration 4 %constant,k,fortheparticleandthefinaleqautioniswrittenas 5 %functionF. 6 7 functionF=Function(r) 8 9 globalCsD12Nr0rhotVm 10 11 %CALCULATIONOFCONSTANT,c 12 %Solubilityo fthesolute,Cs 13 Cs=1.33;%g/mL 14 %DissolutionMediumVolume,Vm 15 Vm=1;%1mLH2O 16 %Constantwithrespecttotime,c 17 c=nthroot((r0.^3. (3.*Cs*Vm)./(N.*rho*4*pi)),3); 18 19 %CONSTANTOFINTEGRATION,k 20 k=(1./(6*c)).*log((c.^3 r0.^3.)/(c r0).^3.) 21 (1./(sqrt(3)*c)).*(atan((2*r0+c)./(sqrt(3)*c))); 22 23 %FINALFUNCTIONRELATINGTIMEANDRADIUS 24 F=(((1./(6*c)).*log((c.^3 r.^3.)./(c r).^3.) 25 (1./(sqrt(3)*c))*(atan((2*r+c)./(sqrt(3)*c)))) k) (((D12*N*4*pi)*t)/(3*Vm)); 26 27 end 1 %MONDISPERSEPARTICLE DISSOLUTION 2 %CONCCONV.m2007 3 %CONCCONVcalculatestheconcentrationandconversionusingthe 4 %radiusfoundinFINDRADIUS. 5 6 globalCsD12Nr0rhotVm 7 8 %InitialMassofParticles,Mo 9 Mo=(r0.^3)*(4/3)*rho.*N*pi; 10 %MassofParticlesRem aining,M 11 M=N*rho*(4/3)*pi*r.^3; 12 %Concentrationofsolid,C1 13 C1=(M)/Vm;%g/mL 14 %Conversion,X 15 Md=(Mo M); 16 if(Md==0) 17 X=1; 18 else 19 X=((Mo M)/Vm)/(Mo/Vm); 20 end 1 %MONDISPERSEPARTICLEDISSOLUTION 2 %PRINTRESULTS.m2007 3 %PRINTRESULTS convertstimeintominutesthendisplays 4 %concentrationandconversioninatable.Thenplotsofthe 5 %concentrationandconversionofcitricacidaredisplayed.

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140 AppendixB:(Continued) 6 globalCsD12Nr0rhotVm 7 8 %TimeConvertedto Minutes 9 TimeMin=Time/60; 10 11 PRINTRESULTS 12 plot(TimeMin,C1,' r','LineWidth',3) 13 14 ModelConcentrationversusTime');gridon 15 holdon; 16 figure; 17 plot(TimeMin,X,'Color',[10.20 .2],'LineWidth',3) 18 xlabel('Time(min)');ylabel('ConversionofCitricAcid(X)');title('MonodisperseModel 19 ConversionversusTime');gridon 20 ylim([01.1]); 21 xlim([020]); 22 holdoff; Twographsareproducedbythisprogramwhicharedisplayedinthecommand window.The firstgraphshowstheconcentrationofcitricacidasafunctionoftime: FigureB. 1 :MATLABresultingplotforconcentrationformonodispersemodel.

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141 AppendixB:(Continued) Thesecondgraphshows theconversionofcitricacidasafunctionoftime: FigureB. 2 :MATLABresultingplotforcon version formonodispersemodel. Thetotaltimeforconversionwasunder8minutesforthemonodispersemodel.

