! ! ! ! A!RAPID!TESTING!INSTRUMENT!TO!ESTIMATE!THERMAL!PROPERTIES!OF!FOOD! MATERIALS!AT!ELEVATED!TEMPERATURES!DURING!NONISOTHERMAL!HEATING! ! By! ! Dharmendra!Kumar!Mishra! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! A!DISSERTATION! ! Submitted!to! Michigan!State!University! In!partial!fulfillment!of!the!requirements! for!the!degree!of! ! Biosystems!Engineering!J!Doctor!of!Philosophy! ! 2013! ! ! ! ! ! ABSTRACT' A!RAPID!TESTING!INSTRUMENT!TO!ESTIMATE!THERMAL!PROPERTIES!OF!FOOD! MATERIALS!AT!ELEVATED!TEMPERATURES!DURING!NONISOTHERMAL!HEATING! By! Dharmendra!Kumar!Mishra' Modeling! kinetics! of! thermal! degradation! of! nutrients! for! food! quality! or! kinetics! of! microbial! reduction! for! food! safety! requires! reliable! estimates! of! the! thermal!properties.!Thermophysical!properties,!especially!thermal!conductivity!and! specific! heat,! are! important! in! establishing! thermal! processes! for! food! manufacturing,!especially!at!higher!processing!temperatures.!!Hence,!in!this!study,!a! novel!instrument!(TPCell)!was!designed!and!developed!using!principles!of!intrinsic! verification! and! inverse! heat! conduction.! An! intrinsic! verification! method! was! developed! to! ascertain! the! parameter! identifiability! in! the! model! and! to! check! the! accuracy! of! the! numerical! codes! used! to! solve! the! partial! differential! equation! for! heat! conduction.! The! concept! of! intrinsic! sum! was! introduced,! which! is! sum! of! all! the!scaled!sensitivity!coefficients!in!the!model.!The!intrinsic!sum!was!derived!using! dimensionless! derivation! of! scaled! sensitivity! coefficients.! The! design! of! the! instrument! was! based! on! the! insight! gained! from! the! dimensionless! scaled! sensitivity! coefficients! and! the! intrinsic! sum.! With! the! instrument,! thermal! conductivity! can! be! measured! from! room! temperature! to! higher! processing! temperature! of! 140oC.! Several! food! materials! were! tested! using! the! instrument.! Sweet!potato!puree!thermal!conductivity!was!measured!to!be!0.539!W/moC!at!20oC! and! 0.574! W/moC! at! 140oC.! The! experimental! time! with! TPCell! is! less! than! a! minute,! as! compared! to! 5J6! hours! with! quasiJisothermal! method! employed! by! currently!available!instruments.!TPCell!has!advantages!over!traditional!methods,!as! it! avoids! the! decomposition! of! materials! that! result! when! achieving! the! quasiJ isothermal! state! at! higher! temperatures.! TemperatureJdependent! thermal! properties! were! used! to! estimate! the! kinetic! parameters! of! nutrient! degradation! during! aseptic! and! conventional! retort! processing.! Vitamin! C! and! thiamin! were! selected!as!model!nutrients!for!degradation!study.!Sweet!potato!puree!was!used!as!a! food!matrix.!Aseptic!processing!had!50%!higher!retention!of!Vitamin!C!as!compared! to!retort!processing.!Thiamin!retention!could!not!be!quantified,!as!it!survived!well!in! aseptic!as!well!as!retort!processing.!The!rate!of!reaction!for!ascorbic!acid!in!aseptic! processing!and!retort!processing!was!0.0073!minJ1!and!0.0114!minJ1!at!a!reference! temperature! of! 127oC,! respectively.! The! activation! energy! for! ascorbic! acid! in! aseptic! processing! and! retort! processing! was! 26.62! KJ/gJmol! and! 3.43! KJ/gJmol,! respectively.! The! kinetic! parameter! of! thiamin! could! not! be! estimated! due! to! insufficient!degradation!in!aseptic!as!well!as!in!retort!processing.! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! DEDICATION' ! To! my! father,! Sri! Ras! Bihari! Mishra! and! my! mother,! Smt! Sabita! Mishra,! for! their! selfless!sacrifice!and!for!believing!in!me!and!providing!all!the!assistance!I!needed!at! all! times.! To! my! sister! Usha! Pandey! and! to! my! brothers! Binod! K! Mishra,! Ashok! K! Mishra! and! Ravindra! K! Mishra! for! their! love! and! affection! and! to! the! younger! generation.!To!Patnarin!Benyathiar!for!her!affection!and!support!and!to!Sri!Balram! Singh!who!taught!me!the!concept!of!belief.! ' ! ! ! iv ACKNOWLEDGEMENTS' ! I!would!like!to!express!my!utmost!gratitude!to!my!advisor,!Dr.!Kirk!Dolan,!for! his!consistent!support!and!guidance!throughout!my!M.S.!and!Ph.D.!research!period.! His!mentorship!has!helped!me!become!a!better!person.!Dr.!Dolan’s!high!motivation! and! expectations! inspired! me! to! achieve! beyond! my! capabilities.! Under! his! mentorship!I!gained!exposure!to!numerous!problemJsolving!techniques!not!only!in! academics!but!also!in!personal!and!professional!life!as!well.!I!would!like!to!thank!Dr.! Ferhan! Ozadali! for! his! relentless! support.! He! jump! started! my! professional! career! and!provided!much!needed!guidance.!Because!of!him!and!with!generous!help!from! Gene!Ford,!my!Ph.D.!project!was!funded!by!Nestle.!!I!extend!my!thanks!to!Gene!Ford! for!his!continued!support!for!my!research!and!also!to!Dr.!Curt!Emenhiser.!I!would! also!like!to!thank!Dr.!James!Beck!for!his!mentorship!and!selfless!help!throughout!my! research.!He!taught!me!the!invaluable!lessons,!which!I!am!going!to!need!all!my!life.!I! would!like!to!thank!my!committee!members!Dr.!Bradley!Marks!and!Dr.!Lijian!Yang! for!their!valuable!time!and!effort!in!the!completion!of!this!research!work!and!were! the! source! of! invaluable! advice! through! our! engaging! discussions.! I! would! like! to! express! my! gratitude! to! Dr.! Mark! Uebersax,! Dr.! Neil! Wright! and! Dr.! Robert! J.! Tempelman!for!providing!opportunities!that!have!helped!me!immensely.!One!more! wonderful!person!whom!I!could!not!forget!is!Dr.!R.!Paul!Singh!for!paving!my!way!to! higher!studies.! v My! sincere! thanks! goes! to! Jonathan! R.! Althouse! and! Mark! Ven! Ee! for! his! dedicated!help!in!designing!the!data!logging!system!for!the!instrument!and!to!Phil! Hill! for! his! help! with! the! instrument! design.! My! heartfelt! thanks! extend! to! my! colleagues!at!Nestle!Nutrition,!PTC!Fremont!for!their!encouragement!and!help.! ! My!deepest!thanks!to!lovely!Patnarin!Benyathiar!(Oh)!for!her!selfless!help!all! the!time.!I!would!not!be!able!to!complete!my!research!without!her!being!there!for! me.!My!appreciation!also!goes!to!Dr.!Winnie!ChiangJDolan!for!her!guidance,!support! and! valuable! advices.! My! very! warm! thanks! to! Dr.! Rabiha! Sulaiman! for! her! passionate!help!with!my!project!and!to!Hayati!Samsudin!who!was!always!there!for! me!when!I!needed!help.!! I!am!also!grateful!to!the!faculty!and!staff!of!Biosystems!Engineering!as!well!as! Food!Science,!MSU.!I!am!also!thankful!to!all!my!friends!and!fellow!graduate!students! who!have!provided!me!the!wonderful!friendly!atmosphere!throughout!my!research.! Special!thanks!to!my!lab!mates!at!room!129!in!Food!Science!Trout!Building!for!their! love!and!encouragement!all!the!time.! Finally,!I!would!like!to!thank!my!parents!and!all!family!members,!as!they!are! the!inspiration!for!what!I!am!today.!!! ! ! ! vi TABLE'OF'CONTENTS' ! LIST!OF!TABLES!.........................................................................................................................................!x! LIST!OF!FIGURES!.....................................................................................................................................!xii! . KEY!TO!SYMBOLS!AND!ABBREVIATIONS!.....................................................................................!xv! Chapter!1!.......................................................................................................................................................!1! Introduction!............................................................................................................................................!1! 1.1!Introduction!................................................................................................................................!1! 1.2!Objectives!of!the!study!...........................................................................................................!2! 1.3!Overview!of!the!dissertation!...............................................................................................!3! 1.4!Literature!Review!.....................................................................................................................!4! 1.4.1!Predictive!model!..............................................................................................................!4! 1.4.2!Thermal!conductivity!devices!....................................................................................!4! 1.4.2.1!Line!heat!source!method!......................................................................................!5! 1.4.2.2!Modified!Fitch!method!..........................................................................................!7! 1.4.3!Thermal!properties!at!elevated!temperatures!....................................................!7! 1.5!Conclusions!.................................................................................................................................!8! REFERENCES!.......................................................................................................................................!11! Chapter!2!....................................................................................................................................................!12! Use!of!Scaled!Sensitivity!Coefficient!Relations!for!Intrinsic!Verification!of! Numerical!Codes!and!Parameter!Estimation!.........................................................................!12! Abstract!..................................................................................................................................................!12! 2.1!Introduction!.............................................................................................................................!12! 2.1.1!Numerical!code!verification!.....................................................................................!13! 2.1.2!Parameter!estimation!and!sensitivity!coefficient!...........................................!15! 2.2!Derivation!of!sensitivity!coefficients!in!oneJdimensional!heat!conduction! problem!.............................................................................................................................................!18! 2.2.1!Case!1:!OneJdimensional!transient!heat!conduction!in!a!flat!plate!with! heat!flux!on!one!side!and!insulated!on!another!(X22BJ0T1)!................................!18! 2.2.1.1!Dimensionless!analysis!of!sensitivity!coefficient!...................................!22! 2.2.2!Case!2:!OneJdimensional!transient!heat!conduction!in!a!flat!plate!with! time!varying!temperature!on!one!side!and!insulated!on!another!(X12BJ0T1) !..........................................................................................................................................................!31! 2.2.3!Case!3:!Scaled!sensitivity!relation!for!oneJdimensional!transient!heat! conduction!in!a!cylindrical!coordinate!system!for!boundary!condition!of! second!kind!(R22B10T1)!......................................................................................................!34! 2.2.4!Case!4:!Scaled!sensitivity!relation!for!oneJdimensional!transient!heat! conduction!in!a!cylindrical!coordinate!system!for!boundary!condition!of!first! kind!(R12B10T1)!.....................................................................................................................!38! 2.2.5!Case!5:!Scaled!sensitivity!relation!for!oneJdimensional!transient!heat! conduction!in!a!cylindrical!coordinate!system!for!boundary!condition!of!third! kind!(R32B10T1)!.....................................................................................................................!41! vii 2.3!Conclusions!..............................................................................................................................!45! REFERENCES!.......................................................................................................................................!47! Chapter!3!....................................................................................................................................................!49! Intrinsic!Verification!in!Parameter!Estimation!Problems!for!TemperatureJ Dependent!Thermal!Properties!...................................................................................................!49! Abstract!..................................................................................................................................................!49! 3.1!Introduction!.............................................................................................................................!50! 3.2!Case!1:!OneJdimensional!transient!heat!conduction!in!a!flat!plate!with!heat! flux!on!one!side!and!insulated!on!another!(X22B10T1)!..............................................!53! 3.3!Case!2:!Transient!heat!conduction!in!a!hollow!cylinder!with!heat!flux!on! inside!and!insulated!on!the!outside!(R22B10T1)!...........................................................!63! 3.4!Conclusions!..............................................................................................................................!71! REFERENCES!.......................................................................................................................................!73! Chapter!4!....................................................................................................................................................!76! A!Novel!Instrument!for!Rapid!Estimation!of!TemperatureJDependent!Thermal! Properties!up!to!140oC!....................................................................................................................!76! Abstract!..................................................................................................................................................!76! 4.1!Introduction!.............................................................................................................................!77! 4.2!Methodology!............................................................................................................................!81! 4.2.1!Mathematical!model!and!numerical!code!verification!.................................!81! 4.2.2!Equipment!design!.........................................................................................................!90! 4.3!Results!......................................................................................................................................!100! 4.3.1!Instrument!calibration!..............................................................................................!100! 4.3.2!Instrument!experimental!result!...........................................................................!103! 4.4!Conclusions!............................................................................................................................!111! APPENDIX!...........................................................................................................................................!113! REFERENCES!.....................................................................................................................................!151! Chapter!5!..................................................................................................................................................!154! Nutritional!Values!Of!The!Food!Material!And!Comparison!Of!Degradation!In! Aseptic!And!Conventional!Thermal!Processing!..................................................................!154! Abstract!................................................................................................................................................!154! 5.1!Introduction!...........................................................................................................................!155! 5.2!Materials!and!Methods!......................................................................................................!159! 5.2.1!Sample!preparation!...................................................................................................!159! . 5.2.2!Aseptic!and!retort!trials!...........................................................................................!159! 5.2.3!Analytical!methods!.....................................................................................................!163! 5.3!Mathematical!model!...........................................................................................................!163! 5.3.1!Sterilization!value!.......................................................................................................!163! 5.3.2!Thermal!degradation!kinetics!...............................................................................!164! 5.4.1!Aseptic!Experiment!....................................................................................................!169! 5.4.2!Retort!Experiments!....................................................................................................!178! REFERENCES!.....................................................................................................................................!191! viii Chapter!6!..................................................................................................................................................!194! Overall!Conclusions!........................................................................................................................!194! . 