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142 AppendixC:COMSOLSampleSourceCode ThissectiondescribeshowtoimportthedataresultscalculatedbyMATLABintoa predesignedfunctioninCOMSOLScriptwhichcreatesthree dimensionalvisualizationsofthe shrinkingsphericalparticle. Thefirstst epinvolvescopyingthedatavectors forvariables r0 and Pr fromthe Workspace windowinMATLABandinputtingthe data intothe CommandPromptwindowin COMSOL.Thefollowingfiguredemonstratesthistechnique. -------------------------------------------------------COMSOLScript1.1.0.511 Copyright(c)COMSOLAB1994 2007 Type'help'toseeavailablefunctions. --------------------------------------------------------CParticleR=[0.06 0.057891 0.055721 0.053481 0.051161 0.048748 0.046 228 0.04358 0.040778 0.037787 0.034557 0.031009 0.027018 0.022345 0.016411 0.0063142 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]; C r0=0.06; FigureC. 1 :CommandpromptwindowinCOMSOLScript. Afterthedatavaluesareinputted,thefunction MonodisperseMOVIE iscalled.This functionparametricallyconvertstheradiusdatainputtedinthe ParticleR variablevector.Italso displaysthethree dimensionalsphereandcorrespondingplotforagivetime.Animation capabilitiesarealsoavailable,sinceeachframeissavedinaviformatandcanbeplayedusing WindowsMediaPlayer. Thefollowingisas amplesourcecodeforthefunctionwhichgeneratesthree dimensionalspherical animationsoftheparticlesforagiventimerange: 1 %MONODISPERSEMODELMOVIE 2 globalr0ParticleR 3 phi=0:pi/20:pi; 4 theta=0:pi/10:2*pi; 5 [Phi,Theta]=meshgrid(phi,theta); 6 %Equatio nsusedinparametrizationofsphere 7 X=sin(Phi).*cos(Theta); 8 Y=sin(Phi).*sin(Theta); 9 Z=cos(Phi);

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143 AppendixC:(Continued) 10 %Plotthesurface,andscaletheaxessothatitlookslikeasphere 11 h=surf(X,Y,Z,'EdgeColor','interp','FaceColor',[10.20.2]); 12 ax is([ r0r0 r0r0 r0r0]); 13 %Savetheframestocreateamovie 14 m=movie('width',800,'height',600); 15 fori=1:60 16 delete(h) 17 a=ParticleR(i); 18 X1=a*X; 19 Y1=a*Y; 20 Z1=a*Z; 21 h=surf(X1,Y1,Z1,'EdgeColor','interp','FaceColor',[10.20.2]); 22 axis([ r0r0 r0r0 r0r0]); 23 title(['Particleatt=',num2str(i),'sec']); 24 gridon; 25 m.addFrame; 26 %Creationofanimationfile 27 m.generate('MonodisperseModelMovie'); 28 end Theresulting plot andanimationwillberepresentedgraphicallyontheagridresemblingtheone seenbelow. FigureC. 2 : COMSOLScriptvisualizationofsphericalparticle.

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144 Appendix D :ExperimentalProcedure usingReactionMicrocalorimeter Forthedissolutionstudiesofsoldparticlesinliquid,a n OmniCalTechnologies SuperCRC20 305 2.4 reactionmicrocalorimeterwas used. Individualsamplesofcitricacid wereweigheddirectlyintothe 16 mLglassvials. Anhydrouscitricacid,C 6 H 8 O 7, wasobtained fromFisher Chemicals. ATeflon linedscrew top wasusedtosealthevialsandusedtoavoid anylossofmaterialduringtheexperiment.Vial pickingneedleswereusedtoplacethevialsinto theCRCunit.Theliquidsamples,1mLofwater,weremeasuredusingintotwosyringesand placedintothesy ringesamplebarrels.TheCRCunitisturnedonandthesamplesandsyringe barrelswereallowedtocometothermalequilibriumwiththecalorimeterheatsinktemperature, usuallyatleast30minutesweregivenforthisstep. TheCRCcalorimeteroperates on linewiththeprogramsoftwareWinCRC Turbo,which isexecutedafterthesampleshavebeeninsertedintotheunitandtheunitisturnedon. The operationalprogramWinCRCTurboisthemicrocalorimeteroperationalprogram.Thesetup windowisshownin FigureD.1. WinCRCTurbowasusedforbothdataacquisitionand conversioncalculations.Samplesweremonitoreduntilasmoothbaselineintheheatflowwas maintained.Thentheliquidwasinjectedintoboththesampleandreferencevial,andthe detec tionandcollectionofdatawasmaintaineduntilnochangeswereseenintheheatflowforat least30minutes. AnexampleofaresultingheatcurveisshowninFigureD.2.

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145 AppendixD:(Continued) FigureD.1:WinCRCTurbomicrocalorimeterprogram setupwindow. FigureD.2:Heatflowcurvefromreaction.

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146 AppendixD:(Continued) Thefollowingstepswerepreformedsothatadynamiccorrectioncouldlaterbe calculatedforthedata.Sincetheheatratecurvesweretobeusedforthereactionkine tics,itwas importanttohavethemostaccurateheatflowpeakvaluesandcurvefittingoftheheatflowdata. Therefore,thethermallag,orthermalinertiaoftheheatconductionreactorstationsystem,was measuredandaccountedfor [51] .Afterthere actionheatflowsignalremainsatthebaselinefor 30minutes,asthepreviousstepmentioned,thecalibrationheaterwasturnedonfor30minutes andtheheatflowwasallowedtostabilized.Thentheheaterwasswitchedoffand10minutesare allottedf orthecalibrationheatflowsignaltoreturntothebaseline.Thedatacollectionisthen stoppedandsaved. FigureD.3showsanexampleoftheresultinggraph. FigureD.3:Dynamiccorrectionusingheatcurveoption.

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147 AppendixD:(Continued) T hedatawasanalyzedintheWinCRCTurboandthecurvewasimportedfromthe operationalpanel.Inthecurveoperationmenu,theredcurserwasdraggedtotheposition5 minutesinfrontoftheheatingpeakandthegreencursorwasmovedto5minutesaft erthepeak. Inthedynamiccorrectionssection,valuesforthe1 st 1 and2 nd 2 timeconstantsare manipulateduntilasquarecurveresults.Thetaucorrectedcurveisthensaved.Toapplythe determinedcorrectiontotheactualheatflow data,theredcursorismovedto5minutesinfrontof theactualreactionheatpeakandgreencursorismovedto5minutesafterthepeak.Usingthe samecorrectionconstants,thedynamiccorrectioniconinthetoolbarisclickedandacorrected heatrat ecurvewillbegenerated ,asshowninFigureD.4 .Thetaucorrectedheatflowdata shouldthenbesaved. FigureD.4:Applyingtaucorrectiontoreactionheatflowcurve.

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148 AppendixD:(Continued) Inanotherprogrammenu,openthesaveddataand importthecurve.Thestartandendpointsof thecurveshouldbeselected.Intheintegrationmenu,thebaselineconstructionselectionshould bepoint to pointnonlinear.Theintegrationshouldbecalculatedandthetotalshouldbe recorded.Thedata shouldbetrimmed,startingfromthepointofinjectionandendingabout30 minutesaftertheheatpeak,usingthecurveoperation asshowninFigureD.5 .Thetrimmed curvewillhavetobesavedasareconstructedcurve. FigureD.5:Trimmedcorrec tedheatflowcurve. Anotherprogrammenumustbeopenedandthecurveshouldbereopened.Usingtheredcursor selectthestartingpointandwiththegreencursorselecttheendpoint.Theprocessofintegration shouldberepeated,withtheconstructio nofabaselinepoint to pointnonlinear.

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149 AppendixD:(Continued) FigureD.6:Integratedheatflowcurveandconversion. Theintegrationshouldberecalculatedandcomparedtothetotalintegrationpreviouslyfound,so thatthevaluecanbevalidated.Ifthevaluecalculatedisreasonable,theFilemenuisopenedand thesaveconversiondataoptionisselected. Theconvers iondatacanalsobeseenintheprogram windowasablackline,asshowninFigureD.6.