6.1!Conclusions!............................................................................................................................!194! 6.2!Recommendations!for!future!work!.............................................................................!196! ! ' ix LIST'OF'TABLES' Table!2.1!Solutions!to!the!heat!transfer!problems!in!case!1:!X22B10T1!with!first! order!approximation!...................................................................................................................!28! Table!2.2!Solutions!to!the!heat!transfer!problems!in!case!1:!X22B10T1!with!second! order!approximation!...................................................................................................................!28! Table!2.3!Solutions!to!the!heat!transfer!problems!in!Case!2:!X12B10T1!.......................!33! Table!2.4!Solutions!to!the!heat!transfer!problems!in!Case!3:!R22B10T1!.......................!37! Table!2.5!Solutions!to!the!heat!transfer!problems!in!Case!4:!R12B10T1!.......................!44! Table!3.1!Intrinsic!Sum!and!dimensionless!scaled!sensitivity!coefficients!for!case!1:! X22B10T1!.........................................................................................................................................!61! Table!3.2!Intrinsic!Sum!and!dimensionless!scaled!sensitivity!coefficients!for!case!1:! X22B10T1!with!T2!=!300oC!......................................................................................................!62! Table!3.3!Intrinsic!Sum!and!dimensionless!scaled!sensitivity!coefficients!for!case!2:! R22B10T1!........................................................................................................................................!68! Table!3.4!Intrinsic!Sum!and!dimensionless!scaled!sensitivity!coefficients!for!case!2:! R22B10T1!with!increased!heat!flux!.....................................................................................!71! . Table!4.1!Composition!of!sweet!potato!puree!from!USDA!nutrient!database!(U.S.! Department!of!Agriculture!2013)!........................................................................................!106! Table!4.2!Estimated!values!of!thermal!conductivities!and!statistical!indices!for!sweet! potato!puree!..................................................................................................................................!107! Table!4.3!Estimated!values!of!thermal!conductivities!and!statistical!indices!for! several!food!materials!...............................................................................................................!107! Table!A.1!Data!recorded!from!TPCell!for!sweet!potato!test!1!...........................................!114! Table!A.2!Data!recorded!from!TPCell!for!sweet!potato!test!2!...........................................!120! Table!A.3!Data!recorded!from!TPCell!for!sweet!potato!test!3!...........................................!126! Table!A.4!Data!recorded!from!TPCell!for!sweet!potato!test!4!...........................................!132! Table!A.5!Data!recorded!from!TPCell!for!sweet!potato!test!5!...........................................!138! x Table!A.6!Data!recorded!from!TPCell!for!sweet!potato!test!6!...........................................!144! Table!5.1!Viscosity!of!sweet!potato!puree!and!Reynolds!number!..................................!170! . Table!5.2!TimeJtemperature!data!for!aseptic!experiment!..................................................!173! Table!5.3!Parameter!estimates!and!statistical!indices!for!kinetic!parameters!of! vitamin!C!degradation!in!aseptic!system!..........................................................................!174! Table!5.4!TimeJtemperature!and!vitamin!C!data!for!retort!experiments!....................!180! Table!5.5!Parameter!estimates!and!statistical!indices!for!kinetic!parameters!of! vitamin!C!degradation!in!retort!............................................................................................!182! ' ' xi LIST'OF'FIGURES' Figure!1.1!A!thermal!conductivity!probe!(Sweat!1995)!...........................................................!6! Figure!2.1!Dimensionless!scaled!sensitivity!coefficient!for!k!and!C!in!case!1:! X22B10T0,!for!k!=!0.5!W/mK,!C!=!3.5x106!J/m3K,!delta!=!0.0001,!x/L!=0,! Δx / L = 0.02 (For!interpretation!of!the!references!to!color!in!this!and!all!other! figure,!the!reader!is!referred!to!the!electronic!version!of!this!dissertation.)!.....!27! Figure!2.2!Plot!of!IS!in!case!1:!X22B10T0,!using!2ndJorder!finite!difference!................!29! Figure!2.3!Plot!of!IS!in!case!1:!X22B10T0!with!refined!and!unrefined!finite!element! code!.....................................................................................................................................................!31! Figure!2.4!Scaled!sensitivity!coefficients!for!k!and!C!in!case!2:!X12B10T0!...................!33! Figure!2.5!Scaled!sensitivity!coefficients!for!k!and!C!in!case!3:!R22B10T0!...................!37! Figure!2.6!Plot!of!IS!in!case!3:!R22B10T0!with!refined!and!unrefined!finite!element! code.!....................................................................................................................................................!38! Figure!2.7!Scaled!sensitivity!coefficient!for!k!and!C!in!case!4:!R12B10T0!.....................!40! Figure!2.8!Scaled!sensitivity!coefficients!for!k,'C'and'h!in!case!5:!R32B10T0!with!h!=! 1000!W/m2JK!.................................................................................................................................!43! Figure!2.9!Plot!of!IS'in!case!5:!R32B10T0!.....................................................................................!44! Figure!3.1!Dimensionless!scaled!sensitivity!coefficient!for!the!temperatureJ dependent!parameters!of!heat!transfer!problems!in!case!1!(X22B10T0).!T1!=! 25°C!and!T2!=!130oC.!...................................................................................................................!59! Figure!3.2!Intrinsic!sum!for!the!heat!transfer!problems!in!case!1!(X22B10T0).!........!60! Figure!3.3!Dimensionless!scaled!sensitivity!coefficient!for!the!temperatureJ dependent!parameters!of!heat!transfer!problems!in!case!1!(X22B10T0).'T1!=! 25°C!and!T2!=!300oC.!...................................................................................................................!62! Figure!3.4!Dimensionless!scaled!sensitivity!coefficient!for!the!temperatureJ dependent!parameters!of!heat!transfer!problems!in!case!2!(R22B10T0).'T1!=! 25°C!and!T2!=!130oC,!q!=!2.4!x104!W/m2!..........................................................................!67! xii Figure!3.5!Intrinsic!sum!for!the!heat!transfer!problems!in!case!2!(R22B10T0).!........!69! Figure!3.6!Dimensionless!scaled!sensitivity!coefficient!for!the!temperatureJ dependent!parameters!of!heat!transfer!problems!in!case!2!(R22B10T0).'T1!=! 25°C!and!T2!=!130oC,!q!=!3.8x104!W/m2!...........................................................................!70! Figure!4.1!Simulated!heating!profile!of!test!material!.............................................................!88! Figure!4.2!Plot!of!Scaled!sensitivity!coefficients!of!the!parameters!in!the!model!given! by!Eq.!(4.1),!using!simulated!temperature!data.!.............................................................!89! Figure!4.3!Plot!of!Intrinsic!Sum!as!given!by!Eq.!(4.21)!...........................................................!90! Figure!4.4!Schematic!of!the!electronics!of!the!temperatureJdependent!thermal! property!measurement!instrument!......................................................................................!95! Figure!4.5!Thermal!property!measurement!instrument!with!the!data!acquisition! devices!...............................................................................................................................................!96! Figure!4.6!Calibration!curve!of!resistance!and!temperature!of!the!heating!element !.............................................................................................................................................................!101! Figure!4.7!Simulated!heating!curves!of!glycerol!at!two!different!locations!for!a!given! power!input!...................................................................................................................................!103! Figure!4.8!Experimental!and!predicted!heating!profile!of!sweet!potato!puree!for!a! given!power!input!of!24!W.!.....................................................................................................!104! Figure!4.9!Residuals!of!sweet!potato!puree!for!a!given!power!input!of!24!W.!..........!105! Figure!4.10!Scaled!sensitivity!coefficients!for!thermal!conductivities!at!T1!and!T2!108! Figure!4.11!Sequential!estimation!of!thermal!conductivities!at!T1!and!T2!.................!110! Figure!4.12!Sequential!estimation!of!thermal!conductivities!at!T1!and!T2!for!second! half!of!the!experimental!time!.................................................................................................!111! Figure!5.1!Vertical!still!water!immersion!retort!.....................................................................!161! Figure!5.2!Microthermics!equipment!for!aseptic!processing!............................................!162! Figure!5.3!Chromatogram!for!HPLC!analysis!of!vitamin!C!as!ascorbic!acid!................!168! Figure!5.4!Chromatogram!for!HPLC!analysis!of!thiamin!.....................................................!168! xiii Figure!5.5!Viscosity!of!sweet!potato!puree!measured!at!different!temperatures,!Q!=!2! lpm!and!D!=!0.43!in.!...................................................................................................................!170! . Figure!5.6!Experimental!and!predicted!temperature!profile!of!sweet!potato!puree! (Test!2)!as!it!goes!through!various!sections!of!aseptic!system.!..............................!172! Figure!5.7!Experimental!and!predicted!degradation!of!vitamin!C!in!aseptically! processed!sweet!potato!puree!..............................................................................................!175! . Figure!5.8!Residuals!of!vitamin!C!in!aseptic!processing!......................................................!176! Figure!5.9!Sequential!parameter!estimates!of!vitamin!C!in!aseptic!processing!........!177! Figure!5.10!Scaled!sensitivity!coefficient!of!kr!and!Ea!in!the!kinetic!degradation! model!of!vitamin!C!in!sweet!potato!puree!processed!in!retort!...............................!178! Figure!5.11!Simulated!temperature!profile!of!sweet!potato!puree!in!a!glass!jar! processed!in!retort!.....................................................................................................................!181! Figure!5.12!Simulated!temperature!profile!of!sweet!potato!puree!at!gauss!points!in!a! glass!jar!processed!in!retort!...................................................................................................!182! Figure!5.13!Experimental!and!predicted!degradation!of!vitamin!C!in!retort! processing!......................................................................................................................................!184! Figure!5.14!Residuals!of!vitamin!C!degradation!in!retort!experiment!..........................!185! Figure!5.15!Normalized!sequential!parameter!estimates!of!vitamin!C!in!retort!......!186! Figure!5.16!Scaled!sensitivity!coefficient!of!kr!and!Ea!in!the!kinetic!degradation! model!................................................................................................................................................!187! xiv KEY'TO'SYMBOLS'AND'ABBREVIATIONS' δ! constant!for!calculation!of!sensitivity!coefficient!  x ! dimensionless!axial!distance!  r ! dimensionless!radial!distance!  Xk ' dimensionless!scaled!sensitivity!coefficient!of!thermal!conductivity!  T dimensionless!temperature! !  t! dimensionless!time! C! massJaverage!concentration!!mg/100g! β! parameter! ˆ Xk ' scaled!sensitivity!coefficient!of!thermal!conductivity,!oC! (C C0 ) ! ( ) vitamin!retention,!where! 0 ≤ C C0 ≤ 1 )!!!!!!!!!!!!!  X C !' dimensionless!scaled!sensitivity!coefficient!of!volumetric!heat!capacity! ˆ X C !! scaled!sensitivity!coefficient!of!volumetric!heat!capacity,!oC! SumSSC !absolute!sum!of!scaled!sensitivity!coefficients,!oC!  x  h ! dimensionless!axial!distance! ! dimensionless!heat!transfer!coefficient! β! parameter!  X h '' dimensionless!scaled!sensitivity!coefficient!of!heat!transfer!coefficient!  X C !' dimensionless!scaled!sensitivity!coefficient!of!volumetric!heat!capacity! xv A! constant! C'' volumetric!heat!capacity,!J/KgJoC! C1'' volumetric!heat!capacity,!J/KgJoC!at!temperature!T1! C2'' volumetric!heat!capacity,!J/KgJoC!at!temperature!T2! Co! initial!massJaverage!concentration,!mg/100g! D! diameter!of!pipe,!in! Ea ! activation!energy,!J/gJmol! Fo! Sterilization!value,!min! g!! power,!W/m3! h!! heat!transfer!coefficient,!W/m2JoC!! IS ' ' intrinsic!sum,!oC! K! Consistency!coefficient,!Pa.sn! k'' thermal!conductivity,!W/mJ!oC! k1'' thermal!conductivity!of!sample,!W/mJ!oC!at!temperature!T1! k2'' thermal!conductivity!of!sample,!W/mJ!oC!at!temperature!T2! kh'' thermal!conductivity!of!heater,!W/mJ!oC!! kr ! rate!constant!at!reference!temperature!Tr,!minJ1! xvi L!! thickness!of!slab,!m! n! flow!behavior!index! NRe,PL!Reynolds!number! q!! heat!flux,!W/m2! Q'' power!generated!by!heater,!W/m! r!! radial!position,!m! R'' resistance!of!the!heater,!ohm! Rg!!!! gas!constant!(J/gJmole!K)! R0'' Inner!radius!of!the!heater,!m! R1'' Interface!of!heater!and!sample,!m! R2'' Outer!radius!of!the!cup,!m! T! temperature!(K)! t! time,!sec! T!! temperature,!oC! T∞! steam!temperature,!K! T0! initial!temperature,!oC! t0 ! time!correction!factor,!sec! t1 ! start!of!heating!time,!sec! T1!! initial!temperature!for!thermal!properties,!oC! xvii t2 ! end!of!heating!time,!sec! ! T2!! final!temperature!for!thermal!properties,!oC! Tr! reference!temperature,!K! ū! velocity!of!product!in!pipe! X! sensitivity!matrix! x!! axial!position,!m! z! temperature!required!to!reduce!microorganism!by!one!log! z'! axial!position! β! parameter! ρ! product!density,!Kg/m3! ' ! xviii Chapter'1 ' Introduction' ' 1.1'Introduction' ! ! Thermal! processing! of! food! products! relies! on! timeJtemperature! data! obtained! from! either! actual! measurements! or! from! mathematical! models.! A! sufficient! lethality,! calculated! based! on! timeJtemperature! data,! needs! to! be! accumulated!to!make!sure!that!the!food!is!safe!for!distribution!and!consumption.!In! the!case!where!no!direct!measurement!can!be!made,!the!timeJtemperature!data!can! be!obtained!from!simulation!with!the!mathematical!model.!This!procedure!requires! two!things!to!be!reasonably!accurate:!1)!the!numerical!code!that!is!used!to!solve!the! problem,! and! 2)! thermal! properties! of! the! product.! Numerical! code,! either! developed!individually!or!commercial!code,!must!be!checked!so!that!it!can!provide! accurate! results.! Sometimes,! not! using! the! correct! settings! in! the! commercially! available! software! can! also! lead! to! significant! deviation! in! results.! Hence,! verification!of!numerical!code!is!an!important!step!in!the!modeling.! ! Thermal!properties!are!often!temperatureJdependent,!as!they!can!change!as! the!temperature!is!changed.!It!is!relatively!easy!to!measure!thermal!parameters!at! room! temperature.! However,! if! one! requires! the! thermal! parameters! at! elevated! temperatures,! there! is! no! instrument! currently! available! that! can! be! used.! The! quasiJisotherm!way!of!measuring!thermal!properties!is!good!if!the!data!are!needed! only!a!one!particular!temperature,!as!the!instrument!needs!to!be!in!equilibrium!with! 1 the! sample! that! is! being! tested.! This! requirement! becomes! even! more! difficult! if! temperature! is! >! 100oC,! as! the! whole! system! needs! to! be! pressurized.! Achieving! equilibrium!conditions!at!such!a!high!temperature!is!also!difficult,!and!by!the!time!it! achieves! equilibrium,! the! sample! would! be! sufficiently! degraded! to! provide! little! useful! information.! Hence,! an! instrument! is! needed! that! can! measure! thermal! properties!over!a!wide!range!of!temperatures!in!a!short!time.!! ! Food!quality!is!also!affected!by!the!thermal!processing.!Most!often!there!is!a! negative! effect! of! higher! temperatures! on! nutrients! of! the! food! materials.! Hence,! temperatureJdependent! thermal! properties! are! needed! to! design! the! thermal! processes! accurately.! Nutritional! degradation! kinetics! can! be! reliable! if! the! mathematical! models! include! the! temperature! dependence! of! the! thermal! properties.! ! 1.2'Objectives'of'the'study' ! ! There!were!three!objectives!of!the!study:! 1. Develop!a!method!for!intrinsic!verification!in!parameter!estimation!problems! and!check!for!accuracy!in!numerical!codes.! 2. Design! and! develop! an! instrument! that! can! measure! thermal! properties! at! elevated!temperatures.! 3. Compare! nutrient! degradation! in! an! aseptic! processing! system! and! conventional!retort!processing.! ! 2 1.3'Overview'of'the'dissertation' ! The! dissertation! is! divided! in! four! different! chapters,! excluding! the! current! chapter.!Each!chapter!covers!topics!related!to!the!objective!mentioned!above.!! 1. Chapter! 