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150 A ppendix E : SampleData Thissectionincludes sample data obtainedexperimentallycomparedwithvalues calculatedfrom thecomputer mo dels Dataforcitricacidamountsof0.02 0. 2 0gramsin1mLH2Oat298Kwith nostirringisgiveninthefollowingtables. Datafor0.02gramsofcitricacid : Time(mins) Experiment Model1:Mono disperse Model2:Poly disperse Residual1 Residual2 0 0.0001 0 0 0.0001 0.0001 0.5 0.0108 0.10545 0.094921 0.09465 0.084121 1 0.179 0.2066 0.18576 0.0276 0.00676 1.5 0.4091 0.30334 0.27239 0.10576 0.13671 2 0.5841 0.39554 0.35461 0.18856 0.22949 2.5 0.7054 0.48303 0.43218 0.22237 0.27322 3 0.789 0.56563 0.50471 0.22337 0.28429 3.5 0.8478 0.64311 0.57132 0.20469 0.27648 4 0.8901 0.71518 0.63255 0.17492 0.25755 4.5 0.9211 0.78148 0.68829 0.13962 0.23281 5 0.943 0.84153 0.73724 0.10147 0.20576 5.5 0.9587 0.8947 0.777 0.064 0.1817 6 0.97 0.94 0.81281 0.03 0.15719 6.5 0.9778 0.97582 0.84459 0.00198 0.13321 7 0.9841 0.99834 0.8711 0.01424 0.113 7.5 0.9889 1 0.88992 0.0111 0.09898 8 0.9926 1 0.90692 0.0074 0.08568 8.5 0.9956 1 0.92246 0.0044 0.07314 9 0.9983 1 0.93637 0.0017 0.06193 9.5 1.0003 1 0.94833 0.0003 0.05197 10 1.0019 1 0.95745 0.0019 0.04445 10.5 1.0037 1 0.96325 0.0037 0.04045 11 1.0049 1 0.96863 0.0049 0.03627 11.5 1.0058 1 0.97355 0.0058 0.03225 12 1.0067 1 0.97793 0.0067 0.02877 12.5 1.0072 1 0.98169 0.0072 0.02551 13 1.0079 1 0.98458 0.0079 0.02332 13.5 1.0086 1 0.98612 0.0086 0.02248 14 1.0091 1 0.98761 0.0091 0.02149 14.5 1.0097 1 0.98904 0.0097 0.02066 15 1.01 1 0.9904 0.01 0.0196 SumofResiduals 1.69373 3.259341

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151 AppendixE:(Continued) Datafor0.04 gramsofcitricacid Time(mins) Experiment Model1:Mono disperse Model2:Poly disperse Residual1 Residual2 0 0.0001 0 0 0.0001 0.0001 0.5 0.0108 0.10554 0.094988 0.09474 0.084188 1 0.179 0.20692 0.18601 0.02792 0.00701 1.5 0.4091 0.30401 0.27291 0.10509 0.13619 2 0.5841 0.39665 0.35547 0.18745 0.22863 2.5 0.7054 0.48465 0.43341 0.22075 0.27199 3 0.789 0.5678 0.5063 0.2212 0.2827 3.5 0.8478 0.64583 0.5732 0.20197 0.2746 4 0.8901 0.71842 0.63474 0.17168 0.25536 4.5 0.9211 0.78517 0.69067 0.13593 0.23043 5 0.943 0.84555 0.73953 0.09745 0.20347 5.5 0.9587 0.89885 0.77921 0.05985 0.17949 6 0.97 0.944 0.81511 0.026 0.15489 6.5 0.9778 0.97917 0.84683 0.00137 0.13097 7 0.9841 1 0.87305 0.0159 0.11105 7.5 0.9889 1 0.89144 0.0111 0.09746 8 0.9926 1 0.90847 0.0074 0.08413 8.5 0.9956 1 0.92402 0.0044 0.07158 9 0.9983 1 0.93787 0.0017 0.06043 9.5 1.0003 1 0.94966 0.0003 0.05064 10 1.0019 1 0.95816 0.0019 0.04374 10.5 1.0037 1 0.96397 0.0037 0.03973 11 1.0049 1 0.96935 0.0049 0.03555 11.5 1.0058 1 0.97424 0.0058 0.03156 12 1.0067 1 0.97858 0.0067 0.02812 12.5 1.0072 1 0.98224 0.0072 0.02496 13 1.0079 1 0.98481 0.0079 0.02309 13.5 1.0086 1 0.98636 0.0086 0.02224 14 1.0091 1 0.98786 0.0091 0.02124 14.5 1.0097 1 0.98929 0.0097 0.02041 15 1.01 1 0.99066 0.01 0.01934 SumofResiduals 1.6678 3.225288