2! –! This! chapter! deals! with! the! topic! of! intrinsic! verification.! The! dimensionless!derivation!of!sensitivity!coefficients!is!presented!with!several! case! studies.! The! dimensionless! derivation! is! a! simple! and! straightforward! approach! for! deriving! the! intrinsic! sum,! which! is! the! sum! of! all! scaled! sensitivity! coefficients! in! the! model.! The! intrinsic! sum! has! two! significant! advantages:! a. It!can!help!to!identify!if!all!parameters!can!be!estimated!in!a!specific! mathematical!model.! b. It!can!be!used!to!verify!the!large!numerical!codes.! 2. Chapter!3!–!The!principles!presented!in!this!chapter!are!based!on!chapter!2,! but! has! been! extended! to! thermal! parameters! that! are! temperatureJ dependent.! The! derivation! is! presented! with! several! case! studies! for! heat! transfer!problems!in!Cartesian!and!cylindrical!coordinate!system.!! 3. Chapter! 4! –! A! thermal! properties! measurement! instrument! (TPCell)! is! presented! with! the! design! and! mathematical! model.! The! advantages! of! the! instrument! are! discussed! as! to! how! it! compares! with! currently! available! instruments.! Several! food! materials! were! tested! and! the! results! are! presented!in!this!chapter.! 3 4. Chapter! 5! –! This! chapter! is! focused! around! the! nutritional! studies! in! an! aseptic! processing! system! compared! to! those! in! conventional! retort! processing.! Kinetic! degradation! parameters! were! analyzed! for! vitamin! degradation!and!the!results!are!presented.!! 1.4'Literature'Review' ! Thermal!properties!of!a!food!material!are!important!for!several!reasons,!such! as! designing! the! thermal! process! and! processing! equipment.! Thermal! conductivity! measurement! poses! a! challenge! as! it! depends! on! the! structural! arrangements! as! well!as!chemical!composition!of!the!food!material!(Sweat!1995).!Thermal!properties! also!depend!on!the!temperature!history!of!the!product.!Previous!studies!have!shown! that! thermal! properties! can! be! either! predicted! using! a! predictive! model! or! measured!by!using!equipment.!! 1.4.1'Predictive'model'' ! There! are! several! predictive! equations! proposed! for! the! prediction! of! thermal!conductivity!based!on!the!composition!of!the!food!material.!Choi!and!Okos! proposed!the!predictive!equation!for!thermal!conductivity!based!on!the!composition! of!food!materials!as!given!in!Eq.!(1.1)J(1.7)!(Choi!and!Okos!1986).!However,!this!is! valid!only!in!the!temperature!range!of!0!–!90oC.!! ! k water = 0.57109 + 1.7625 × 10−3T − 6.7036 × 10−6 T 2 !! (1.1)' ! kCHO = 0.20141 + 1.3874 × 10−3T − 4.3312 × 10−6 T 2 !! (1.2)' 4 ! k protein = 0.17881 + 1.1958 × 10−3T − 2.7178 × 10−6 T 2 !! (1.3)' ! k fat = 0.18071 − 2.7604 × 10−3T − 1.7749 × 10−7 T 2 !! (1.4)' ! kash = 0.32961 + 1.4011 × 10−3T − 2.9069 × 10−6 T 2 !! (1.5)' ! kice = 2.2196 − 6.2489 × 10−3T + 1.0154 × 10−6 T 2 !! (1.6)' ! k fiber = 0.18331 + 1.2497 × 10−3T − 3.1683 × 10−6 T 2 !! (1.7)' 1.4.2'Thermal'conductivity'devices' 1.4.2.1'Line'heat'source'method' ! Devices! used! for! thermal! conductivity! measurement! of! food! materials! are! mainly! based! on! the! line! heat! source! method.! Other! methods! such! as! guarded! hot! plate!is!not!suitable!for!food!materials!due!to!long!temperature!equilibration!time,! moisture!migration!in!sample!and!the!need!for!large!sample!size!(Sweat!1995).!The! line! heat! source! method! requires! small! sample! size! and! is! recommended! for! food! applications!(Sweat!1995;!Monsenin!1980).!Improvement!on!the!construction!of!the! probe! and! linearity! of! the! temperature! versus! logarithm! of! time! was! done! to! improve! the! accuracy! of! thermal! conductivity! measurements! (BagheJKhandan! and! others!1981).!A!thermal!conductivity!probe!is!shown!in!Figure!1.1.!! 5 Heater& Leads 3.9$cm 0.66mm%O.D. Thermocouple+ Junction Figure!1.1!A!thermal!conductivity!probe!(Sweat!1995)! ! ! ! The!theory!behind!the!line!heat!source!method!is!that!the!probe!heats!a! sample,!initially!at!uniform!temperature.!The!temperature!at!the!surface!of!the! probe!is!monitored.!After!a!brief!transient!phase,!the!plot!of!temperature!versus!the! logarithm!of!time!is!linear.!The!slope!of!this!line!is!given!by! Q / 4π k .!Thermal! conductivity!can!be!calculated!by!Eq.!(1.8).! ! k=Q ln[(t2 − t0 ) / (t1 − t0 )] !! 4π (T2 − T1) (1.8)' Several! researchers! have! reported! that! the! position! of! thermocouple! inside! the! probe! does! not! influence! the! thermal! conductivity! measurement! (Lentz! 1952;! Hooper!and!Lepper!1950;!Sweat!1995).!Also,!for!a!probe!diameter!of!less!than!0.66! mm,!it!was!concluded!that!the!time!correction!factor!was!negligible!(Sweat!1995).! 6 This! type! of! probe! was! calibrated! with! water! with! 0.5%! agar! solution! and! with! glycerol.!! 1.4.2.2'Modified'Fitch'method' The Fitch device is based on the principles of heat transfer from a sample, kept at uniform temperature at one side and standard copper on the other. Copper is insulated on the other sides. The lumped heat transfer in copper is monitored. The solution to the heat transfer problem is simplified by assuming quasi-steady-state heat transfer through the sample. The thermal conductivity is determined by analyzing the liner portion of the temperature rise (U.S. Department of Agriculture 2013). 1.4.3'Thermal'properties'at'elevated'temperatures' ! Compilations! of! thermal! conductivity! of! food! products! are! presented! in! several!publications!(Qashou!and!others!1972;!Choi!and!Okos!1983).!Most!of!these! values! are! based! on! either! using! the! predictive! model! using! compositions! of! food! materials! or! by! using! the! line! heat! source! method.! However,! measurements! of! thermal!properties!of!food!materials!at!elevated!temperatures,!especially!>90°C!are! scarce! (Nesvadba! 2005).! The! problem! at! elevated! temperatures! is! the! moisture! migration! in! and! moisture! loss! from! the! sample.! However,! the! moisture! migration! and!loss!can!be!minimized!by!using!a!pressurized!system.!The!work!that!has!been! done! at! elevated! temperatures! requires! a! pressurized! sample! holder! in! which! the! line! heat! source! can! be! inserted! (Shrivastava! and! Datta! 1999).! One! important! criterion! of! the! line! heat! source! method! is! that! the! sample! and! heater! must! be! in! equilibrium! before! starting! the! test! (Sweat! 1995).! The! drift! in! initial! temperature! 7 will! produce! erroneous! thermal! conductivity! values! (Sweat! 1995).! The! line! heat! source! method! has! advantages! in! terms! of! mathematical! processing! of! the! results! and! the! control! of! experimental! conditions! (Sahin! and! Sumnu! 2006).! The! disadvantages! of! this! method! include! a! long! time! to! achieve! a! condition! of! equilibrium! and! moisture! migration! during! long! tests! (Sahin! and! Sumnu! 2006).! Using!an!oil!bath!to!control!the!temperature!inside!the!pressurized!sample!cup!can! take!an!hour!to!achieve!equilibrium.!The!equilibrium!condition!time!increases!as!the! temperature! of! the! bath! is! increased! to! elevated! temperatures,! such! as! 140°C.! For! example,!a!test!that!include!measurement!at!25°C!and!one!measurement!at!140°C,! can!take!at!least!2!hours.!If!several!other!temperatures!are!added!to!the!experiment! then! the! time! could! be! >6! hours.! One! severe! disadvantage! of! this! method! is! that! because! thermal! properties! also! depend! on! temperature! history,! the! state! of! the! sample! may! have! been! changed! dramatically! by! holding! the! product! at! such! high! temperatures! for! significantly! long! time! (hours).! Therefore,! there! is! a! need! for! a! reliable!instrument!that!can!measure!thermal!conductivity!of!food!materials!in!less! time! and! at! elevated! temperature! that! can! cover! the! entire! processing! range! for! commercial!processes!(25oC!J!140oC).!! 1.5'Conclusions' Whenever! a! mathematical! model! is! used! to! establish! a! thermal! process! for! food!processing,!accurate!and!reliable!thermal!properties!are!needed!to!ensure!good! quality!and!safety!of!food.!Traditional!methods!of!measuring!thermal!properties!at! elevated! temperatures! may! not! be! accurate! as! the! sample! degrades! significantly! 8 before! the! measurement! can! be! made.! However,! there! is! no! instrument! currently! available! that! can! be! used! to! measure! thermal! conductivity! at! elevated! temperatures!in!a!short!time!as!compared!to!six!hours!using!traditional!methods.!! ! ! 9 REFERENCES' 10 REFERENCES' ' BagheJKhandan! MS,! Choi! Y,! Okos! MR.! 1981.! Improved! Line! Heat! Source! Thermal! Conductivity!Probe.!Journal!of!Food!Science!46(5):1430J2.! Choi!Y,!Okos!M.!1983.!Thermal!properties!of!liquid!foods:!review.!American!Society! of!Agricultural!Engineers,!Chicago,!Paper!No.!83J6516.! Choi! Y,! Okos! M.! 1986.! Effects! of! temperature! and! composition! on! the! thermal! properties!of!foods.!Food!engineering!and!process!applications!1:93J101.! Hooper! F,! Lepper! F.! 1950.! Transient! heat! flow! apparatus! for! the! determination! of! thermal!conductivities:!National!Emergency!Training!Center.! Lentz!C.!1952.!A!Transient!Heat!Flow!Method!of!Determining!Thermal!Conductivity:! Application! to! Insulating! Materials.! Canadian! Journal! of! Technology! 30(6):153J66.! Monsenin! N.! 1980.! Thermal! properties! of! foods! and! other! agricultural! materials:! CRC!Press.! Nesvadba! P.! 2005.! Thermal! Properties! of! Unfrozen! Foods.! In:! Rao! MA,! Rizvi! SS,! Datta!AK,!editors.!Engineering!properties!of!foods.!Boca!Raton,!FL:!CRC!Press.! Qashou! M,! Vachon! R,! Touloukian! Y.! 1972.! Thermal! conductivity! of! foods.! ASHRAE! Trans!78(1):165J83.! Sahin! S,! Sumnu! Slm.! 2006.! Thermal! Properties! of! Foods.! Physical! Properties! of! Foods:!Springer!New!York.!p.!107J55.! Shrivastava! M,! Datta! AK.! 1999.! Determination! of! specific! heat! and! thermal! conductivity!of!mushrooms!(Pleurotus!florida).!Journal!of!Food!Engineering! 39(3):255J60.! Sweat! VE.! 1995.! Thermal! properties! of! foods.! In:! Rao! MA,! Rizvi! SSH,! editors.! Food! Science!and!Technology.!New!York:!Marcel!Dekker,!Inc.!p.!99J135.! 11 Chapter'2 ' Use'of'Scaled'Sensitivity'Coefficient'Relations'for'Intrinsic'Verification'of' Numerical'Codes'and'Parameter'Estimation' ! Abstract' ! ! ! Numerical! codes! are! important! in! providing! solutions! to! partial! differential! equations!in!many!areas,!such!as!the!heat!transfer!problem.!However,!verification!of! these!codes!is!very!critical.!!A!methodology!is!presented!in!this!paper!as!an!intrinsic! verification!method!to!the!solution!to!the!partial!differential!equation.!Derivation!of! dimensionless!form!of!scaled!sensitivity!coefficient!has!been!presented,!and!the!sum! of!scaled!sensitivity!coefficients!has!been!used!in!the!dimensionless!form!to!provide! a!method!for!verification.!Intrinsic!verification!methodology!is!demonstrated!using! examples! of! heat! transfer! problems! in! Cartesian! and! cylindrical! coordinate.! The! intrinsic! verification! method! presented! here! is! applicable! to! analytical! as! well! as! numerical!solutions!to!partial!differential!equations.! ! Keywords:! Parameter! estimation,! Intrinsic! verification,! Sensitivity! coefficients,! Inverse!problems,!Heat!transfer,!Intrinsic!sum! 2.1'Introduction' ! ! Verification!of!large!numerical!codes,!such!as!finite!element!and!finite! control!volume!method,!is!an!important!issue.!The!accuracy!of!such!programs!needs! to! be! assured! (Beck! and! others! 2006).! For! example,! engineers! often! rely! on! simulation! for! various! engineering! problems! and! hence! the! numerical! code! that! 12 they! use! has! to! be! reliable.! Accuracy! of! numerical! code! is! also! important! in! parameter!estimation!problem.!! 2.1.1'Numerical'code'verification' ! Coding! mistakes! can! lead! to! serious! flaws! in! the! result! (Salari! and! Knupp! 2000).! There! are! two! aspects! of! verification;! 1)! code! verification,! and! 2)! code! validation.! The! method! presented! in! this! paper! deals! with! the! code! verification.! However,!it!has!potential!for!the!code!validation!as!well.!! ! Some!of!the!methods!that!have!been!used!in!code!verification!are:! ! 1. Trend!–!This!method!is!applied!to!see!any!trend!in!the!numerical!solution!of!a! problem.! The! solution! accuracy! is! not! checked! in! this! method.! This! method! provides! only! a! means! to! say! that! the! change! in! solution! is! in! the! right! direction! when! a! specific! parameter! in! the! model! is! changed! (Salari! and! Knupp!2000).! ! 2. Symmetry!–!The!symmetry!method!is!a!check!for!symmetry!in!the!solution.!A! problem!can!be!set!up!so!that!it!provides!a!symmetric!solution.!For!example,! an! axiJsymmetric! problem! for! cylindrical! heating! can! be! employed! in! the! numerical! code.! If! the! solution! does! not! produce! a! symmetrical! result,! then! there!is!some!error!in!the!code!(Salari!and!Knupp!2000).!! ! 13 3. Comparison! –! In! this! method,! the! existing! code! is! compared! with! another! wellJestablished!code.!A!problem!is!solved!using!both!codes!and!results!are! compared!to!check!for!accuracy!(Salari!and!Knupp!2000).! ! 4. Exact!Solution!(MES)!–!This!is!a!widely!used!method!for!code!verification.!If! an! exact! solution! to! the! partial! differential! equation! can! be! obtained,! it! is! compared!with!the!numerical!result!from!the!code.!A!criterion!is!set!for!the! code!to!achieve!to!pass!the!verification!process!(Roy!2005;!Salari!and!Knupp! 2000).! ! 5. Manufactured! solution! (MMS)! –! Also! widely! accepted! and! used! for! code! verification.!In!this!method,!a!manufactured!solution!to!a!problem!is!defined! and!the!code!is!used!to!solve!this!manufactured!solution.!Acceptance!criteria! should! be! defined! before! starting! the! test! (Roy! 2005;! Salari! and! Knupp! 2000).!! ! ! Only!MES!and!MMS!are!appropriate!for!code!verification! (Salari!and!Knupp! 2000).! The! disadvantage! of! MES! is! that! one! must! know! the! exact! solution! to! compare! it! with! the! numerical! solution.! However,! in! many! situations! finding! an! exact!solution,!such!as!problems!involving!nonlinear!problems,!can!be!very!difficult! or!not!possible!at!all.!There!are!some!disadvantages!of!MMS!for!code!verification!as! well.! MMS! requires! arbitrary! source! terms! that! have! to! be! incorporated! into! the! code! (Roy! 2005).! Hence,! MMS! is! codeJintrusive! and! cannot! be! performed! on! large! software!where!code!is!not!accessible.!! 14 ! ! ! In!such!cases,!there!is!a!need!for!a!method!that!can!be!relied!on!to!assure!the! accuracy! of! the! code.! That! is! where! the! concept! of! intrinsic! verification! method! (IVM)! provides! a! tool! to! check! the! accuracy! of! the! numerical! code.! Intrinsic! verification!methods!(IVM)!provide!a!convenient!way!to!check!accuracy!of!solutions! for! such! numerical! codes! (Beck! and! others! 2006).! In! this! paper,! IVM! for! several! cases!for!checking!the!accuracy!of!numerical!codes!has!been!presented.!The!idea!of! IVM!presented!in!this!article!is!based!on!the!scaled!sensitivity!coefficients.!A!modelJ specific! identity! for! a! partial! differential! equation! can! be! derived! based! on! dimensionless! analysis! of! scaled! sensitivity! coefficients.! A! good! feature! of! the! dimensionless!analysis!of!scaled!sensitivity!coefficient!is!that!it!is!simple!and!can!be! replicated! for! various! models! with! less! work! than! other! methods.! Also,! a! major! advantage! of! the! presented! method! is! that! the! exact! solution! of! the! partial! differential!equation!is!not!needed.!Mistakes!in!discretization!can!be!detected!with! the! IVM! presented! in! this! article.! ! Some! examples! cases! are! shown! to! prove! the! effectiveness!of!this!method.!This!method!also!demonstrates!that!any!minor!coding! error,!such!as!typographical!mistakes,!can!be!detected.! 2.1.2'Parameter'estimation'and'sensitivity'coefficient' ! The! IVM! presented! in! this! paper! is! a! combination! of! sensitivity! coefficients! used! in! verification! of! numerical! codes! and! parameter! estimation.! The! sensitivity! coefficient!of!a!parameter!is!the!first!partial!derivative!of!the!function!involving!the! parameter,!with!respect!to!the!parameter!(Beck!1970).!Consider!a!simple!function:! 15 T = f (k,C, x,t) ! ! (2.1)! where!k!and!C!are!parameters!of!the!function!T.!! The! sensitivity! coefficients! of! k! and! C' are ∂T ∂T ! and! ,! respectively.! After! ∂C ! ∂k multiplying! sensitivity! with! its! parameter,! the! scaled! sensitivity! coefficients! are! represented!by! ! ! ∂T ˆ ∂T ˆ ! Xk = k , XC = C ∂k ∂C (2.2)! for!k!and!C!respectively.! ! The! importance! and! use! of! sensitivity! coefficients! has! been! discussed! and! presented!by!Beck!and!others!(Beck!and!Arnold!1977a;!Blackwell!and!others!1999;! Sun! and! others! 2001;! Chen! and! Tong! 2004).! Sensitivity! coefficients! provide! considerable! insight! into! the! parameter! estimation! problem! (Beck! 1967;! Dowding! and!others!1999b).!Some!of!the!applications!of!sensitivity!coefficients!are!in!optimal! experimental! design! (Beck! 1969;! Beck! and! Woodbury! 1998)! and! parameter! estimation!(Beck!and!Arnold!1977a;!Dolan!and!others!2012;!Koda!and!others!1979).! Some! other! insight! can! be! gained.! An! example! is,! if! the! sensitivity! coefficient! is! a! function!of!the!parameter,!then!the!estimation!problem!is!nonlinear!and!should!be! solved! using! nonlinear! regression! techniques! (Beck! and! Arnold! 1977a).! In! that! regard,! it! is! also! noted! that! a! linear! partial! differential! equation! can! produce! a! nonlinear!estimation!problems.! ! 16 ! It!can!be!shown,!using!the!solution!of!the!heat!transfer!problem,!whether!the! parameters!can!be!estimated!individually!or!as!a!group.!One!of!the!conditions!that!is! necessary! in! parameter! estimation! is! that! the! sum! of! the! scaled! sensitivity! coefficients!is!not!equal!to!zero.!Scaled!sensitivity!coefficients!must!not!have!linear! dependence! among! them.! Specifically,! if! the! measured! quantity! is! temperature! T,! the!linear!dependence!relation! ! A1β1 ∂T ∂T ∂T + A2β 2 ++ Ap β p = 0 !! ∂β1 ∂β 2 ∂β p (2.3)! ! where! at! least! one! of! the! coefficients! Ai' ! is! not! zero.! The! ith! scaled! sensitivity! coefficient!is!defined!to!be! ! βi = βi ∂T !! ∂β i (2.4)! ! The! units! must! be! consistent;! one! way! to! have! this! consistency! is! to! have! each! Ai! coefficient! be! equal! to! unity.! The! Eq.! (2.3)! shows! that! such! a! linear! relationship! can! occur;! when! it! does! occur,! all! the! parameters! cannot! be! simultaneously! and! independently! estimated.! ! Hence,! relationships! between! the! sensitivity! coefficients! are! important.! They! are! discussed! in! detail! in! following! sections.!The!relationship!can!also!be!used!to!provide!intrinsic!verification!as!well.! ! There!are!three!novel!concepts!presented!in!this!paper:! ! 1. Dimensionless!derivation!of!sums!of!scaled!sensitivity!coefficients.! 17 2. Intrinsic! verification! of! numerical! codes! using! sum! of! scaled! sensitivity! coefficients.! 3. Number!of!parameters!that!can!be!estimated!in!a!model.! ! Several! case! studies! are! presented! based! on! transient! heat! conduction! problems! in! Cartesian! and! cylindrical! coordinate! system! with! various! boundary! conditions.!Scaled!sensitivity!summation!relations!are!derived!using!dimensionless! analysis,!and!its!effectiveness!is!shown!for!verifying!numerical!codes.! 2.2'Derivation'of'sensitivity'coefficients'in'oneSdimensional'heat'conduction' problem' In!the!following!section,!two!derivations!of!scaled!sensitivity!coefficients!are! given.!The!first!derivation!is!the!regular!way!of!deriving!sensitivity!coefficient!and! the!second!one!is!a!derivation!using!dimensionless!analysis.!The!numbering!system! for! heat! conduction! problems! (Beck! and! Litkouhi! 1988)! has! been! used! for! each! of! the!case!studies.! 2.2.1'Case'1:'OneSdimensional'transient'heat'conduction'in'a'flat'plate'with' heat'flux'on'one'side'and'insulated'on'another'(X22BS0T1)' ! The!mathematical!model!for!oneJdimensional!transient!heat!conduction!in!a! plate!can!be!given!by,! ! k ∂2 T ∂x 2 =C ∂T ∂t 0 < x < L, t > 0 ! where!boundary!and!initial!conditions!are! 18 (2.5)! −k ! ∂T ∂T (0,t) = q0 f (t) (L,t) = 0 T (x,0) = T0 ! ∂x ∂x (2.6)! Note!that!T!is!a!function!of! (x, L,t,C,T0 ,q0 ). !!!! ! The!scaled!sensitivity!coefficients!are!derived!based!on!the!derivatives!of!Eq.! (2.5)! (Beck! and! Arnold! 1977a).! The! dimensionless! derivation! is! discussed! later! in! this! section.! The! sensitivity! coefficient! for! k ! can! be! computed! by! taking! the! derivative!of!Eq.!(2.5)!with!respect!to! k !as!follows,! Xk = ! ∂2 T ∂x 2 +k ∂2 ⎛ ∂T ⎞ ∂ ⎛ ∂T ⎞ ⎜ ∂k ⎟ = C ∂t ⎜ ∂k ⎟ ! ⎝ ⎠ ∂x 2 ⎝ ⎠ (2.7)! ! and!the!initial!and!boundary!conditions!can!be!found!by!taking!the!derivatives!with! respect!to! k ,of!the!original!boundary!conditions!given!in!Eq.!(2.6).! ! − ∂T ∂ ⎛ ∂T ⎞ (0,t) − k ⎜ ⎟ (0,t) = 0 ∂x ∂x ⎝ ∂k ⎠ ∂ ⎛ ∂T ⎞ ( L,t) = 0 ∂x ⎜ ∂k ⎟ ⎝ ⎠ ∂T (x,0) = 0 ! (2.8)! ∂k ! Repeating!the!calculation!steps!as!above!for!the!sensitivity!coefficient!for!parameter! C gives;! ∂2 ⎛ ∂T ⎞ ∂T ∂ ⎛ ∂T ⎞ XC = k = +C ⎜ ! 2 ⎜ ∂C ⎟ ∂t ∂t ⎝ ∂C ⎟ ⎠ ⎠ ∂x ⎝ ! (2.9)! ! ! −k ∂ ⎛ ∂T ⎞ (0,t) = 0 ∂x ⎜ ∂C ⎟ ⎝ ⎠ ∂ ⎛ ∂T ⎞ (L,t) = 0 ∂x ⎜ ∂C ⎟ ⎝ ⎠ ! by!multiplying!and!adding! kX k + CX C ,!the!result!is;! 19 ∂T (x,0) = 0 ! ∂C (2.10)! ! ∂2 T ∂2 ⎛ ∂T ∂T ⎞ ∂T ∂ ⎛ ∂T ∂T ⎞ ⇒ k +k k +C ⎟ = C ∂t + C ∂t ⎜ k ∂k + C ∂C ⎟ 2 2 ⎜ ∂k ∂C ⎠ ⎝ ⎠ ∂x ∂x ⎝ ! Since k ∂2 T ∂x 2 =C ∂T ∂t ! these!two!terms!can!be!eliminated!on!the!left!and!right!sides,!respectively:! ! ∂2 ⎛ ∂T ∂T ⎞ ∂ ⎛ ∂T ∂T ⎞ ⇒ k k +C ⎟ = C ∂t ⎜ k ∂k + C ∂C ⎟ ! 2 ⎜ ∂k ∂C ⎠ ⎝ ⎠ ∂x ⎝ ! (2.11)! ! the!boundary!and!initial!conditions!are!reJstated!from!Eq.!(2.8)!and!(2.10):! ! − ∂T ∂ ⎛ ∂T ⎞ (0,t) − k ⎜ ⎟ (0,t) = 0 ∂x ∂x ⎝ ∂k ⎠ ∂ ⎛ ∂T ⎞ (L,t) = 0 ∂x ⎜ ∂k ⎟ ⎝ ⎠ ∂T (x,0) = 0 ! (2.12)! ∂k ! ! −k ∂ ⎛ ∂T ⎞ (0,t) = 0 ∂x ⎜ ∂C ⎟ ⎝ ⎠ ∂ ⎛ ∂T ⎞ (L,t) = 0 ∂x ⎜ ∂C ⎟ ⎝ ⎠ ∂T (x,0) = 0 ! ∂C (2.13)! ! For!x!=!0,!again!multiply!and!add!kXk+CXC:! ! ! −k ∂T ∂ ⎛ ∂T ∂T ⎞ (0,t) − k ⎜ k +C (0,t) = 0 ! ∂x ∂x ⎝ ∂k ∂C ⎟ ⎠ ! Setting!the!first!term!equal!to!the!flux!from!Eq.!(2.6)!yields:! ! 20 (2.14)! ! ⇒ −k ∂ ⎛ ∂T ∂T ⎞ ⎜ k ∂k + C ∂C ⎟ (0,t) = −qo ! ∂x ⎝ ⎠ (2.15)! at! x = L ! ∂ ⎛ ∂T ∂T ⎞ ⎜ k ∂k + C ∂C ⎟ (L,t) = 0 ! ∂x ⎝ ⎠ (2.16)! ⎛ ∂T ∂T ⎞ ⎜ k ∂k + C ∂C ⎟ (x,0) = 0 ! ⎝ ⎠ ! (2.17)! and!at! t = 0 ! ! Using!the!initial!condition,!Eq.!(2.17)!can!be!written!as!follows,! ! ! k ∂T ∂T +C = −(T − T0 ) ! ∂k ∂C (2.18)! ! Equation! (2.18)! provides! an! important! relationship! of! scaled! sensitivity! coefficients! of! the! parameters! in! the! model! given! by! Eq.! (2.5).! The! larger! the! difference!in! T (x,t) − To ,!the!greater!will!be!the!magnitude!of!the!scaled!sensitivity! coefficients.! Larger! scaled! sensitivity! coefficients! are! desired! to! have! a! good! estimate!of!the!parameters!with!lower!standard!error.!Because!the!right!side!of!Eq.! (2.18)! is! nonJzero,! it! also! suggests! that! the! sensitivity! coefficients! might! be! uncorrelated!and!hence!could!be!identified!uniquely.!When!right!side!of!Eq.!(2.18)!is! equal! to! zero,! it! is! not! possible! to! independently! estimate! k! and! C! from! data! obtained!from!the!related!experiment.!! ! ! 21 2.2.1.1'Dimensionless'analysis'of'sensitivity'coefficient' ! In! this! section,! dimensionless! derivation! of! sensitivity! coefficient! is! presented.!The!heat!transfer!model!is!made!dimensionless!by!using!dimensionless! groups! and! the! sensitivity! coefficients! are! derived.! For! the! same! model! and! boundary!conditions!given!by!Eq.!(2.5),!the!model!in!a!dimensionless!form!and!then! derive!the!sensitivity!relation.!Let! ! !  x≡ x  kt  T − T0 !! ,t≡ ,T≡ L q0 L CL2 k (2.19)! The!dimensionless!scaled!sensitivity!coefficient!is!represented!by!Eq.!(2.20).! !  X= ( βi ∂T !! q0 L k ∂β i ) (2.20)! With! model! and! with! a! known! and! fixed! boundary! heat! flux,! the! dimensionless! temperature!is!given!symbolically!by! ! !     T = T ( x, t ) !! (2.21)! ! Now!the!partial!derivatives!of!temperature!T!are!found.!Notice!that! ! ! q L    T − T0 = 0 T ( x, t (k,C) ) !! k ! Using!the!chain!rule!of!differentiation,!the!derivative!with!respect!to!k!is! ! 22 (2.22)!   q L  q L ⎡ ∂T ∂t ⎤ ∂T =− 0 T+ 0 ⎢ ⎥  ∂k C k ⎢ ∂t ∂k ⎥ k2 ⎣ ⎦ C ! !! k (2.23)!  q L  q L ⎡ ∂T ⎤ ∂T  = − 0 T + 0 ⎢t ⎥  ∂k C k k ⎢ ∂t ⎥ ⎣ C⎦ ! Similarly,!the!scaled!sensitivity!coefficients!for!C!is! ! ! C  q L ⎡ ∂T ⎤ ∂T  = 0 ⎢ −t ⎥ !!  ∂C k k ⎢ ∂t ⎥ ⎣ k⎦ (2.24)! ! Adding!Eqs.!(2.23)!and!(2.24)!gives! ! ! k   q L  q L ⎡ ∂T ⎤ q0 L ⎡ ∂T ∂T ∂T   +C = − 0 T + 0 ⎢t ⎥+ ⎢ −t   ∂k C ∂C k k k ⎢ ∂t ⎥ k ⎢ ∂t ⎣ ⎣ C⎦ ⎤ q L  ⎥ = − 0 T !! (2.25)! k ⎦ k⎥ ! which!can!be!expressed!as,! ! k ∂T ∂T +C = −(T − T0 ) !! ∂k C ∂C k (2.26)! ! Equation! (2.18)! is! same! as! the! Eq.! (2.26),! which! was! derived! using! dimensionless! analysis.! Dimensionless! analysis! is! very! powerful,! as! one! does! not! need! to! find! exact! derivatives! for! complex! problems.! It! simplifies! the! problem! of! obtaining! the! relationship! of! sum! of! scaled! sensitivity! coefficients! as! shown! in! Eq.! (2.26).! A! number! of! applications! of! Eq.! (2.26)! can! be! cited.! It! can! be! used! in! 23 parameter! estimation.! It! indicates! that! it! might! be! possible! to! simultaneously! estimate!all!of!the!thermal!properties,!k!and!C!in!a!related!experiment.! Also,!Eq.!(2.18)!can!be!written!as,! ! ! k ∂T ∂T +C + (T − T0 ) = 0 ! ∂k ∂C (2.27)! ! In!this!form,!Eq.!(2.27)!is!termed!as!Intrinsic!Sum!(IS).!It!is!a!relation!that!can! be!used!to!provide!intrinsic!verification!of!finite!element!and!finite!control!volume! computer! codes! for! heat! conduction.! Note! that! Eq.! (2.3)! (sum! of! scaled! sensitivity! coefficients!=!0)!is!not!satisfied!by!Eq.!(2.26),!since!the!sum!of!the!scaled!sensitivity! coefficients!of!k!and!C!is!not!zero.! ! ! An!important!application!is!to!verify!large!numerical!codes.!The!verification! can! be! obtained! by! using! the! code! to! generate! the! sensitivity! coefficients! by! using! finite!differences,!such!as!! k ! ∂T T (x,t,(1+ δ )k,C) − T (x,t,k,C) ≈k ∂k C (1+ δ )k − k !! T (x,t,(1+ δ )k,C) − T (x,t,k,C) ≈ δ (2.28)! ! where! δ ! is! a! small! value! such! as! 0.0001.! This! is! a! forward! difference! first! order! approximation.!!A!more!accurate!central!difference!could!be!used.!The!second!order! approximation!for!the!scaled!sensitivity!coefficient!(Dunker!1984)!can!be!expressed! as,! 24 ! k ∂T T (x,t,(1+ δ )k,C) − T (x,t,(1− δ )k,C) ≈ !! ∂k C 2δ (2.29)! ! For!simplicity!in!notation,!let! ! ! Tδ k = T (x,t,(1+ δ )k,C) !! (2.30)! ! Then!using!the!forward!difference!approximation!as!given!by!Eq.!(2.28)!in!Eq.!(2.26)! gives! ! ! Tδ k − T Tδ C − T + + (T − T0 ) ≈ 0 !! δ δ (2.31)! ! Tδ k Tδ C 2T + − + (T − T0 ) ≈ 0 !! δ δ δ (2.32)! Tδ k + Tδ C − 2T + δ (T − T0 ) ≈ 0 !! (2.33)! or! ! Finally!after!reJarranging,!! ! ! ! Recall! that! Eq.! (2.33)! is! for! the! case! when! the! heat! flux! is! known! and! the! thermal! properties! are! constant.! Equation! (2.33)! is! the! important! equation! to! use! for! intrinsic! verification! for! this! problem.! It! requires! only! three! complete! computations,! one! for! T,! one! for! Tδ k ,! and! finally! for! Tδ C .! The! summation! can! be! evaluated!over!the!complete!domain!of!the!problem!to!see!if!it!indeed!is!nearly!equal! to!zero!for!all!locations!and!times!of!the!numerical!solution.! 25 For!the!situations!where!the!step!size! δ !can!be!very!small,!a!more!accurate! sensitivity!coefficient!can!be!obtained!using!method!of!complex!variables!(Martins! and!others!2000).!! ! k ∂T Im(T (x,t,(1+ iδ )k,C)) ≈ !! ∂k C δ (2.34)! ! One! of! the! major! advantages! of! using! complex! variable! for! calculating! sensitivity! coefficients! is! that! the! truncations! error! is! minimized,! as! there! is! no! difference! involved! in! numerator! of! Eq.! (2.34)! as! compared! to! finite! difference! method! (Martins! and! others! 2000).! However,! in! this! article! finite! difference! methods!have!been!used!to!calculate!sensitivity!coefficients.! ! ! ! The! following! values,! which! are! typical! for! foods,! are! considered! as! an! example! for! each! of! the! cases.! Dimensionless! scaled! sensitivity! coefficients! are! calculated!for!k!=!0.5!W/mK,!C!=!3.5x106!J/m3K,!T0!=!20oC,! δ !=!0.0001,!x/L!=0,!L!=! 20! mm,! Δx / L = 0.02, αΔt / L = 3.5714 × 10−05. ! The! maximum! temperature! rise! =! 100°C.! ! A!finite!element!code!(COMSOL®)!is!used!to!demonstrate!the!concepts.!An! analytical! solution! is! not! needed! for! this! intrinsic! verification! case.! Problem! explained!in!case!1!is!solved!numerically!and!the!relationship!given!by!eq.!(2.33)!is! calculated.!! ! ! ! Dimensionless!scaled!sensitivity!coefficients!for!k!and!C!are!shown!in!Fig.!2.1! at! the! heated! surface.! Note! that! the! shapes! of! two! curves! are! equal! for! a! 26 dimensionless! time! of! 0.2.! This! suggests! that! they! are! linearly! dependent! or! correlated.!Hence,!an!experiment!performed!for!dimensionless!time!<0.2,!it!will!not! be!possible!to!estimate!both!parameters.!For!dimensionless!time!>0.2,!the!value!of!   X k ! goes! to! a! constant! value! and! X C ! keeps! increasing.! Hence,! to! estimate! both!  parameters,!the!experiment!has!to!be!performed!at!least!for! t !of!0.6.!However,!since!   X C is!larger!the!standard!error!will!be!lower!than!that!for! X k .!! ! ! 0 ˜ Xk ˜ XC −0.1 ˜ SSC X −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 0 0.1 0.2 0.3 ˜ t 0.4 0.5 0.6 Figure!2.1!Dimensionless!scaled!sensitivity!coefficient!for!k!and!C!in!case!1:! X22B10T0,!for!k!=!0.5!W/mK,!C!=!3.5x106!J/m3K,!delta!=!0.0001,!x/L!=0,! Δx / L = 0.02 (For!interpretation!of!the!references!to!color!in!this!and!all!other!figure,! the!reader!is!referred!to!the!electronic!version!of!this!dissertation.)! 27 ! Table! 2.1! Solutions! to! the! heat! transfer! problems! in! case! 1:! X22B10T1! with! first! order!approximation! t ' 0.1! 0.21! 0.31! 0.41! 0.52! x ' δ ! 0! 0! 0! 0! 0! 0.0001! 0.0001! 0.0001! 0.0001! 0.0001!  T !  Xk !  XC ! IS ! 0.36386368!J0.18189423!J0.18194214! J0.0032773! 0.51386064!J0.25315515!J0.26066516!J0.00483891! 0.63351296!J0.29462794!J0.33883058!J0.00653256! 0.74245078!J0.31578265!J0.42659962!J0.00822166! 0.84751112!J0.32568253!J0.52174708!J0.00978089! ! ! Table! 2.1! shows! the! scaled! sensitivity! calculation! using! the! first! order! derivative! mentioned! in! Eq.! (2.28).! Scaled! sensitivity! coefficients! values! are! presented! along! with! the! time,! position,! delta! and! temperature! rise.! In! the! last! column! the! absolute! sum! of! scaled! sensitivity! coefficient! is! presented! after! subtracting!with!the!temperature!rise!(Eq.!(2.26)).!The!values!are!very!close!to!zero! as!expected,!which!suggests!that!the!finite!element!program!has!passed!the!intrinsic! verification!test.!! Table!2.2!Solutions!to!the!heat!transfer!problems!in!case!1:!X22B10T1!with!second! order!approximation! t ' 0.1! 0.21! 0.31! 0.41! 0.52! x ' δ ! 0! 0! 0! 0! 0! 0.0001! 0.0001! 0.0001! 0.0001! 0.0001!  T !  Xk !  XC ! IS ! 0.36386368!J0.18190788! J0.1819558! 0.00000029! 0.51386064!J0.25317494! J0.2606857! 0.00000046! 0.63351296!J0.29465296!J0.33886001!0.00000067! 0.74245078!J0.31581137!J0.42663942! 0.0000009! 0.84751112!J0.32571349!J0.52179764!0.00000113! ! However,!the!values!of!IS!can!be!improved!by!using!a!more!accurate!second! order! approximation! as! mentioned! in! Eq.! (2.29).! The! values! for! the! second! order! 28 approximation! are! presented! in! Table! 2.2.! The! IS! is! plotted! in! Figure! 2.2! and! note! that!the!values!are!increasing!over!time.!Comparing!the!values!of!IS!in!Table!2.1!for! first! order! and! Table! 2.2! for! second! order! approximation,! the! difference! is! quite! large.! Second! order! approximation! provides! better! accuracy! than! the! first! order! approximation.! Again,! the! finite! element! code! has! passed! the! intrinsic! verification! process.! −6 Intrinsic Sum ( I S ) 1.4 x 10 1.2 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 ˜ t 0.4 0.5 0.6 Figure!2.2!Plot!of!IS!in!case!1:!X22B10T0,!using!2ndJorder!finite!difference! ! So! far! IVM! has! worked! well! and! showed! that! the! finite! element! model! was! adequate.! To! show! the! strength! of! the! IVM,! consider! a! small! imperfection! in! the! finite!element!method.!To!demonstrate!this,!an!unrefined!model!is!considered!with! 29 the! refined! model! in! case! 1.! The! initial! finer! time! step! from! the! finite! element! program! was! eliminated! to! create! the! unrefined! model.! In! the! refined! model,! this! initial! fine! time! step! size! was! 0.0001! sec.! However,! make! this! as! 0.1! sec,! which! is! same! as! the! time! step! of! solution! method.! The! result! from! this! imperfection! is! presented! in! Figure! 2.3.! There! is! substantial! error! in! the! unrefined! model! as! compared!to!the!refined!model.!This!error!would!not!have!been!obvious!without!the! use! of! IVM.! Also,! during! the! parameter! estimation! problem! the! unrefined! model! would!not!provide!good!estimates!of!the!parameters!and!would!have!a!signature!in! residuals,! large! root! mean! squared! error! and! large! standard! error! of! the! parameters.!! ! 30 Intrinsic Sum ( I S ) 0.2 0 −0.2 −0.4 −0.6 −0.8 0 Unrefined Code Refined Code 0.1 0.2 0.3 ˜ t 0.4 0.5 0.6 Figure!2.3!Plot!of!IS!in!case!1:!X22B10T0!with!refined!and!unrefined!finite!element! code! 2.2.2'Case'2:'OneSdimensional'transient'heat'conduction'in'a'flat'plate'with' time'varying'temperature'on'one'side'and'insulated'on'another'(X12BS0T1)' ! In! this! case,! IS! is! derived! with! a! different! boundary! condition.! If! the! nonJ homogeneous!boundary!condition!of!a!given!heat!flux!in!Eq.!(2.6)!were!replaced!by! ! ! T (0,t) = T1 + (T0 − T1) f (t) !! (2.35)!  A!significantly!different!relation!than!Eq.!(2.26)!is!obtained.!In!this!case!the! T ! function!is!defined!by! !  T − T1 !! T= T0 − T1 31 (2.36)! ! Notice!that!now!Eq.!(2.22)!becomes! ! !    T − T1 = (T0 − T1)T ( x, t (k,C) ) !! (2.37)! ! The!scaled!sensitivity!coefficient!for!k!is! ! ⎡ ∂T ∂t ⎤   ∂T = (T0 − T1) ⎢ ⎥  ∂k C ⎢ ∂t C ∂k ⎥ ⎣ ⎦ ! !! k (2.38)! ⎡ ∂T ⎤  ∂T  = (T0 − T1) ⎢ −t ⎥ !!  ∂C k ⎢ ∂t k ⎥ ⎣ ⎦ (2.39)! ∂T ∂T +C = 0 !! ∂k C ∂C k (2.40)! ⎡ ∂T ⎤  ∂T  = (T0 − T1) ⎢t ⎥  ∂k C ⎢ ∂t C ⎥ ⎣ ⎦ ! The!sensitivity!coefficient!for!C!is! ! ! C ! and!Eq.!(2.26)!becomes! ! k ! Comparing! Eq.! (2.40)! with! (2.26),! the! only! difference! is! that! the! rightJhand! side!of!Eq.!(2.40)!is!zero.!Equation!(2.40)!demonstrates!linear!dependence!as!given! by!Eq.!(2.3).!Hence,!in!this!case!the!parameters!k!and!C!cannot!be!estimated!uniquely! and!simultaneously!(Figure!2.4).!Only!a!combination!of!parameters,!such!as!thermal! diffusivity,!can!be!estimated.!The!numerical!values!are!represented!in!Table!2.3.! 32 ! ˜ Xk ˜ XC 0.2 ˜ SSC X 0.1 0 −0.1 −0.2 0 0.1 0.2 0.3 ˜ t 0.4 0.5 0.6 Figure!2.4!Scaled!sensitivity!coefficients!for!k!and!C!in!case!2:!X12B10T0!! Table 2.3 Solutions!to!the!heat!transfer!problems!in!Case!2:!X12B10T1! !  T ! t ' x ' δ ! 0.1! 0.21! 0.31! 0.41! 0.52! 0.5! 0.5! 0.5! 0.5! 0.5! 0.0001! 0.0001! 0.0001! 0.0001! 0.0001!  Xk !  XC ! IS ! 0.72316561!0.07492805!J0.07492805!0.00000003! 0.76868147!0.06123446!J0.06123446!0.00000013! 