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152 AppendixE:(Continued) Datafor0.06gramsofcitricacid Time(mins) Experiment Model1:Mono disperse Model2:Poly disperse Residual1 Residual2 0 0.0001 0 0 0.0001 0.0001 0.5 0.0377 0.10536 0.094838 0.06766 0.057138 1 0.1721 0.20627 0.18545 0.03417 0.01335 1.5 0.3222 0.30263 0.27174 0.01957 0.05046 2 0.4506 0.39435 0.35354 0.05625 0.09706 2.5 0.5546 0.48129 0.43066 0.07331 0.12394 3 0.6382 0.5633 0.50276 0.0749 0.13544 3.5 0.7054 0.64019 0.569 0.06521 0.1364 4 0.7598 0.7117 0.62985 0.0481 0.12995 4.5 0.8036 0.77752 0.68533 0.02608 0.11827 5 0.839 0.83721 0.73435 0.00179 0.10465 5.5 0.8685 0.89021 0.77428 0.02171 0.09422 6 0.8928 0.93564 0.80996 0.04284 0.08284 6.5 0.9128 0.97207 0.84179 0.05927 0.07101 7 0.9293 0.99637 0.86872 0.06707 0.06058 7.5 0.9433 1 0.88805 0.0567 0.05525 8 0.9546 1 0.90498 0.0454 0.04962 8.5 0.9641 1 0.92052 0.0359 0.04358 9 0.9719 1 0.93449 0.0281 0.03741 9.5 0.9781 1 0.94663 0.0219 0.03147 10 0.9832 1 0.95632 0.0168 0.02688 10.5 0.9873 1 0.96235 0.0127 0.02495 11 0.9908 1 0.96773 0.0092 0.02307 11.5 0.9935 1 0.97267 0.0065 0.02083 12 0.996 1 0.97711 0.004 0.01889 12.5 0.9981 1 0.98097 0.0019 0.01713 13 0.9996 1 0.98405 0.0004 0.01555 13.5 1.0013 1 0.98583 0.0013 0.01547 14 1.0026 1 0.98731 0.0026 0.01529 14.5 1.0037 1 0.98873 0.0037 0.01497 15 1.0047 1 0.99009 0.0047 0.01461 SumofResiduals 0.90983 1.700378

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153 AppendixE:(Continued) Datafor0.08grams ofcitricacid Time(mins) Experiment Model1:Mono disperse Model2:Poly disperse Residual1 Residual2 0 0.0002 0 0 0.0002 0.0002 0.5 0.0617 0.10527 0.094755 0.04357 0.033055 1 0.2249 0.20593 0.18514 0.01897 0.03976 1.5 0.3901 0.30191 0.27109 0.08819 0.11901 2 0.5247 0.39315 0.35248 0.13155 0.17222 2.5 0.6293 0.47954 0.42914 0.14976 0.20016 3 0.71 0.56097 0.50078 0.14903 0.20922 3.5 0.7721 0.63726 0.56667 0.13484 0.20543 4 0.82 0.7082 0.62713 0.1118 0.19287 4.5 0.8566 0.77353 0.68234 0.08307 0.17426 5 0.8847 0.83285 0.73136 0.05185 0.15334 5.5 0.9065 0.88566 0.77152 0.02084 0.13498 6 0.9238 0.93117 0.80707 0.00737 0.11673 6.5 0.9379 0.96814 0.83891 0.03024 0.09899 7 0.949 0.99395 0.86618 0.04495 0.08282 7.5 0.9579 1 0.88616 0.0421 0.07174 8 0.9649 1 0.90302 0.0351 0.06188 8.5 0.9705 1 0.91854 0.0295 0.05196 9 0.9748 1 0.93256 0.0252 0.04224 9.5 0.9784 1 0.94486 0.0216 0.03354 10 0.9812 1 0.95495 0.0188 0.02625 10.5 0.9837 1 0.96144 0.0163 0.02226 11 0.9858 1 0.96682 0.0142 0.01898 11.5 0.9877 1 0.97178 0.0123 0.01592 12 0.9892 1 0.97627 0.0108 0.01293 12.5 0.9906 1 0.98021 0.0094 0.01039 13 0.9919 1 0.98346 0.0081 0.00844 13.5 0.9932 1 0.98553 0.0068 0.00767 14 0.9941 1 0.987 0.0059 0.0071 14.5 0.9952 1 0.98841 0.0048 0.00679 15 0.996 1 0.98977 0.004 0.00623 SumofResiduals 1.33113 2.337365