0.79366186!0.06322003!J0.06322003!0.00000019! 0.81201048!0.06426641!J0.06426641!0.00000023! 0.8261412! 0.0621557! J0.0621557! 0.00000026! ! ' 33 2.2.3'Case'3:'Scaled'sensitivity'relation'for'oneSdimensional'transient'heat' conduction'in'a'cylindrical'coordinate'system'for'boundary'condition'of' second'kind'(R22B10T1)' ! So! far,! case! studies! with! the! dimensionless! analysis! and! scaled! sensitivity! coefficient!relationships!in!the!Cartesian!coordinate!system!are!presented.!Given!the! boundary!conditions!are!same,!the!relation!still!holds!true!for!cylindrical!coordinate! system.! This! will! be! demonstrated! by! showing! different! cases.! Let! us! consider! a! hollow!cylinder!with!inner!radius!R1!and!outer!radius!R2.!! For!the!model;! ! k ∂ ⎛ ∂T ⎞ ∂T ⎜ r ∂r ⎟ = C ∂t r ∂r ⎝ ⎠ R1 < r < R2 , t > 0 ! (2.41)! ! Where!boundary!and!initial!conditions!are! ! ! −k ∂T (R ,t) = qo ∂r 1 ∂T (R ,t) = 0 ∂r 2 T (r,0) = T1 ! (2.42)! ! Note!that!T!is!a!function!of! (r, R,t,C,T0 ,q0 ). !!!! ! !  r≡ r  ,R ≡ R1 2 R2 kt  T − T1 !!  ,t≡ ,T≡ R1 q0 R1 CR12 k (2.43)! With!this!model!and!with!a!known!and!fixed!boundary!heat!flux,!the!dimensionless! temperature!is!given!symbolically!by! ! 34      T = T ( r, R2 , t ) !! ! (2.44)! Now!the!partial!derivatives!of!temperature!T!are!found.!Notice!that! ! ( ) q R     T − T1 = 0 1 T r, R2,t (k,C) !! k ! (2.45)! The!derivative!with!respect!to!k!is! !   q R  q R ⎡ ∂T ∂t ⎤ ∂T = − 0 1T + 0 1 ⎢ ⎥  ∂k C k ⎢ ∂t ∂k ⎥ k2 ⎣ ⎦ C ! !! k (2.46)!  q R  q R ⎡ ∂T ⎤ ∂T  = − 0 1 T + 0 1 ⎢t ⎥  ∂k C k k ⎢ ∂t ⎥ ⎣ C⎦ ! The!sensitivity!coefficients!for!C!is! ! ! C  q R ⎡ ∂T ∂T  = 0 1 ⎢ −t  ∂C k k ⎢ ∂t ⎣ ⎤ ⎥ !! ⎦ k⎥ (2.47)! ! Adding!Eqs.!(2.46)!and!(2.47)!gives! ! !k   q R  q R ⎡ ∂T ⎤ q0 R1 ⎡ ∂T ⎤ q R  ∂T ∂T   +C = − 0 1 T + 0 1 ⎢t ⎥+ ⎢ −t ⎥ = − 0 1 T !!(2.48)!  ⎥  ⎥ ∂k C ∂C k k k ⎢ ∂t k ⎢ ∂t k ⎣ ⎣ C⎦ k⎦ ! which!can!be!expressed!as,! ! ! k ∂T ∂T +C = −(T − T1) !! ∂k C ∂C k 35 (2.49)! Even! in! the! case! of! a! cylindrical! coordinate! system! for! same! boundary! conditions,!the!relationship!of!scaled!sensitivity!(Eq.!(2.49))!coefficients!is!the!same! as!in!the!Cartesian!coordinate!system!(Eq.!(2.26)).!In!this!case!as!well,!since!the!sum! of! scaled! sensitivity! coefficients! is! not! zero,! the! parameters! can! be! estimated!  uniquely!and!simultaneously.!It!can!be!seen!from!Fig.!2J5!that!the!shapes!of! X k !are!   very! different! form! that! of! X C .! Also,! the! magnitude! of! X k ! keeps! increasing! with!  time! more! than! X C ,! suggesting! that! in! this! case! thermal! conductivity! can! be! estimated! with! good! accuracy.! Also,! as! compared! to! the! X22B10T1! problem,! R22B10T1!provides!a!solution!where!the!estimation!of!k!and!C!might!be!possible!for!   even! smaller! times.! This! is! possible! because! of! the! shape! of! X k and! X C .! For! the!   dimensionless! time! of! <0.2,! X k and X C ! are! correlated! in! X22B10T1! and! uncorrelated!in!R22B10T1.! ! The!intrinsic!verification!identity!given!by!Eq.!(2.49)!is!shown!in!Table!2.4!as! IS.!The!finite!element!code!again!has!done!well!in!this!case.!The!values!of!the!identity! are!very!small!and!can!be!safely!assumed!to!be!approximately!zero.!For!checking!the! effectiveness! of! this! identity,! an! unrefined! model! was! created! by! not! refining! the! initial!time!steps.!The!resulting!values!are!plotted!in!Fig.!2.6.!The!unrefined!model! will! not! produce! good! results! and! also! will! not! do! very! well! when! one! tries! to! estimate! parameters! based! on! this! model.! These! kinds! of! imperfections! in! large! numerical!code!can!be!easily!detected!by!the!use!of!IVM.!! 36 Table 2.4 Solutions!to!the!heat!transfer!problems!in!Case!3:!R22B10T1! ! x ' 0.1! 0.21! 0.31! 0.41! 0.52! δ ! 0! 0! 0! 0! 0! t ' 0.0001! 0.0001! 0.0001! 0.0001! 0.0001!  T !  Xk !  XC ! IS ! 0.38364793!J0.24094645!J0.14270148!0.00000025! 0.50257048!J0.28473387!J0.21783661!0.00000043! 0.60869499!J0.29227133!J0.31642366!0.00000053! 0.71356111!J0.29333138!J0.42022974!0.00000059! 0.81830343! J0.293467! J0.52483643!0.00000063! ! ! 0 ˜ Xk ˜ XC −0.1 ˜ SSC X −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 0 0.1 0.2 0.3 ˜ t 0.4 0.5 0.6 Figure!2.5!Scaled!sensitivity!coefficients!for!k!and!C!in!case!3:!R22B10T0!! 37 Intrinsic Sum ( I S ) 0.4 0.2 0 −0.2 −0.4 −0.6 0 Unrefined Code Refined Code 0.1 0.2 0.3 ˜ t 0.4 0.5 0.6 ! Figure!2.6!Plot!of!IS!in!case!3:!R22B10T0!with!refined!and!unrefined!finite!element! code.! ' 2.2.4'Case'4:'Scaled'sensitivity'relation'for'oneSdimensional'transient'heat' conduction'in'a'cylindrical'coordinate'system'for'boundary'condition'of'first' kind'(R12B10T1)' In!this!case,!scaled!sensitivity!relation!with!a!different!boundary!condition!is! investigated.!If!the!nonJhomogeneous!boundary!condition!of!a!given!heat!flux!in!Eq.! (2.42)!were!replaced!by! ! ! T (R,t) = T1 + (T0 − T1) f (t) !! ! 38 (2.50)!  In!this!case!the! T !function!is!defined!by! !  T − T1 !! T= T0 − T1 ! (2.51)! ! Notice!that!now!Eq.!(2.22)!becomes! ! ! ( )     T − T1 = (T0 − T1)T r, R2 , t (k,C) !! (2.52)! ! The!scaled!sensitivity!coefficients!for!k!is! ! ⎡ ∂T ∂t ⎤   ∂T = (T0 − T1) ⎢ ⎥  ∂k C ⎢ ∂t C ∂k ⎥ ⎣ ⎦ ! !! k (2.53)! ⎤ ⎥ !! ⎦ k⎥ (2.54)! ⎡ ∂T ⎤  ∂T  = (T0 − T1) ⎢t ⎥  ⎥ ∂k C ⎢ ∂t C ⎦ ⎣ ! The!sensitivity!coefficients!for!C!is! ! ! C ⎡ ∂T  ∂T  = (T0 − T1) ⎢ −t  ∂C k ⎢ ∂t ⎣ ! Adding!Eqss.!(2.53)!and!(2.54)!gives! ! ! k ∂T ∂T +C = 0 !! ∂k C ∂C k 39 (2.55)! Comparing! Eq.! (2.55)! with! (2.40),! the! sum! of! scaled! sensitivity! coefficients! are! equal! to! zero.! The! cylindrical! coordinate! system! and! the! Cartesian! coordinate! system!have!same!result.!Both!of!these!equations!show!linear!dependence!as!given! by! Eq.! (2.3).! Hence,! in! this! case! also! the! parameters! k! and! C! cannot! be! estimated! simultaneously! (Fig.! 2.7).! Only! a! combination! of! parameters,! such! as! thermal! diffusivity,! can! be! estimated.! Sum! of! scaled! sensitivity! coefficients! as! given! by! Eq.! (2.55)!is!zero!and!it!is!shown!in!Table!2.5.!! ! 0.2 ˜ Xk ˜ XC ˜ SSC X 0.1 0 −0.1 −0.2 0 0.1 0.2 0.3 ˜ t 0.4 0.5 0.6 Figure!2.7!Scaled!sensitivity!coefficient!for!k!and!C!in!case!4:!R12B10T0!! ! ! ! 40 2.2.5'Case'5:'Scaled'sensitivity'relation'for'oneSdimensional'transient'heat' conduction'in'a'cylindrical'coordinate'system'for'boundary'condition'of'third' kind'(R32B10T1)' In! this! case,! the! IS! is! derived! with! a! convective! boundary! condition.! If! the! nonJhomogeneous! boundary! condition! of! a! given! heat! flux! in! Eq.! (2.42)! were! replaced!by! In! this! case,! the! IS! is! derived! with! a! convective! boundary! condition.! If! the! nonJhomogeneous! boundary! condition! of! a! given! heat! flux! in! Eq.! (2.42)! were! replaced!by! !! −k ∂T (R ,t) = h T (R1,t) − T∞ !! ∂r 1 ( ) (2.56)! !  In!this!case!the! T !function!is!defined!by! !  T !! T= T∞ (2.57)! ! Notice!that!by!using!the!dimensionless!form!of!equations,!Eq.!(2.56)!becomes! ! !  ∂T   (1, t ) = h !!  ∂r  hR Where,! h = 1 k ! ! Which!leads!to!the!following!temperature!relation:! ! 41 (2.58)! ( ) ( )      T = T∞ T r, R2 , t (k,C), h !! ! (2.59)! ! The!derivative!with!respect!to!k!is! ! ⎡ ∂T     ∂T ∂t ∂T ∂h ⎤ ⎥ = T∞ ⎢ +   ∂k C,h ⎢ ∂t C,h ∂k ∂ h C,h ∂k ⎥ ⎣ ⎦ ! !! k (2.60)! ⎡ ∂T ⎤   ∂T  ∂T  ⎥ = T∞ ⎢t −h   ∂k C,h ∂ h C,h ⎥ ⎢ ∂t C,h ⎣ ⎦ ! The!scaled!sensitivity!coefficients!for!C!is! ! ! C ⎡ ∂T ⎤  ∂T  ⎥ !! = T∞ ⎢ −t  ∂C k,h ⎢ ∂t k,h ⎥ ⎣ ⎦ (2.61)! ! The!scaled!sensitivity!coefficients!for!h!is! ! ! h ⎡ ∂T ⎤  ∂T  ⎥ !! = T∞ ⎢ h  ∂h k,C ⎢ ∂ h k,C ⎥ ⎣ ⎦ (2.62)! ! Adding!Eqss.!(2.60),!(2.61)!and!(2.62)!provides,! ! ! k ∂T ∂T ∂T +C +h = 0 !! ∂k C,h ∂C k,h ∂h C,h ! 42 (2.63)! With! the! convective! boundary! condition,! the! sum! of! scaled! sensitivity! coefficients!is!equal!to!zero!(Eq.!(2.63)).!Hence,!in!this!case!the!parameters!k,!C!and!h' cannot!be!estimated!uniquely!and!simultaneously.!For!a!value!of!h!=!1000!W/m2JK,!    the! X k , X C !and! X h !are!shown!in!Fig.!2.8.!For!very!large!values!of!h,!the!boundary! condition! tends! to! be! same! as! a! temperature! boundary! condition! and! in! that! case!    the! X h ! is! very! small! and! X k ! and! X C ! are! highly! correlated.! The! sum! of! scaled! sensitivity!coefficients!is!presented!in!Fig.!2.9.!! ! ! 0.15 0.1 ˜ SSC X 0.05 0 −0.05 ˜ Xk ˜ XC −0.1 ˜ Xh −0.15 −0.2 0 0.1 0.2 0.3 ˜ t 0.4 0.5 0.6 Figure!2.8!Scaled!sensitivity!coefficients!for!k,'C'and'h!in!case!5:!R32B10T0!with!h!=! 1000!W/m2JK! 43 −7 Intrinsic Sum ( I S ) 3.5 x 10 3 2.5 2 1.5 1 0.5 0 0 0.1 0.2 0.3 ˜ t 0.4 0.5 0.6 Figure!2.9!Plot!of!IS'in!case!5:!R32B10T0!! ! ! ! Table!2.5!Solutions!to!the!heat!transfer!problems!in!Case!4:!R12B10T1!  T ! t ' x ' δ ! 0.1! 0.21! 0.31! 0.41! 0.52! 0.5! 0.5! 0.5! 0.5! 0.5! 0.0001! 0.0001! 0.0001! 0.0001! 0.0001!  Xk !  XC ! IS ! 0.67490936!0.08902885!J0.08902885!0.00000001! 0.7435452! 0.11325045!J0.11325045! 0! 0.79155742! 0.1223164! J0.1223164! 0.00000005! 0.82624201!0.11776184!J0.11776184!0.00000012! 0.8513107! 0.1063334! J0.1063334! 0.00000017! ' ' 44 2.3'Conclusions' The!dimensionless!derivation!of!scaled!sensitivity!coefficients!was!presented.! Two! important! applications! were! discussed.! The! first! application! is! related! to! the! idea! of! intrinsic! verification! of! large! numerical! codes.! The! identity! given! by! scaled! sensitivity!relations!for!heat!transfer!problems!provides!a!method!for!checking!the! accuracy! of! a! computer! code! at! interior! and! boundary! points! and! at! any! time.! Several! equations! are! presented! in! each! case! that! gives! a! suggested! way! to! implement! the! concept.! The! concept! is! quite! general! and! is! not! restricted! to! heat! conduction! or! linear! problems.! The! concept! may! be! very! important! for! intrinsic! verification! of! computer! codes! for! various! engineering! problems.! The! second! application! is! related! to! the! problems! in! parameter! estimation.! The! scaled! sensitivity! coefficient! can! provide! useful! insight! in! to! the! parameter! estimation! problem.!It!can!show!if!all!the!parameters!in!the!model!can!be!estimated!and!with! what!accuracy.!With!the!scaled!sensitivity!relation,!it!has!been!shown!that!in!certain! boundary! conditions,! not! all! the! parameters! in! the! model! can! be! estimated.! As! a! general!rule,!if!the!sum!of!scaled!sensitivity!coefficients!is!equal!to!zero!then!not!all! the! parameters! in! the! model! can! be! estimated! uniquely! and! simultaneously.!! Instead,!only!a!combination!of!the!parameters,!such!as!a!ratio,!can!be!estimated.! ' ' 45 REFERENCES 46 REFERENCES' Beck!JV.!1967.!Transient!sensitivity!coefficients!for!the!thermal!contact!conductance.! International!Journal!of!Heat!and!Mass!Transfer!10(11):1615J7.! Beck! JV.! 1969.! Determination! of! optimum,! transient! experiments! for! thermal! contact! conductance.! International! Journal! of! Heat! and! Mass! Transfer! 12(5):621J33.! Beck! JV.! 1970.! Nonlinear! estimation! applied! to! the! nonlinear! inverse! heat! conduction! problem.! International! Journal! of! Heat! and! Mass! Transfer! 13(4):703J16.! Beck!JV,!Arnold!KJ.!1977.!Parameter!Estimation.!New!York:!Wiley.! Beck!JV,!Litkouhi!B.!1988.!Heat!conduction!numbering!system!for!basic!geometries.! International!Journal!of!Heat!and!Mass!Transfer!31(3):505J15.! Beck!JV,!McMasters!R,!Dowding!KJ,!Amos!DE.!2006.!Intrinsic!verification!methods!in! linear! heat! conduction.! International! Journal! of! Heat! and! Mass! Transfer! 49(17‚Äì18):2984J94.! Beck! JV,! Woodbury! KA.! 1998.! Inverse! problems! and! parameter! estimation:! integration! of! measurements! and! analysis.! Measurement! Science! and! Technology!9(6):839.! Blackwell! BF,! Dowding! KJ,! Cochran! RJ.! 1999.! Development! and! implementation! of! sensitivity! coefficient! equation! for! heat! conduction! problems.! Numerical! Heat!Transfer,!Part!B:!Fundamentals!36(1):15J32.! Chen!B,!Tong!L.!2004.!Sensitivity!analysis!of!heat!conduction!for!functionally!graded! materials.!Materials!&!Design!25(8):663J72.! Dolan! K,! Valdramidis! V,! Mishra! D.! 2012.! Parameter! estimation! for! dynamic! microbial!inactivation;!which!model,!which!precision?!Food!Control.! Dowding! KJ,! Blackwell! BF,! Cochran! RJ.! 1999.! Application! of! sensitivity! coefficients! for! heat! conduction! problems.! Numerical! Heat! Transfer,! Part! B:! Fundamentals!36(1):33J55.! 47 Dunker! AM.! 1984.! The! decoupled! direct! method! for! calculating! sensitivity! coefficients!in!chemical!kinetics.!The!Journal!of!Chemical!Physics!81(5):2385J 93.! Koda! M,! Dogru! AH,! Seinfeld! JH.! 1979.! Sensitivity! analysis! of! partial! differential! equations! with! application! to! reaction! and! diffusion! processes.! Journal! of! Computational!Physics!30(2):259J82.! Martins! J,! Kroo! IM,! Alonso! JJ.! 2000.! An! automated! method! for! sensitivity! analysis! using!complex!variables.!AIAA!paper!689:2000.! Roy!CJ.!2005.!Review!of!code!and!solution!verification!procedures!for!computational! simulation.!Journal!of!Computational!Physics!205(1):131J56.! Salari!K,!Knupp!P.!2000.!Code!Verification!by!the!Method!of!Manufactured!Solutions.! Other!Information:!PBD:!1!Jun!2000.!p.!Medium:!P;!Size:!124!pages.! Sun! N,! Sun! NZ,! Elimelech! M,! Ryan! JN.! 2001.! Sensitivity! analysis! and! parameter! identifiability! for! colloid! transport! in! geochemically! heterogeneous! porous! media.!Water!Resources!Research!37(2):209J22.! ! ! ! ! ! ! ! ! ! 48 Chapter'3 ' Intrinsic'Verification'in'Parameter'Estimation'Problems'for'TemperatureS Dependent'Thermal'Properties' ' Abstract' ! ! ! ' ! Verification! of! numerical! codes! is! important,! because! the! accuracy! of! the! code!not!only!affects!the!estimation!of!parameters!in!the!inverse!problem,!but!also! the! model! prediction! while! solving! the! forward! problem.! The! current! study! is! focused!on!the!numerical!verification!of!numerical!codes!in!the!context!of!parameter! estimation.!The!inverse!heat!conduction!problem!is!solved!in!the!Cartesian!as!well! as! the! cylindrical! coordinate! system! with! the! temperatureJdependent! thermal! properties.! Dimensionless! derivation! of! sensitivity! coefficient! is! presented! and! the! Intrinsic!Sum!is!derived!for!each!case.!The!intrinsic!Sum!for!the!cases!presented!in! this! article! shows! that! it! is! possible! to! estimate! thermal! conductivity! and! specific! heat!simultaneously.!It!also!shows!that!verification!of!the!numerical!code!is!possible! with!this!identity.!! ! Keywords:!Numerical!code!verification,!sensitivity!analysis,!inverse!problems,!heat! transfer,!intrinsic!sum! ! ! 49 3.1'Introduction' ! Thermal!properties!are!important!for!conduction!heat!transfer!problems!for! the! prediction! of! temperature.! However,! knowledge! of! these! properties! is! often! limited! for! new! materials.! Whether! temperatureJdependent! thermal! conductivity! and!specific!heat!can!be!estimated!simultaneously!depends!on!the!defined!boundary! conditions!and!initial!condition!(Beck!and!Arnold!1977c).!TemperatureJdependent! thermal!properties!are!especially!important!to!predict!the!true!temperature!field!at! specific! processing! temperatures.! Estimation! of! these! properties! involves! inverse! heat! conduction! problems! (IHCP).! Temperature! dependence! of! thermal! properties! makes! the! heat! conduction! problem! nonlinear.! Since! the! IHCP! are! often! illJposed,! estimation! of! temperatureJdependent! properties! is! difficult.! Estimated! parameter! accuracy!depends!on!the!measurement!accuracy!and!the!inverse!approach!(Cui!and! others! 2012).! If! a! numerical! solution! is! used! for! the! inverse! problem,! then! its! accuracy! also! affects! the! accuracy! of! the! estimated! parameters.! Hence,! the! numerical!solution!must!be!verified!before!using!it!for!the!inverse!problems.!!! ! Estimation! of! temperatureJdependent! thermal! properties! has! received! considerable! attention! and! many! methods! have! been! proposed! for! the! inverse! problem!(Chang!and!Payne!1990;!Cui!and!others!2012;!Hays!and!Curd!1968;!Imani! and!others!2006;!Kevin!J.!Dowding!1999;!Kim!2001;!Kim!and!others!2003a;!Kim!and! others! 2003b;! Mierzwiczak! and! Kołodziej! 2011;! Yang! 1998;! Yang! 1999;! Yang! 2000b;!Huang!and!Ozisik!1991;!Huang!and!JanJYuan!1995;!Chen!and!others!1996;! Dowding! and! others! 1999;! Beck! and! Osman! 1990;! Dowding! and! Blackwell! 1999;! 50 Dowding!and!others!1998;!Emery!and!Fadale!1997).!!Huang!and!Ozisik!(Huang!and! Ozisik! 1991)! proposed! a! direct! integration! method! for! simultaneously! estimating! temperatureJdependent!thermal!conductivity!and!specific!heat.!They!also!used!this! method! to! accurately! provide! the! initial! guesses! of! the! parameters.! Due! to! the! nonlinearity! of! the! problem! with! temperatureJdependent! thermal! properties! estimation,! an! exact! solution! is! not! possible.! Hence,! there! are! several! numerical! techniques! that! have! been! used! as! a! solution! to! this! problem! (Huang! and! Ozisik! 1991).!However,!an!exact!solution!have!been!proposed!for!the!case!where!thermal! diffusivity! is! constant! (Lesnic! and! others! 1995).! Simultaneous! estimation! of! temperatureJdependent! thermal! properties! using! a! oneJdimensional! heat! conduction! problem! was! solved! using! nonlinear! estimation! for! the! Carbon/Epoxy! material! during! curing! (Scott! and! Beck! 1992).! TemperatureJdependent! thermal! conductivity!was!estimated!using!the!oneJdimensional!heat!conduction!problem!by! a! linear! inverse! model! (Yang! 1998).! TemperatureJdependent! thermal! conductivity! was!estimated!using!nonlinear!estimation!(Yang!1999).!Simultaneous!estimation!of! temperatureJdependent! thermal! conductivity! and! specific! heat! was! performed! using!a!nonlinear!method!and!nonisothermal!experiments!(Yang!2000b).!Dowding! and! Blackwell! considered! the! linear! variation! in! thermal! conductivity! and! specific! heat! and! proposed! an! optimal! experimental! design! for! simultaneous! estimation! of! temperatureJdependent!parameters!(Dowding!and!Blackwell!1999).!! Verification!of!numerical!codes!has!been!an!issue!where!no!exact!solution!to! the! heat! transfer! problem! exists! (Salari! and! Knupp! 2000).! In! this! paper,! a! 51 verification! method! is! proposed! with! a! potential! to! verify! the! accuracy! of! the! numerical!code!used!for!estimating!temperatureJdependent!thermal!properties.!The! intrinsic! verification! method! (IVM)! presented! in! this! paper! is! based! on! dimensionless! derivation! of! sensitivity! coefficients.! The! sensitivity! coefficient! of! a! parameter!is!the!first!partial!derivative!of!the!function!involving!the!parameter,!with! respect!to!the!parameter!(Beck!and!Arnold!1977c;!Bennie!F.!Blackwell!1999).!Sum! of! the! scaled! sensitivity! coefficients! should! not! be! equal! to! zero! in! the! parameter! estimation!problem.!That!means!there!should!not!be!any!linear!dependence!among! the! sensitivity! coefficients.! If! the! measured! quantity! is! temperature! T,! the! linear! dependence!relation!can!be!given!by,! ! ! A1β1 ∂T ∂T ∂T + A2β 2 ++ Ap β p = 0 !! ∂β1 ∂β 2 ∂β p (3.1)! ! where! at! least! one! of! the! coefficients! Ai' ! is! not! zero.! The! ith! scaled! sensitivity! coefficient!is!defined!to!be! ! ∂T ˆ !! βi = βi ∂β i (3.2)! The!units!of!each!term!in!Eq.!(2.3)!must!be!consistent.!One!way!to!have!this! consistency! is! to! let! each! Ai! coefficient! be! equal! to! unity.! This! result! for! Eq.! (2.3)! shows! that! when! such! a! linear! relationship! occurs,! not! all! the! parameters! can! be! simultaneously! and! independently! estimated.! Hence,! relationships! among! the! sensitivity!coefficients!are!important!for!parameter!identifiability.! ! 52 The! objective! of! this! paper! is! to! demonstrate! a! verification! method! for! numerical!codes!using!examples!of!temperatureJdependent!thermal!properties.!The! heat!conduction!problem!in!Cartesian!and!cylindrical!coordinate!systems!with!initial! and! boundary! conditions! are! chosen.! The! intrinsic! sum! is! derived! using! the! dimensionless!derivation!of!sensitivity!coefficients.!! 3.2'Case'1:'OneSdimensional'transient'heat'conduction'in'a'flat'plate'with'heat' flux'on'one'side'and'insulated'on'another'(X22B10T1)' ! The!oneJdimensional!transient!heat!conduction!equation!for!temperatureJ variable!thermal!conductivity!and!volumetric!heat!capacity!(caused!by!changes!in! the!specific!heat)!can!be!given!as! ! ! ∂ ⎡ ∂T ⎤ ∂T ⎢ k1 f k ∂x ⎥ = C1 fC ∂t , 0 < x < L, t > 0 !! ∂x ⎣ ⎦ (3.3)! !  k  C k = 2 , C = 2 !! k1 C1 (3.4)! where! Linear!functions!of!temperature!for!the!thermal!conductivity!and!volumetric!heat! capacity!are!considered!(Beck!1964);!they!are! ! !  f k (T − T0 , k ) = 1+ (T − T0 ) − (T1 − T0 ) ( k − 1) !! (T2 − T0 ) − (T1 − T0 ) (3.5)! !  fC (T − T0 , C) = 1+ (T − T0 ) − (T1 − T0 ) (C − 1) !!  T2 − T0 ) − (T1 − T0 ) ( (3.6)! 53 which!gives!the!thermal!conductivity!value!of!k1!at!T1!and!k2!at!T2.! ! The! temperature! dependence! causes! Eq.! (3.3)! to! be! nonlinear! in! T.! The! boundary! conditions!are! ! ∂T (0,t) = q0 f (t) !! ∂x (3.7)! ∂T (L,t) = 0 !! ∂x −k1 f k ! (3.8)! ! where!the!function!on!the!right!of!Eq.!(3.7)!is!known;!also!the!q0!value!is!known.!The! initial!condition!is! T (x,0) = T0 !! ! (3.9)! ! ! The!above!problem!is!now!put!in!a!dimensionless!form.!Let! !  x≡ kt x    T − T0 , k ≡ ,t≡ 1 ,T≡ 2 L q0 L C1L k1 k2  C2  T1 − T0  T2 − T0 ,C≡ ,T ≡ , T2 ≡ !!(3.10)! k1 C1 1 q0 L q0 L k1 k1 Then!the!describing!differential!equation!becomes! ! !       ⎞ ∂T ⎤ ⎛ ⎞ ∂T T − T1  T − T1  ∂ ⎡⎛   ( k − 1)⎟ (C − 1)⎟ , 0 < x < 1, t > 0 !! (3.11)! ⎢⎜ 1+  ⎥ = ⎜ 1+     ⎢ ⎥ ∂ x ⎣⎝ T2 − T1 ⎠ ∂ x ⎦ ⎝ T2 − T1 ⎠ ∂t ! The!boundary!conditions!become! ! !    ⎛ ⎞ ∂T T − T1    − ⎜ 1+  ( k − 1)⎟ (0, t ) = f (t ) !!   ⎝ T2 − T1 ⎠ ∂x 54 (3.12)!  ∂T  (1, t ) = 0 !!  ∂x ! (3.13)! The!initial!temperature!distribution!is!!   T ( x,0) = 0 !! ! ! (3.14)! ! With!this!model!and!with!a!known!and!fixed!boundary!heat!flux!in!Eq.!(3.12)! the!dimensionless!temperature!is!given!symbolically!by! ! !         T = T ( x, t , k, C, T1, T2 ) !! (3.15)! ! Now!the!partial!derivatives!of!temperature!T!are!found.!Notice!that! ! q L      T − T0 = 0 T x, t (k1,C1), k(k1,k2 ), C(C1,C2 ),T1(k1,T1 − T0 ),T2 (k1,T2 − T0 ) !(3.16)! k1 ( ) ! The!derivative!with!respect!to!k1!is! !       q L  q L ⎡ ∂T ∂t ∂T ∂ k ∂T ∂T1 ∂T ∂T2 ⎤ ∂T =− 0 T+ 0 ⎢ +  + + ⎥  2 ∂k1 k1 ⎣ ∂t ∂k1 ∂ k ∂k1 ∂T1 ∂k1 ∂T2 ∂k1 ⎦ k1 ! !! (3.17)!     q L  q L ⎡ ∂T  ∂T ∂T ∂T ∂T ⎤ ˆ  X k = k1 = − 0 T + 0 ⎢t − k  + T1 + T2 ⎥  1 ∂k1 k1 k1 ⎣ ∂t ∂T1 ∂T2 ⎦ ∂k ! Repeat!for!the!derivative!with!respect!to! k2 to!get!scaled!sensitivity!coefficient! !  ∂T q0 L ⎡  ∂T ⎤ ˆ X k = k2 = k  ⎥ !! 2 ∂k2 k1 ⎢ ∂ k ⎦ ⎣ ! 55 (3.18)! The!scaled!sensitivity!coefficients!for!C1!and!C2!are! !   q L ⎡ ∂T ∂T ∂T ⎤ ˆ  !! X C = C1 = + 0 ⎢ −t −C  1 ∂C1 k1 ⎣ ∂t ∂C ⎥ ⎦ (3.19)! !  q L ⎡ ∂T ⎤ ∂T ˆ !! X C = C2 = 0 ⎢C 2 ∂C2 k1 ⎣ ∂C ⎥ ⎦ (3.20)! ! Notice! that! sum! of! Eqs.! (2.24)! and! (3.20)! is! negative! if! the! temperature! is! increasing!with!time.!Hence!the!effect!of!increasing!the!volumetric!heat!capacity!is! to! decrease! the! computed! temperature.! Next! the! derivative! with! respect! to! T1FT0! and!T2FT0!is!found;!it!is! !  q L  ∂T ∂T ˆ X (T −T ) = (T1 − T0 ) = 0 T1  !! 1 0 ∂(T1 − T0 ) k1 ∂T1 (3.21)! !  q L  ∂T ∂T ˆ X (T −T ) = (T2 − T0 ) = 0 T2  !! 2 0 ∂(T2 − T0 ) k1 ∂T2 (3.22)! ! Adding!Eqs.!(2.23),!(3.18),!(2.24)!and!(3.20)!while!subtracting!Eq.!(3.21)!and!(3.22)! gives! ! ! ( ) ˆ ˆ ˆ ˆ ˆ ˆ X k + X k + X C + X C − X (T −T ) − X (T −T ) + T − T0 = 0 !! 1 2 1 2 1 0 2 0 (3.23)! !  If!the!perfect!insulation!boundary!condition!at! x = 1 !given!by!Eq.!(3.13)!were! replaced!by!the!isothermal!condition!of! !   T (1, t ) = 0 !! the!relation!given!by!Eq.!(3.23)!is!still!valid.! 56 (3.24)! Intrinsic! Sum! (IS)! for! this! case! is! given! by! Eq.! (3.23).! The! four! sensitivity! coefficients!are!also!for!fixed!x!and!t'values.!!The!final!two!derivatives!are!the!rate!of! change!in!the!computed!temperature!when!the!specified!temperature!T1! and'T2!is! changed.!Equation!(3.23)!indicates!that!the!four!parameters!k1,!k2,!C1!and!C2!may!be! simultaneously! estimated! when! temperatures! are! measured! in! the! plate! and! the! heat!flux!is!prescribed.!!The!IS!relation!can!be!used!to!provide!intrinsic!verification! of! finite! element! and! finite! control! volume! computer! codes! for! heat! conduction.! Note!that!Eq.!(2.3)!(sum!of!scaled!sensitivity!coefficients!=!0)!is!not!satisfied!by!Eq.! (3.23),!since!the!sum!of!the!scaled!sensitivity!coefficients!of!k1,!k2,!C1!and!C2!is!not! zero.! Scaled! sensitivity! coefficients! of! (T1! –! T0)! and! (T2! –! T0)! are! extra! terms! that! appear! because! of! the! temperature! dependence! of! the! thermal! conductivity! and! specific!heat.!! To! implement! the! IS,! the! scaled! sensitivity! coefficients! can! be! evaluated! numerically.!Scaled!sensitivity!can!be!computed!as!a!forward!difference!first!order! approximation.! However,! a! more! accurate! central! difference! could! be! used.! The! second! order! approximation! for! the! scaled! sensitivity! coefficient! can! be! expressed! as,! ! βi ∂T T ((1+ δ )β i ) − T ((1− δ )β i ) !! ≈ ∂β i 2δ ! where! δ !is!a!small!value!such!as!0.0001.!! 57 (3.25)! The! following! values! are! considered! as! an! example! for! this! case.! Dimensionless!scaled!sensitivity!coefficients!are!calculated!for!k1!=!0.5!W/mK,!k2!=! 0.55!W/mK,!C1!=!3.5x106!J/m3K,!C2!=!3.9x106!J/m3K,!T0!=!20oC,!x/L!=0,!L!=!5!mm,! Δx / L = 0.02 .! The! maximum! temperature! rise! is! ~100°C! at! the! heated! surface.! ! A! finite! element! code! (COMSOL®,! (COMSOL! 2012))! is! used! to! demonstrate! the! concepts.!An!analytical!solution!is!not!needed!for!this!intrinsic!verification!case.!The! problem!explained!in!case!1!is!solved!numerically!and!the!relationship!given!by!Eq.! (3.23)!is!calculated.!Dimensionless!scaled!sensitivity!coefficients!are!calculated!as! ! ! β ∂T ˆ Xβ = i !! i q0 L ∂β i k1 (3.26)! ! Dimensionless! scaled! sensitivity! coefficients! for! all! the! parameters! are! plotted!in!Figure!3.1.!The!dimensionless!scaled!sensitivities!of!k1!and!C1!are!larger! than!those!for!k2!and!C2.!It!is!also!important!to!notice!that!the!dimensionless!scaled! sensitivities! of! k1! and! C1! are! almost! same! for! dimensionless! time! less! than! 0.05.! Hence,! it! would! not! be! possible! to! estimate! them! uniquely! if! an! experiment! is! performed!only!for!dimensionless!time!up!to!0.05.!This!information!is!important!for! device!design!related!to!estimating!temperatureJdependent!thermal!properties.!The! accuracy!of!the!estimated!parameters!depends!on!the!absolute!magnitude!of!scaled! sensitivity!coefficients!as!compared!to!the!temperature!rise.!In!this!case,!the!order!of! 58 accuracy!for!estimated!parameters!would!be!C1,!k1,!k2,!and!C2.!The!parameters!(T1!–! T0)!and!(T2!–!T0)!are!nuisance!parameters;!the!lower!the!values!of!these,!the!better! the!accuracy!of!thermal!parameters.! ! 0.1 ˜ Xk1 ˜ Xk2 ˜ X C1 ˜ X C2 ˜ X T 1 −T 0 ˜ X T 2 −T 0 ˜ SSC X 0 −0.1 −0.2 −0.3 −0.4 0 0.2 0.4 ˜ t 0.6 0.8 1 Figure! 3.1! Dimensionless! scaled! sensitivity! coefficient! for! the! temperatureJ dependent!parameters!of!heat!transfer!problems!in!case!1!(X22B10T0).!T1!=!25oC! and!T2!=!130oC.! ! Intrinsic!Sum!(Eq.!(3.23))!is!plotted!in!Figure!3.2.!The!values!of!IS!are!on!the! order!of!10J7,!a!small!number!that!can!be!considered!approaching!zero.!This!result! confirms!the!identity!relationship!of!the!IVM,!and!show!that!the!numerical!code!is! 59 accurate.! Any! imperfection! in! the! model! or! numerical! code! will! result! in! larger! IS! than! the! value! shown! in! Figure! 3.2.! The! results! are! shown! in! Table! 3.1.! This! is! a! good!test!to!perform!before!doing!any!inverse!problems!or!even!forward!problem! where! accuracy! in! prediction! is! very! important.! Even! while! developing! models! in! the! commercial! numerical! codes,! it! is! important! to! do! this! verification! to! avoid! coding!errors!in!the!software.!! −7 Intrinsic Sum ( I S ) 0 x 10 −1 −2 −3 −4 0 0.1 0.2 0.3 ˜ t 0.4 0.5 0.6 Figure!3.2!Intrinsic!sum!for!the!heat!transfer!problems!in!case!1!(X22B10T0).! ! ! ! 60 Table!3.1!Intrinsic!Sum!and!dimensionless!scaled!sensitivity!coefficients!for!case!1:! X22B10T1! t !! ! 0.01! 0.11! 0.22! 0.32! 0.42! 0.53!   X !!!!! X k1 ! !!!!! k2 ! %0.0565! %0.0023! %0.1408! %0.0418! %0.1620! %0.0817! %0.1576! %0.1199! %0.1372! %0.1558! %0.1095! %0.1891!     −7 X X X X !!! C2 ! !!!! C2 !!!!!!!! (T1−T0 ) ! ! (T2 −T0 ) ! !! I S × 10 ! !! %0.0582! %0.0006! 0.0006! 0.0003! %0.7! %0.1580! %0.0248! 0.0015! 0.0066! %1.0! %0.2031! %0.0507! 0.0019! 0.0132! %1.3! %0.2444! %0.0800! 0.0021! 0.0200! %1.7! %0.2859! %0.1164! 0.0022! 0.0273! %2.2! %0.3233! %0.1615! 0.0023! 0.0354! %2.7! ! The!choices!of!T1!and!T2!also!affect!the!sensitivity!coefficients!of!parameters! and! eventually! the! parameter! estimates.! In! Figure! 3.1,! the! temperatures! for! evaluation! were,! T1! =! 25oC! and! T2! =! 130oC.! The! values! for! T1! and! T2! are! in! the! range! of! the! experimental! temperatures;! T1' is! close! to! the! initial! temperature! and! T2! is! close! to! the! final! temperature! of! the! product.! For! example,! Figure! 3.3! shows! the! effect! on! dimensionless! scaled! sensitivity! coefficients! if! the! value! of! T2! is! changed! to! 300oC,! which! is! much! higher! than! the! maximum! temperature! attained! by! the! product.! Results! of! this! case! are! presented! in! Table! 3.2.! The! dimensionless! scaled!sensitivity!coefficients!of!k2!and!C2!are!very!small,!so!these!parameters!might! not! be! estimated! with! good! accuracy.! Also,! k1! and! C1! are! correlated! for! dimensionless! time! of! 0.1,! which! means! that! the! experiment! must! be! performed!>0.1!sec!to!estimated!both!k1!and!C1.! 61 ! Table!3.2!Intrinsic!Sum!and!dimensionless!scaled!sensitivity!coefficients!for!case!1:! X22B10T1!with!T2!=!300!oC! t !! ! 0.01! 0.11! 0.22! 0.32! 0.42! 0.53! !       X X X X X !!!!! X k1 ! !!!!! k2 ! !!! C2 ! !!!! C2 !!!!!!!! (T1−T0 ) !! (T2 −T0 ) ! I S %0.0583! %0.0009! %0.0590! %0.0002! 0.0002! 0.0001! %0.1703! %0.0168! %0.1774! %0.0099! 0.0007! 0.0026! %0.2186! %0.0334! %0.2418! %0.0207! 0.0009! 0.0052! %0.2392! %0.0498! %0.3051! %0.0331! 0.0011! 0.0081! %0.2413! %0.0657! %0.3735! %0.0487! 0.0012! 0.0112! %0.2338! %0.0809! %0.4438! %0.0684! 0.0013! 0.0146! × 10−7 ! %0.8! %1.9! %2.5! %3.2! %3.8! %4.0! ! 0.1 ˜ Xk1 ˜ Xk2 ˜ X C1 ˜ X C2 ˜ X T 1 −T 0 ˜ X T 2 −T 0 ˜ SSC X 0 −0.1 −0.2 −0.3 −0.4 0 0.2 0.4 ˜ t 0.6 0.8 1 Figure! 3.3! Dimensionless! scaled! sensitivity! coefficient! for! the! temperatureJ dependent!parameters!of!heat!transfer!problems!in!case!1!(X22B10T0).'T1!=!25oC! and!T2!=!300oC.! 62 3.3'Case'2:'Transient'heat'conduction'in'a'hollow'cylinder'with'heat'flux'on' inside'and'insulated'on'the'outside'(R22B10T1)' ! One! important! point! about! the! Eq.! (3.23)! is! that! it! holds! true! also! for! the! cylindrical! heat! transfer! problems! in! r! and! z.! In! this! section! we! are! going! to! demonstrate!that!Eq.!(3.23)!is!applicable!to!cylindrical!heat!transfer!problems.!The! oneJdimensional! transient! heat! conduction! equation! for! temperatureJvariable! thermal! conductivity! and! volumetric! heat! capacity! (caused! by! changes! in! the! specific!heat)!can!be!given!as! ! ! 1 ∂ ⎡ ∂T ⎤ ∂ ⎡ ∂T ⎤ ∂T ⎢ k1 f k r ∂r ⎥ + ∂z ⎢ k1 f k (T , K ) ∂z ⎥ = C1 fC ∂t r ∂r ⎣ !! ⎦ ⎣ ⎦ (3.27)! R1 < r < R2 ,0 < z < Z, t > 0 where!  k  C k = 2 , C = 2 !! k1 C1 ! (3.28)! Linear! functions! of! temperature! for! the! thermal! conductivity! and! volumetric! heat! capacity!are!considered;!they!are! ! !  f k (T − T0 , k ) = 1+ (T − T0 ) − (T1 − T0 ) ( k − 1) !! (T2 − T0 ) − (T1 − T0 ) (3.29)! !  fC (T − T0 , C) = 1+ (T − T0 ) − (T1 − T0 ) (C − 1) !!  (T2 − T0 ) − (T1 − T0 ) (3.30)! ! which!gives!the!thermal!conductivity!value!of!k1!at!T1!and!k2!at!T2.! 63 ! ! The!temperature!dependence!causes!Eq.!(3.27)!to!be!nonlinear!in!T.!The!boundary! conditions!are! −k1 f k ! ∂T (R , z,t) = 0 ∂r 2 ! ∂T (R , z,t) = q0 f (t) !! ∂r 1 ∂T (r,0,t) = 0 ∂z (3.31)! ∂T (r,Z,t) = 0 !! ∂z (3.32)! where!the!function!on!the!right!of!Eq.!(3.31)!is!known;!also!the!q0!value!is!known.! The!initial!condition!is! T (r, z,0) = T0 !! ! (3.33)! ! The!above!problem!is!now!put!in!a!dimensionless!form.!Let! ! !  r≡ r  ,R ≡ R1 2 R2 kt z    T − T0 , k ≡  , z ≡ ,t ≡ 1 , T ≡ 2 R1 R1 q0 R1 C1R1 k1 k2  C2 ,C ≡ !! k1 C1 (3.34)! and! ! ! !  T −T  T −T T1 ≡ 1 0 , T2 ≡ 2 0 !! q0 R1 q0 R1 k1 k1 (3.35)! With!this!model!and!with!a!known!and!fixed!boundary!heat!flux!in!Eq.!(3.31)! the!dimensionless!temperature!is!given!symbolically!by! ! !           T = T ( r, R2 , z , t , k, C, T1, T2 ) !! ! Now!the!partial!derivatives!of!temperature!T!are!found.!Notice!that! 64 (3.36)! ! q L        T − T0 = 0 T r, R2 , z , t (k1,C1), k(k1,k2 ), C(C1,C2 ),T1(k1,T1 − T0 ),T2 (k1,T2 − T0 ) !(3.37)! k1 ( ) ! The!important!point!to!note!in!Eq.!(3.37)!is!that!it!is!similar!to!Eq.!(2.22)!with!   the! extra! terms! of! R2 ! and! z .! However,! when! we! take! the! derivative! of! Eq.! (3.37)! with!respect!to!k1,!k2,!C1,!C2,'T1FT0!and!T2FT0,!we!will!get!the!same!result!as!given! by!Eq.!(3.23).!Hence,!this!derivation!shows!that!the!Eq.! (3.23)!does!not!depend!on! the!coJordinate!system.!This!result!is!demonstrated!below.! The!derivative!with!respect!to!k1!is! ! !     q R  q R ⎡ ∂T  ∂T ∂T ∂T ⎤ ˆ  X k = − 0 1 T + 0 1 ⎢t − k  + T1 + T2 ⎥ !!  1 k1 k1 ⎣ ∂t ∂T1 ∂T2 ⎦ ∂k (3.38)! ! Repeat!for!the!derivative!with!respect!to! k2 to!get! ! !  q R ⎡  ∂T ⎤ ˆ X k = 0 1 ⎢ k  ⎥ !! 2 k1 ⎣ ∂ k ⎦ (3.39)! ! The!sensitivity!coefficients!for!C1!and!C2!are! ! !   q R ⎡ ∂T ∂T ⎤ ˆ  !! X C = 0 1 ⎢ −t −C  1 k1 ⎣ ∂t ∂C ⎥ ⎦ (3.40)! !  q R ⎡ ∂T ⎤ ˆ !! X C = 0 1 ⎢C 2 k1 ⎣ ∂C ⎥ ⎦ (3.41)! ! 65 Next!the!derivative!with!respect!to!T1FT0!and!T2FT0!is!found;!it!is! !  q R  ∂T ˆ X (T −T ) = 0 1 T1  !! 1 0 k1 ∂T1 (3.42)! !  q R  ∂T ˆ X (T −T ) = 0 1 T2  !! 2 0 k1 ∂T2 (3.43)! ! Adding!Eqs.!(3.38),!(3.39),!(3.40)!and!(3.41)!while!subtracting!Eq.!(3.42)!and!(3.43)! gives! ! ! ! ( ) ˆ ˆ ˆ ˆ ˆ ˆ X k + X k + X C + X C − X (T −T ) − X (T −T ) + T − T0 = 0 !! 1 2 1 2 1 0 2 0 (3.44)! ! In!the!case!of!cylindrical!coordinates!and!boundary!conditions!similar!to!case! 1,!the!IS!is!the!same!as!shown!by!Eqs.!(3.23)!and!(3.44).!Even!though!the!IS!is!same! in!both!cases,!the!scaled!sensitivity!coefficients!might!not!be!the!same.!However,!the! sum!of!all!the!scaled!sensitivity!coefficients!in!both!the!cases!would!be!same!as!the! absolute! value! of! the! temperature! rise.! Eq.! (3.44)! suggests! that! the! sum! of! scaled! sensitivity!coefficients!is!not!equal!to!zero!and!does!not!satisfy!Eq.!(2.3).!Hence,!in! this!case,!it!might!be!possible!to!estimate!parameters!k1,!k2,!C1!and!C2!uniquely!and! simultaneously.!! The! following! values! are! considered! as! an! example! for! this! case.! Dimensionless!scaled!sensitivity!coefficients!are!calculated!for!k1!=!0.5!W/mK,!k2!=! 0.55! W/mK,! C1! =! 3.5x106! J/m3K,! C2! =! 3.9x106! J/m3K,! T0! =! 20oC,! R! =! 5! mm,! 66 Δr / R = 0.02 .!!The!heat!flux!is!2.4!x104!W/m2.!The!dimensionless!scaled!sensitivity! coefficients!for!temperatureJdependent!properties!are!plotted!in!Figure!3.4,!for!T1!