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154 AppendixE:(Continued) Datafor0.10gramsofcitricacid Time(mins) Experiment Model1:Mono disperse Model2:Poly disperse Residual1 Residual2 0 0.0001 0.11214 0.094682 0.11204 0.094582 0.5 0.0541 0.21885 0.18487 0.16475 0.13077 1 0.2059 0.3201 0.27052 0.1142 0.06462 1.5 0.3586 0.41583 0.35155 0.05723 0.00705 2 0.4831 0.50595 0.42781 0.02285 0.05529 2.5 0.5797 0.59031 0.49907 0.01061 0.08063 3 0.6543 0.66874 0.56464 0.01444 0.08966 3.5 0.7123 0.74098 0.62476 0.02868 0.08754 4 0.7576 0.80669 0.67973 0.04909 0.07787 4.5 0.7932 0.86537 0.72871 0.07217 0.06449 5 0.821 0.91629 0.76912 0.09529 0.05188 5.5 0.8429 0.95827 0.80454 0.11537 0.03836 6 0.8602 0.98899 0.83638 0.12879 0.02382 6.5 0.8742 1 0.86389 0.1258 0.01031 7 0.8859 1 0.88451 0.1141 0.00139 7.5 0.8956 1 0.90131 0.1044 0.00571 8 0.9036 1 0.9168 0.0964 0.0132 8.5 0.9103 1 0.93085 0.0897 0.02055 9 0.916 1 0.94327 0.084 0.02727 9.5 0.921 1 0.95365 0.079 0.03265 10 0.9253 1 0.96065 0.0747 0.03535 10.5 0.9291 1 0.96602 0.0709 0.03692 11 0.9323 1 0.97099 0.0677 0.03869 11.5 0.935 1 0.97551 0.065 0.04051 12 0.9373 1 0.97952 0.0627 0.04222 12.5 0.9394 1 0.98289 0.0606 0.04349 13 0.9412 1 0.98527 0.0588 0.04407 13.5 0.9429 1 0.98673 0.0571 0.04383 14 0.9444 1 0.98814 0.0556 0.04374 14.5 0.9459 1 0.98949 0.0541 0.04359 15 0.9473 1 0.99078 0.0527 0.04348 SumofResiduals 2.35881 1.433532

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155 AppendixE:(Continued) Datafor 0.20gramsofcitricacid Time(mins) Experiment Model1:Mono disperse Model2:Poly disperse Residual1 Residual2 0 0.0001 1.36E 16 0 1E 04 0.0001 0.5 0.0465 0.11166 0.094292 0.06516 0.047792 1 0.1784 0.21707 0.18342 0.03867 0.00502 1.5 0.3029 0.31635 0.26749 0.01345 0.03541 2 0.3997 0.40962 0.34659 0.00992 0.05311 2.5 0.4743 0.49695 0.42074 0.02265 0.05356 3 0.5328 0.57838 0.48989 0.04558 0.04291 3.5 0.5799 0.65389 0.55378 0.07399 0.02612 4 0.6186 0.72344 0.61203 0.10484 0.00657 4.5 0.6512 0.78688 0.66555 0.13568 0.01435 5 0.679 0.84398 0.71395 0.16498 0.03495 5.5 0.703 0.89436 0.75617 0.19136 0.05317 6 0.7243 0.93739 0.79078 0.21309 0.06648 6.5 0.7434 0.97195 0.82233 0.22855 0.07893 7 0.7605 0.99548 0.85048 0.23498 0.08998 7.5 0.7761 1 0.87447 0.2239 0.09837 8 0.7905 1 0.89202 0.2095 0.10152 8.5 0.8039 1 0.90728 0.1961 0.10338 9 0.8164 1 0.92135 0.1836 0.10495 9.5 0.8281 1 0.93412 0.1719 0.10602 10 0.8391 1 0.94544 0.1609 0.10634 10.5 0.8495 1 0.95498 0.1505 0.10548 11 0.8593 1 0.96163 0.1407 0.10233 11.5 0.8685 1 0.9666 0.1315 0.0981 12 0.8772 1 0.97122 0.1228 0.09402 12.5 0.8853 1 0.97545 0.1147 0.09015 13 0.893 1 0.97924 0.107 0.08624 13.5 0.9003 1 0.98253 0.0997 0.08223 14 0.9072 1 0.98514 0.0928 0.07794 14.5 0.9136 1 0.98666 0.0864 0.07306 15 0.9196 1 0.98798 0.0804 0.06838 SumofResiduals 3.8154 2.106962