=! 25oC!and!T2!=!130oC.!The!order!of!the!magnitude!of!sensitivity!coefficients!is!k1,!C1,! k2! and! C2,! hence! k1! will! have! the! lowest! relative! error! of! all! the! estimated! parameters.!! ! 0.05 ˜ Xk1 ˜ Xk2 ˜ X C1 ˜ X C2 ˜ X T 1 −T 0 ˜ X T 2 −T 0 ˜ SSC X 0 −0.05 −0.1 −0.15 −0.2 0 0.2 0.4 ˜ t 0.6 0.8 1 Figure! 3.4! Dimensionless! scaled! sensitivity! coefficient! for! the! temperatureJ dependent!parameters!of!heat!transfer!problems!in!case!2!(R22B10T0).'T1!=!25oC! and!T2!=!130oC,!q!=!2.4!x104!W/m2! ! ! 67 Table!3.3!Intrinsic!Sum!and!dimensionless!scaled!sensitivity!coefficients!for!case!2:! R22B10T1!   X t !!!!! X k1 ! !!!!! k2 ! !! ! 0.02! J0.0752! J0.0085! 0.13! J0.1361! J0.0449! 0.24! J0.1537! J0.0733! 0.35! J0.1511! J0.0975! 0.45! J0.1378! J0.1191! 0.56! J0.1201! J0.1391! !  X !!! C2 ! J0.0619! J0.0969! J0.1121! J0.1337! J0.1600! J0.1866!    X X X !!!! C2 !!!!!!!!! (T1−T0 ) !! (T2 −T0 ) ! I S J0.0031! 0.0007! 0.0012! J0.0152! 0.0012! 0.0059! J0.0232! 0.0013! 0.0095! J0.0317! 0.0014! 0.0127! J0.0433! 0.0015! 0.0160! J0.0584! 0.0016! 0.0195! × 10−7 ! J0.9! J0.9! J1.3! J1.9! J2.3! J2.6! One! important! difference! between! case! 1! and! case! 2! is! that! k1! has! a! larger! absolute!scaled!sensitivity!coefficient!in!case!2.!Hence,!if!the!experimental!objective! is! to! estimate! thermal! conductivity,! then! a! cylindrical! geometry! would! provide! a! better!estimate!than!a!plate!geometry.!Intrinsic!Sum!is!plotted!in!Figure!3.5.!The!IS!is! in!the!magnitude!of!10J7,!which!is!very!small.!This!low!value!of!IS!suggests!that!the! numerical!code!is!sufficiently!accurate.!If!an!inverse!problem!is!performed!using!this! code,! the! results! might! be! better! for! the! cylindrical! geometry! than! for! the! plate! geometry.!The!results!are!tabulated!in!Table!3.3.! ! 68 −7 Intrinsic Sum ( I S ) 0 x 10 −0.5 −1 −1.5 −2 −2.5 −3 0 0.1 0.2 0.3 ˜ t 0.4 0.5 0.6 Figure!3.5!Intrinsic!sum!for!the!heat!transfer!problems!in!case!2!(R22B10T0).! ! The! dimensionless! scaled! sensitivity! coefficients! for! parameters! at! higher! temperature!(k2!and!C2)!are!lower!in!magnitude!than!the!ones!at!lower!temperature! (k1! and! C1),! Figure! 3.4.! In! equipment! design,! this! insight! will! be! helpful.! For! example,! if! the! heat! flux! in! case! 2! is! increased! to! 3.8x104! W/m2! as! compared! to! 2.4x104! W/m2,! the! resulting! dimensionless! scaled! sensitivity! coefficients! are!   plotted! in! Figure! 3.6.! It! should! be! noted! that! now! X k ! is! larger! than! X k for! 2 1 dimensionless!time!>0.21.!So,!increasing!the!heat!flux!has!a!positive!influence!on!the! 69 estimation!of!thermal!conductivity!at!higher!temperature.!The!numerical!values!are! presented!in!Table!!!!3.4.!! 0.05 ˜ Xk1 ˜ Xk2 ˜ X C1 ˜ X C2 ˜ X T 1 −T 0 ˜ X T 2 −T 0 ˜ SSC X 0 −0.05 −0.1 −0.15 −0.2 0 0.2 0.4 ˜ t 0.6 0.8 1 Figure! 3.6! Dimensionless! scaled! sensitivity! coefficient! for! the! temperatureJ dependent!parameters!of!heat!transfer!problems!in!case!2!(R22B10T0).'T1!=!25°C! and!T2!=!130°C,!q!=!3.8x104!W/m2! ! !! ! 70 Table!3.4!Intrinsic!Sum!and!dimensionless!scaled!sensitivity!coefficients!for!case!2:! R22B10T1!with!increased!heat!flux!       −7 X X X X X t !!!!! X k1 ! !!!!! k2 ! !!! C2 ! !!!! C2 !!!!!!!! (T1−T0 ) ! (T2 −T0 ) ! I S × 10 ! !! ! 0.02! %0.0670! %0.0156! %0.0575! %0.0068! 0.0006! 0.0022! %0.9! 0.13! %0.1040! %0.0722! %0.0828! %0.0262! 0.0009! 0.0097! %1.4! 0.24! %0.1051! %0.1147! %0.0926! %0.0384! 0.0010! 0.0151! %2.9! 0.35! %0.0901! %0.1498! %0.1080! %0.0516! 0.0010! 0.0198! %4.8! 0.45! %0.0668! %0.1802! %0.1261! %0.0692! 0.0010! 0.0246! %6.7! 0.56! %0.0409! %0.2075! %0.1425! %0.0919! 0.0010! 0.0296! %9.0! ' 3.4'Conclusions' ! Dimensionless! derivation! of! sensitivity! coefficients! has! been! presented! for! temperatureJdependent! thermal! properties.! An! important! aspect! of! the!Intrinsic!Sum,!numerical!code!verification,!is!demonstrated!with!examples!from! transient! conduction! heat! transfer.! The! Intrinsic! Sum! relation! can! also! be! used! to! identify!if!all!the!parameters!can!be!estimated.!The!relative!error!of!the!parameters! can! also! be! assessed! as! it! depends! on! the! magnitude! of! the! scaled! sensitivity! coefficients.!The!larger!the!scaled!sensitivity!coefficient!of!a!parameter!as!compared! to! the! maximum! temperature! rise,! the! lower! the! standard! error! will! be.! Methodologies! presented! in! this! article! provide! great! insight! into! the! inverse! problems!and!parameter!estimation.!Dimensionless!derivation!of!scaled!sensitivity! coefficients!can!be!conveniently!performed!for!various!problems!and!implemented! to!check!the!large!numerical!codes.!! ! ! 71 REFERENCES 72 REFERENCES' Beck! JV.! 1964.! The! Optimum! Analytical! Design! of! Transient! Experiments! for! Simultaneous! Determinations! of! Thermal! Conductivity! and! Specific! Heat.! East!Lansing,!MI:!Michigan!State!University.! Beck! JV,! Arnold! KJ.! 1977c.! Parameter! Estimation! in! Engineering! and! Science. New! York:!Wiley.! Beck!JV,!Osman!AM.!1990.!Sequential!estimation!of!temperatureJdependent!thermal! properties.!High!TemperaturesJJHigh!Pressures(UK)!23(3):255J66.! Bennie! F.! Blackwell! KJDRJC.! 1999.! Development! and! implementation! of! sensitivity! coefficient!equations!for!heat!conduction!problems.!Numerical!Heat!Transfer,! Part!B:!Fundamentals!36(1):15J32.! Chang!KC,!Payne!UJ.!1990.!Analytical!and!numerical!approaches!for!heat!conduction! in! composite! materials.! Mathematical! and! Computer! Modelling! 14(0):899J 904.! Chen! HJT,! Lin! JJY,! Wu! CJH,! Huang! CJH.! 1996.! Numerical! algorithm! for! estimating! temperatureJdependent!thermal!conductivity.!Numerical!Heat!Transfer,!Part! B:!Fundamentals!29(4):509J22.! COMSOL.! 2012.! COMSOL! Multiphysics.' 42a! ed:! COMSOL! Inc.,! Burlington,! Massachusetts,!United!States.! Cui! M,! Gao! X,! Zhang! J.! 2012.! A! new! approach! for! the! estimation! of! temperatureJ dependent! thermal! properties! by! solving! transient! inverse! heat! conduction! problems.!International!Journal!of!Thermal!Sciences!58(0):113J9.! Dowding! K,! Blackwell! B.! 1999.! Design! of! experiments! to! estimate! temperature! dependent! thermal! properties.! 3rd! Int.! Conf.! on! Inverse! Problems! in! Engineering:!Theory!and!Practice.!p.!509J18.! Dowding! KJ,! Beck! JV,! Blackwell! BF.! 1998.! Estimating! temperatureJdependent! thermal! properties! of! carbonJcarbon! composite.! 7th! AIAAJASME! Joint! Thermophysics!and!Heat!Transfer!Conference,!AIAA!paper.!p.!98J2933.! 73 Dowding! KJ,! Beck! JV,! Blackwell! BF.! 1999.! Estimating! TemperatureJDependent! Thermal!Properties.!Journal!of!Thermophysics!and!Heat!Transfer!13(3):328J 36.! Emery! AF,! Fadale! TD.! 1997.! Handling! temperature! dependent! properties! and! boundary! conditions! in! stochastic! finite! element! analysis.! Numerical! Heat! Transfer,!Part!A:!Applications!31(1):37J51.! Hays! DF,! Curd! HN.! 1968.! Heat! conduction! in! solids:! TemperatureJdependent! thermal! conductivity.! International! Journal! of! Heat! and! Mass! Transfer! 11(2):285J95.! Huang! CJH,! JanJYuan! Y.! 1995.! An! inverse! problem! in! simultaneously! measuring! temperatureJdependent! thermal! conductivity! and! heat! capacity.! International!Journal!of!Heat!and!Mass!Transfer!38(18):3433J41.! Huang! CH,! Ozisik! MN.! 1991.! Direct! integration! approach! for! simultaneously! estimating! temperature! dependent! thermal! conductivity! and! heat! capacity.! Numerical!Heat!Transfer,!Part!A:!Applications!20(1):95J110.! Imani! A,! Ranjbar! AA,! Esmkhani! M.! 2006.! Simultaneous! estimation! of! temperatureJ dependent!thermal!conductivity!and!heat!capacity!based!on!modified!genetic! algorithm.!Inverse!Problems!in!Science!and!Engineering!14(7):767J83.! Kevin! J.! Dowding! BFBRJC.! 1999.! Application! of! sensitivity! coefficients! for! heat! conduction! problems.! Numerical! Heat! Transfer,! Part! B:! Fundamentals! 36(1):33J55.! Kim! S.! 2001.! A! simple! direct! estimation! of! temperatureJdependent! thermal! conductivity!with!kirchhoff!transformation.!International!Communications!in! Heat!and!Mass!Transfer!28(4):537J44.! Kim! S,! Chung! BJJ,! Kim! MC,! Kim! KY.! 2003a.! Inverse! estimation! of! temperatureJ dependent!thermal!conductivity!and!heat!capacity!per!unit!volume!with!the! direct! integration! approach.! Numerical! Heat! Transfer,! Part! A:! Applications! 44(5):521J35.! Kim!S,!Kim!MC,!Kim!KY.!2003b.!NonJiterative!estimation!of!temperatureJdependent! thermal! conductivity! without! internal! measurements.! International! Journal! of!Heat!and!Mass!Transfer!46(10):1801J10.! 74 Lesnic! D,! Elliott! L,! Ingham! DB.! 1995.! A! note! on! the! determination! of! the! thermal! properties! of! a! material! in! a! transient! nonlinear! heat! conduction! problem.! International!Communications!in!Heat!and!Mass!Transfer!22(4):475J82.! Mierzwiczak! M,! Kołodziej! JA.! 2011.! The! determination! temperatureJdependent! thermal! conductivity! as! inverse! steady! heat! conduction! problem.! International!Journal!of!Heat!and!Mass!Transfer!54(4):790J6.! Salari!K,!Knupp!P.!2000.!Code!Verification!by!the!Method!of!Manufactured!Solutions.! Other!Information:!PBD:!1!Jun!2000.!p.!Medium:!P;!Size:!124!pages.! Scott! EP,! Beck! JV.! 1992.! Estimation! of! Thermal! Properties! in! Carbon/Epoxy! Composite!Materials!during!Curing.!p.!20J36.! Yang! CJy.! 1998.! A! linear! inverse! model! for! the! temperatureJdependent! thermal! conductivity! determination! in! oneJdimensional! problems.! Applied! Mathematical!Modelling!22(1‚Äì2):1J9.! Yang! CJy.! 1999.! Estimation! of! the! temperatureJdependent! thermal! conductivity! in! inverse! heat! conduction! problems.! Applied! Mathematical! Modelling! 23(6):469J78.! Yang! CJy.! 2000b.! Determination! of! the! temperature! dependent! thermophysical! properties! from! temperature! responses! measured! at! medium‚Äôs! boundaries.!International!Journal!of!Heat!and!Mass!Transfer!43(7):1261J70.! ! ! 75 Chapter'4 ' A'Novel'Instrument'for'Rapid'Estimation'of'TemperatureSDependent'Thermal' Properties'up'to'140oC' ! Abstract' ! Estimating! thermal! properties! for! thick! or! solid! foods! at! temperatures! greater!than!100oC!is!challenging!for!two!reasons:!the!long!time!needed!to!reach!a! constant! temperature,! and! the! pressure! needed! to! be! maintained! in! the! sealed! container.!!An!instrument!(TPCell)!was!developed!based!on!a!rapid!nonJisothermal! method! to! estimate! the! temperatureJdependent! thermal! properties! within! a! range! of!commercial!food!processes!(20!–!140! oC).!The!instrument!was!developed!based! on! simulation! and! insight! from! the! scaled! sensitivity! coefficients.! The! instrument! design!consists!of!a!custom!sample!holder!and!special!fittings!to!accommodate!the! heater! within! a! pressurized! environment.! The! instrument! is! kept! under! pressure! during! the! test.! The! total! time! of! the! experiment! is! less! than! 1min.,! compared! to! existing!isothermal!instruments!requiring!5J6!hours!to!cover!a!similar!temperature! range.!The!sequential!estimation!procedure!is!used!to!estimate!the!parameters!from! a! dynamic! experiment.! Glycerin! is! used! to! calibrate! the! sensor.! Thermal! conductivities!of!different!food!materials!were!estimated!for!the!temperature!range! of! commercial! food! processes.! ! The! novelty! of! the! instrument! lies! in! its! ability! to! analyze!transient!temperature!data!using!a!nonlinear!form!of!the!twoJdimensional! heat!conduction!equations.!!! 76 4.1'Introduction' ! Thermophysical!properties,!especially!thermal!conductivity!and!specific!heat,! are!very!important!in!designing!and!developing!processes!such!as!heat!exchangers,! aseptic!processing!systems,!etc.!Thermal!properties!are!also!critical!in!determining! scheduled! thermal! processes! for! a! specific! product.! Modeling! kinetics! of! thermal! degradation! of! nutrients! and! thermal! inactivation! of! microorganisms! requires! reliable! estimates! of! the! thermal! properties! of! foods.! ! Mathematical! modeling! is! used! for! new! and! novel! processes! to! design! and! optimize! food! quality.! However,! input! of! thermophysical! properties! to! these! models! is! often! a! limiting! step.! For! example,! maximizing! quality! and! ensuring! safety! of! solid! or! thick! foods! requires! tracking!the!food!temperature!during!the!process.!!Thermal!properties!are!needed! to!predict!the!food!temperature.!!The!“isothermal”!(0.5J2oC!temperature!rise)!lineJ source! method! has! been! commonly! used,! because! it! is! fast! at! lower! temperatures.! Yet!determining!thermal!properties!at!higher!temperatures!(>!100oC)!is!challenging,! because! by! the! time! the! entire! food! sample! in! the! container! reaches! a! constant! temperature,!the!quality!is!grossly!degraded.!It!is!at!higher!temperatures!where!rate! of!quality!degradation!and!microbial!inactivation!increases!very!rapidly.!!Therefore,! accurate! thermal! properties! are! critical! for! process! design! of! foods,! as! well! as! for! other!materials,!such!as!biomass,!foams,!pastes,!and!thick!slurries.! ! The! most! common! method! to! estimate! thermal! properties! is! the! hotJwire! probe.!HeatJflux!boundary!conditions!or!volumetric!generation!in!the!heatJtransfer! 77 partial! differential! equation! allows! for! simultaneous! estimation! of! thermal! conductivity! and! volumetric! heat! capacity! (Beck! and! Arnold! 1977a).! A! heat! pulse! method! can! be! used! to! estimate! thermal! properties! (Bristow! and! others! 1994b;! Bristow! and! others! 1994a).! Nahor! and! others! (2001)! performed! temperatureJ independent! simultaneous! estimation! of! thermal! conductivity! and! volumetric! heat! capacity! at! room! temperature,! and! an! optimal! design! of! a! heatJgeneration! profile! was!presented!to!estimate!the!parameters.!The!optimal!design!for!the!placement!of! the! sensor! was! also! studied! for! the! estimation! of! thermal! parameters! (Nahor! and! others!2001).!In!another!study,!a!hotJwire!probe!method!using!nonlinear!regression! was! employed! for! the! simultaneous! estimation! of! thermal! conductivity! and! volumetric! specific! heat! (Scheerlinck! and! others! 2008).! Scheerlinck! et! al.! (2008)! also!studied!the!optimum!heatJgeneration!profiles!and!applied!a!global!optimization! technique!for!optimization!of!the!heating!profile!and!the!position!of!the!sensor.!One! important! issue! to! consider! with! the! hotJwire! probe! method! is! the! design! of! the! probe!and!sources!of!error.!Design!of!a!thermal!conductivity!probe!was!considered! and!the!possible!sources!of!error!were!analyzed!for!the!construction!of!such!a!probe! (Murakami! and! others! 1996).! Carefully! designed! probes! have! higher! accuracy! in! thermal!parameter!estimation.! ! The! thermal! conductivity! and! specific! heat! of! carrots! at! elevated! temperatures!were!estimated!by!linear!regression!using!the!line!heatJsource!probe! by! performing! several! experiments! at! a! predetermined! initial! temperature! of! the! food!material!(Gratzek!and!Toledo!1993).!The!transient!line!heatJsource!technique! 78 was!used!to!estimate!thermal!conductivity!of!potato!granules!and!maize!grits!over!a! temperature! range! of! 30J120oC! (Halliday! and! others! 1995).! The! line! heatJsource! probe! method! was! also! used! to! estimate! thermal! conductivity! of! food! in! a! highJ pressure! (up! to! 400! MPa)! system! (Denys! and! Hendrickx! 1999).! Thermal! conductivity! of! food! material! was! estimated! under! heated! and! pressurized! conditions!using!the!transient!hotJwire!method!(ShariatyJNiassar!and!others!2000).! In!their!study,!thermal!conductivity!of!gelatinized!potato!starch!was!determined!at! 25J80oC,! 50%J80%! moisture,! and! 0.2J10! MPa.! They! also! found! that! the! thermal! conductivity!of!starch!gel!increases!with!temperature!and!moisture!content!up!to!1! MPa!pressure.!A!dualJneedle!probe!was!used!to!estimate!thermal!properties!under! highJpressure!processing!conditions,!and!(Zhu!and!others!2007)!found!that!thermal! conductivity! and! thermal! diffusivity! increased! with! increasing! pressure.! TemperatureJdependent! thermal! conductivity! was! estimated! using! nonlinear! estimation!(Yang!1999).!The!temperature!used!in!the!experiment!was!in!the!range! of!0J30oC.!Simultaneous!estimation!of!temperatureJdependent!thermal!conductivity! and! specific! heat! was! performed! using! a! nonlinear! method! and! nonisothermal! experiments! (Yang! 2000a).! Estimation! of! temperatureJdependent! specific! heat! capacity! of! food! material! by! the! oneJdimensional! inverse! problem! was! solved! for! the!thawing!of!fish!(J40!to!5oC!)!(Zueco!and!others!2004).!! ! ! ! The! inverse! method! is! a! useful! tool! for! parameter! estimation! (Beck! and! Arnold!1977a).!The!inverse!method!was!used!to!estimate!the!thermal!conductivity! 79 of!carrot!puree!during!freezing!(Mariani!and!others!2009)!and!thermal!diffusivity!of! various! foods! (Betta! and! others! 2009;! Mohamed! 2010;! Mishra! and! others! 2008;! Mishra! and! others! 2011).! A! temperatureJdependent! estimation! of! thermal! conductivity! was! done! using! a! polynomial! model! for! sandwich! bread! using! the! cooling!curve!(Monteau!2008).!! ! One!of!the!drawbacks!of!these!methods!is!that!one!has!to!wait!a!long!time!(45! min! minimum)! from! one! temperature! level! to! another! temperature! level,! as! the! probe! and! food! material! must! be! in! equilibrium! to! start! the! experiment.! The! experiment! can! only! be! performed! once! the! heat! source! and! the! sample! are! in! thermal! equilibrium.! Hence,! performing! tests! at! five! or! six! different! temperature! levels! makes! the! experimental! time! unacceptably! long! (5J6! h),! and! unwanted! changes! in! material! properties! occur! because! of! long! durations! at! higher! temperature.!! ! In! the! literature! reviewed,! there! was! no! standard! method! to! estimate! thermal!properties!of!conductionJheated!materials!rapidly!over!a!large!temperature! range! in! one! experiment.! There! is! a! lack! of! research! on! rapid! estimation! of! temperatureJdependent! thermal! properties! covering! the! entire! relevant! food! processing! temperature! range! (25! –! 140oC)! using! a! single! experiment.!! Development!of!a!device!would!be!of!great!use!to!a!variety!of!industries,!such!as!the! food,!pharmaceutical,!and!chemical!industries.!Therefore,!the!objective!of!this!study! was! to! devise! an! inverse! method! and! construct! a! device! to! accurately! estimate! 80 temperatureJdependent!thermal!properties!from!20!to!140oC!using!nonJisothermal! heating.! 4.2'Methodology' 4.2.1'Mathematical'model'and'numerical'code'verification' The!transient!heat!conduction!equation!in!a!hollow!cylinder!for!temperatureJ variable! thermal! conductivity! and! volumetric! heat! capacity! (caused! by! changes! in! the!specific!heat)!with!the!heater!at!the!center!can!be!given!as! ! 1 ∂ ⎡ ∂T ⎤ ∂ ⎡ ∂T ⎤ ∂T ⎢ khr ∂r ⎥ + ∂z ⎢ kh ∂z ⎥ + g0 f (t) = Ch ∂t for R0 < r < R1,0 < z < Z, t > 0 r ∂r ⎣ ⎦ ⎣ ⎦ !(4.1)! 1 ∂ ⎡ ∂T ⎤ ∂ ⎡ ∂T ⎤ ∂T ⎢ k1 f k r ∂r ⎥ + ∂z ⎢ k1 f k (T , K ) ∂z ⎥ = C1 fC ∂t for R1 < r < R2 ,0 < z < Z, t > 0 r ∂r ⎣ ⎦ ⎣ ⎦ ! ! ! This! problem! was! solved! numerically! with! finite! element! software! (COMSOL®,! (COMSOL! 2012)).! Numerical! codes! are! very! important! in! providing! a! solution! to! partial! differential! equations! in! many! areas! of! study,! such! as! the! heat! transfer!problem.!However,!verification!of!these!codes!is!critical!(Salari!and!Knupp! 2000;!Roy!2005).!!In!this!section,!an!intrinsic!verification!method!to!the!numerical! solution! of! the! partial! differential! equation! is! presented.! Derivation! of! the! dimensionless!form!of!scaled!sensitivity!coefficients!is!presented.!!The!sum!of!scaled! sensitivity! coefficients! is! used! in! the! dimensionless! form! to! provide! a! method! for! verification.! 81 ! The! Intrinsic! Verification! Method! (IVM)! is! based! on! the! scaled! sensitivity! coefficients!and!can!be!used!in!verification!of!numerical!codes!as!well!as!to!perform! parameter! estimation.! The! sensitivity! coefficient! of! a! parameter! is! the! first! partial! derivative! of! the! function! involving! the! parameter,! with! respect! to! the! parameter! (Beck!1970).!Consider!a!simple!function;! ! T = f (k,C, x,t) ! ! (4.2)! ! Where!k!and!C!are!parameters!of!the!function!T.!The!sensitivity!coefficient!of!k!and!C' are ∂T ∂T ! and! ,! respectively.! After! multiplying! the! sensitivity! coefficient! by! the! ∂C ! ∂k ∂T ˆ parameter,! we! get! the! scaled! sensitivity! coefficient! represented! by! X k = k and! ∂k ! ∂T ˆ !for!k!and!C,!respectively.! XC = C ∂C For! the! numerical! code! verification,! the! heat! conduction! problem! is! made! dimensionless.!Let,!  k  C k = 2 , C = 2 !! k1 C1 ! (4.3)! Linear!functions!of!temperature!for!the!thermal!conductivity!and!volumetric!heat! capacity!are!considered;!they!are! !  f k (T − T0 , k ) = 1+ (T − T0 ) − (T1 − T0 ) ( k − 1) !! (T2 − T0 ) − (T1 − T0 ) (4.4)! !  fC (T − T0 , C) = 1+ (T − T0 ) − (T1 − T0 ) (C − 1) !!  T2 − T0 ) − (T1 − T0 ) ( (4.5)! 82 ! which!gives!the!thermal!conductivity!value!of!k1!at!T1!and!k2!at!T2.!The!temperature! dependence!causes!Eq.!(4.1)!to!be!nonlinear!in!T.!!The!boundary!conditions!are! ! ∂T (R , z,t) = 0 ∂r 2 ∂T (r,0,t) = 0 ∂z ∂T (r,Z,t) = 0 !! ∂z (4.6)! ! The!initial!condition!is! ! T (r, z,0) = T0 !! (4.7)! ! k  C  kh ≡ h , Ch ≡ h !! k1 C1 (4.8)! Also,! ! The!above!problem!is!now!put!in!a!dimensionless!form.!Let! !  ! r≡ r  ,R ≡ R1 2 R2 kt z   k  C   , z ≡ , t ≡ 1 , k ≡ 2 , C ≡ 2 , kh ≡ R1 R1 k1 C1 C R2 1 1 kh  C , Ch ≡ h !! k1 C1 (4.9)! and! !  T − T0 , T ≡ T1 − T0 , T ≡ T2 − T0 !!   T≡ 2 2 1 g R2 2 g0 R1 g0 R1 0 1 k1 k1 k1 (4.10)! ! With! this! model! and! with! a! known! volumetric! heat! generation,! the! dimensionless!temperature!is!given!symbolically!by! ! !             T = T ( r, R2 , z , t , k, C, kh , Ch , T1, T2 ) !! 83 (4.11)! ! Now!the!partial!derivatives!of!temperature!T!are!found.!Notice!that! ! ! T − T0 = 2   ⎞     gR1  ⎛ r, R2 , z , t (k1,C1), k(k1,k2 ), C(C1,C2 ),... T⎜ ⎟ !! (4.12)!   k1 ⎝ k (k ,k ), C (C ,C ),T (k ,T − T ),T (k ,T − T )⎠ h h 1 h h 1 1 1 1 0 2 1 2 0 The!scaled!sensitivity!coefficient!with!respect!to!k1!is! ! !      g R 2  g R 2 ⎡ ∂T  ∂T  ∂T ∂T ∂T ⎤ ˆ  X k = − 0 1 T + 0 1 ⎢t − k  − kh  + T1 + T2 ⎥ !!  1 k1 k1 ⎢ ∂t ∂T1 ∂T2 ⎥ ∂k ∂ kh ⎣ ⎦ (4.13)! ! Repeat!for!the!derivative!with!respect!to!k2!and!kh!to!get! ! ! 2   ˆ = g0 R1 ⎡ k ∂T ⎤ !! Xk  2 k1 ⎢ ∂ k ⎥ ⎣ ⎦ (4.14)! !  g R 2 ⎡  ∂T ⎤ ˆ X k = 0 1 ⎢ kh  ⎥ !! h k1 ⎢ ∂ kh ⎥ ⎣ ⎦ (4.15)! ! The!scaled!sensitivity!coefficients!for!C1!and!C2!and!Ch!are! ! ! 2⎡    ⎤   ˆ = g0 R1 ⎢ −t ∂T − C ∂T − C ∂T ⎥ !!  XC h ∂C    1 k1 ⎣ ∂t ∂C h⎦ (4.16)! ! 2   ˆ = g0 R1 ⎡C ∂T ⎤ !! XC  2 k1 ⎢ ∂C ⎥ ⎣ ⎦ (4.17)! ! 2  g0 R1 ⎡  ∂T ⎤ ˆ XC = ⎢Ch  ⎥ !! h k1 ⎣ ∂Ch ⎦ (4.18)! ! 84 Next!the!derivative!with!respect!to!T1FT0!and!T2FT0!is!found;!it!is! ! !  g R 2  ∂T ˆ X (T −T ) = 0 1 T1  !! 1 0 k1 ∂T1 (4.19)! !  g R 2  ∂T ˆ X (T −T ) = 0 1 T2  !! 2 0 k1 ∂T2 (4.20)! Adding! Eqs.! (4.13)J(4.18),! while! subtracting! Eqs.! (4.19)! and! (4.20)! gives! the! Intrinsic!Sum,!IS:! ( ) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ X k + X k + X k + X C + X C + X C − X (T −T ) − X (T −T ) + T − T0 = 0 !(4.21)! 1 2 h 1 2 h 1 0 2 0 ! ! ! The! first! six! scaled! sensitivity! coefficients! in! Eq.! (4.21)! are! for! fixed! x! and! t' values.!The!final!two!derivatives!are!the!rate!of!change!in!the!computed!temperature! when!the!specified!temperature!T1!or!T2!is!changed.!As!a!general!rule,!if!the!sum!of! scaled!sensitivity!coefficients!is!equal!to!zero!then!all!the!parameters!in!the!model! cannot!be!estimated!uniquely!and!simultaneously.!Equation!(4.21)!indicates!that!the! four!parameters! k1,k2 ,C1,C2 !may!be!simultaneously!estimated!when!temperatures! are! measured! and! the! volumetric! heat! generation! is! prescribed,! because! sum! of! scaled!sensitivity!coefficients!is!not!zero.!! ! A!simulation!of!the!instrument!was!performed!using!the!finite!element!code! COMSOL®.! The! simulation! was! performed! with! assumed! values! of! temperatureJ dependent!thermal!properties!for!biological!materials.!For!simulation!purposes,!the! assumed! values! of! parameter! were:! k1! =! 0.5! W/mK,! k2! =! 0.6! W/mK,' C1! =! 3.5x106! 85 J/m3K,!C2!=!3.9!x106!J/m3K,!kh!=!3!W/mK,!Ch!=!720!J/m3K,!T0!=!20oC,!T1!=!25oC!and! T2!=!140°C.!The!heater!power!was!5.63!x107!W/m3,!which!is!equivalent!to!a!total! heater!power!of!24!W.!Figure!4.1!represents!the!simulated!profile!of!temperature!in! the!instrument.!For!the!simulation,!the!heater!element!was!powered!for!26!seconds.! The! power! supply! was! automatically! cut! off! once! the! heater! temperature! reached! 140oC.! The! scaled! sensitivity! coefficients! are! plotted! in! Figure! 4.2.! The! scaled! sensitivity! coefficient! of! k1! is! larger! for! time! <23! sec,! and! for! time! >23! sec! k2! is! larger.!!This!information!is!of!great!importance!to!design!the!instrument!for!thermal! properties!measurement.!If!we!wish!to!estimate!k2!accurately,!it!is!important!for!the! experimental!time!for!be!>23!sec,!for!a!given!heater!power!of!24!W.!Increasing!the! heater!power!will!increase!the!scaled!sensitivity!coefficient!of!k2;!however,!there!is! a!constraint!of!maximum!temperature!rise!of!the!product.!Based!on!the!absolute!size! of! the! scaled! sensitivity! coefficients! (Figure! 4.2)! the! order! of! accuracy! (most! to! least)!for!estimation!of!the!parameters!would!be!k2,'k1,!C1!and!C2.!Since!the!scaled! sensitivity! of! T1JT0! and! T2JT0! is! positive,! it! indicates! that! the! temperature! rise! is! decreased! by! the! T1JT0! and! T2JT0! sensitivity.! The! scaled! sensitivity! of! C2! is! relatively!small!compared!to!the!maximum!temperature!rise!and!hence!might!not!be! estimated!with!high!accuracy.!Hence,!even!though!all!the!sensitivity!coefficients!are! different! shapes! and! uncorrelated,! some! of! the! parameters! may! not! be! estimated! 86 well.!To!estimate!all!of!the!four!parameters!in!the!model,!another!thermocouple!may! be! needed! at! a! different! location! from! the! surface! of! the! heater.! Therefore,! it! is! important!to!analyze!the!sensitivity!coefficients!to!obtain!insight!into!the!estimation! problem.! The! IS! is! plotted! in! Figure! 4.3.! The! values! of! IS! is! on! the! order! of! 10J7,! which!indicates!that!the!numerical!code!is!accurate.! ! 87 Figure!4.1!Simulated!heating!profile!of!test!material! ! ! 88 10 ˆ Xk1 ˆ Xk2 ˆ XC 1 ˆ XC 2 ˆ Xkh ˆ XC h ˆ XT 1− T0 ˆ XT 2− T0 ˆ o SSC X, C 0 −10 −20 −30 −40 0 10 20 30 40 t(sec) 50 60 Figure!4.2!Plot!of!Scaled!sensitivity!coefficients!of!the!parameters!in!the!model!given! by!Eq.!(4.1),!using!simulated!temperature!data.! ! 89 −7 o Intrinsic Sum ( I S ), C 0 x 10 −1 −2 −3 −4 0 10 20 t(sec) 30 40 Figure!4.3!Plot!of!Intrinsic!Sum!as!given!by!Eq.!(4.21)! ! 4.2.2'Equipment'design' The! design! of! the! equipment! is! based! on! the! mathematical! model! given! by! Eq.! (4.1).! The! schematic! of! the! probe! design! and! the! container! is! shown! in! Figure! 4.1.! By! using! the! inverse! problem,! the! thermal! parameters! will! be! estimated! simultaneously!within!the!desired!temperature!range,!which!will!be!representative! of! current! processing! temperatures.! The! transient! heat! conduction! problem! with! the! temperatureJdependent! thermal! properties! is! given! by! Eq.! (4.1).! Thermal! conductivity! k(T)! and! specific! heat! capacity! C(T)! were! modeled! as! functions! of! temperature.! The! spatial! domain! (that! is,! the! size! of! the! container)! is! sufficiently! 90 large! so! that! the! outer! boundaries! do! not! affect! the! temperature! distribution! near! the!heater;!hence!the!domain!is!from!the!center!of!probe!to!a!large!radius.!(de!Monte! and!others!2008).!Notice!that!this!condition!includes!the!heat!capacity!of!the!probe,! which! is! not! always! done! but! can! be! treated! using! the! software.! A! sensitivity! analysis! is! done! to! evaluate! the! accuracy! of! the! parameters.! This! method! of! estimation!of!the!thermal!parameters!is!approximately!1!min!or!less!as!opposed!to! 5J6!hours!required!in!the!isothermal!method!of!measurement!to!cover!the!desired! temperature!range.!! There!are!several!components!of!the!instrument.!The!equipment!consists!of! the!following!units:! 1. NI! 9225! 3JCh! +/J300V! Analog! Input! (National! Instruments,! Austin,! Texas):! This! unit! has! been! added! to! the! instrument! to! measure! the! voltage!going!through!the!heater.!It!has!a!full!measurement!range!of!300! Vrms.! 2. NI!9227!4!ch!current!input,!5Amp,!ISO,!50k,!24bit!(National!Instruments,! Austin,! Texas):! This! current! measuring! unit! is! used! in! conjunction! with! the! voltage! measuring! device! to! measure! the! power! and! energy! consumption! for! the! heater! application.! This! module! is! designed! to! measure! 5! Amps! nominal! and! up! to! 14! A! peak.! It! can! sample! at! 50kS/s! per!channel.! 3. NI!9481!4JCh!30!VDC!(2!A),!60!VDC!(1!A),!250!VAC!(2A)!EM!Form!A!SPST! Relay!Module!(National!Instruments,!Austin,!Texas):!The!main!function!of! 91 this!relay!device!is!to!act!as!a!safety!switch.!This!is!programmed!to!switch! off! the! power! supply! to! the! heater! when! the! temperature! rise! in! the! instrument!is!greater!than!140oC.!! 4. cDAQJ9174,! CompactDAQ! chassis! (4! slot! USB)! (National! Instruments,! Austin,! Texas):! This! CompactDAQ! USB! chassis! is! designed! for! small,! portable,! mixedJmeasurement! test! systems.! The! voltage,! current,! relay! and! voltage! output! modules! are! mounted! on! this! CompactDAQ.! It! connects!with!the!computer!through!a!USB!connection!and!programmed! with!the!LabView!interface.!! 5. NI! 9263! 4JChannel,! 100! kS/s,! 16Jbit,! ±10! V,! Analog! Output! Module! (National!Instruments,!Austin,!Texas):!Analog!output!module!is!used!for! providing!specified!voltage!output!to!the!heater.!! 6. Phase! angle! controller,! FC11AL/2! (United! Automation):! Phase! angle! controller! is! used! to! adjust! the! incoming! full! voltage! (110! V)! from! the! main! power! supply.! The! output! signal! from! this! device! is! fed! to! the! voltage!output!device!(NI!9263).!! 7. Sample!Cup:!The!testing!sample!holder!is!made!from!a!316!stainless!steel! and! is! rated! to! hold! pressure! up! to100! psi.! The! tripod! type! legs! are! attached!to!the!bottom!part!of!the!cup!and!provide!the!full!stability!to!the! whole! equipment.! The! sample! holder! cylindrical! cup! radius! is! 23.5! mm,! with!a!length!of!124!mm.! 92 8. Lid:!The!closing!of!the!sample!cup!is!accomplished!from!the!top!lid!with!a! gasket!in!between.!The!lid!is!also!manufactured!from!316!stainless!steel! material.!There!are!three!NPT!(National!Pipe!Thread)!threaded!holes!on! the!lid.!The!first!NPT!is!at!the!center!of!the!lid!and!is!provided!to!insert! the! heater.! The! second! NPT! is! offJcenter! and! is! designed! to! provide! a! pressure! connection! from! a! pressurized! air! tank.! The! third! NPT! is! also! offJcenter! and! is! designed! for! the! pressure! relief! valve.! The! pressure! relief!valve!is!rate!to!50!psi.!! 9. Heating!element:!Total!output!power!of!the!heater!is!30!Watts.!Power!of! heater!is!calculated!based!on!the!desired!temperature!rise!in!the!product.! The! power! output! of! the! heater! is! programmable! with! a! combination! of! the! phase! angle! controller! and! the! voltage! output! module.! The! heater! specifications!are!provided!below:! a. High!Watt!Density:!33.4!w/in2! b. Diameter:!0.125",!High:!3.15mm!Low:!3.05mm!Swage!to!Size! c. Heater!Length:!7",!Tolerance:!±0.210"! d. Watts:!30,!±10%! e. Volts:!120! f. Unheated:!Leads:!4.5",!Cap:0.211"! g. Heated!Length:!2.29"! h. Lead!Length:!48",!±2.4"! i. Lead!Type:!Teflon!260C/500oF[Swaged]! 93 j. Tube!Material:!SS321! k. Potting:!None!(Teflon!plug)! 10. Pressure:!a!compressed!air!tank!supplies!Pressure!to!the!instrument.! 11. Software! Interface:! a! program! developed! using! LabVIEW! (National! Instruments)!software!controls!the!instrument.!! ! A! schematic! of! the! electronics! of! the! instrument! is! presented! in! Figure! 4.4.! The! instrumentation! has! the! capability! to! generate! different! heating! profiles.! To! generate!different!heating!profiles,!a!phase!angle!controller!device!(FC11AL/2)!has! been! added! to! the! circuit.! This! voltage! module! has! different! settings! that! can! be! adjusted! with! the! program.! With! the! settings! from! the! voltage! module,! a! voltage! output!module!(NI!9263))!sends!the!information!to!the!heater.!This!setting!allows! better! control! of! the! heating! rate! in! the! sample.! The! LabVIEW®! program! includes! the! settings! for! the! heating! profile.! The! voltage! and! ampere! loggers! record! the! power! generated! in! the! heater.! A! relay! unit! is! programmed! so! that! if! the! temperature!rise!of!the!heater!is!more!than!a!set!value,!then!the!relay!will!turn!off! the!device.!! ! 94 To#Heater Fuse N V+ G Relay Amp$Meter Volt%Setting G Volt%Meter ! L N Voltage( Control Figure! 4.4! Schematic! of! the! electronics! of! the! temperatureJdependent! thermal! property!measurement!instrument! ! A! custom! stainless! steel! cup! was! manufactured! to! hold! the! sample! and! the! heater.! This! vessel! is! pressurized! to! allow! a! maximum! temperature! of! 140oC.! The! pressure!fittings!accommodate!the!heater!and!thermocouples.!The!pressurized!air!is! connected!from!the!pressurized!tank!to!the!vessel!with!a!threeJway!valve.!A!safety! valve!is!installed!on!the!vessel!to!limit!the!maximum!pressure!rise.!This!safety!valve! is!set!to!a!maximum!pressure!rating!of!50!psi.!The!pressure!rating!of!the!vessel!itself! is! greater! than! 100! psi.! The! complete! setJup! of! the! instrument! is! represented! in! Figure!4.4.!The!output!power!is!recorded!every!0.2!sec!along!with!the!temperature.!! The!complete!set!up!of!the!instrument!is!represented!in!Figure!4.5.! 95 ! Figure! 4.5! Thermal! property! measurement! instrument! with! the! data! acquisition! devices! 4.2.3'Temperature'calibration'and'sample'testing' ! Temperature! measurement! using! a! thermocouple! was! found! to! be! inaccurate,!due!to!conduction!from!the!heater!to!the!thermocouple.!!Therefore,!the! heating! element! of! the! heater! was! used! as! a! temperatureJindicating! device.! The! resistance!of!the!heating!element!is!dependent!on!its!temperature.!This!information! was! used! to! calibrate! the! resistance! of! the! heating! element.! The! calibration! was! 96 performed!using!a!silicon!oil!bath!set!at!predetermined!temperatures.!A!resistance! meter! (Fluke! 741B)! was! used! to! measure! the! resistance! of! heater! at! set! temperatures.! Several! food! materials! were! tested! to! determine! temperature! dependent!thermal!conductivity,!using!the!instrument.!The!foods!tested!were!carrot! puree,!banana!puree,!banana!oat!puree,!and!sweet!potato!puree.! ! 4.2.4'Sequential'estimation'of'parameters' ! The! sequential! method! of! estimation! updates! parameters! as! new! observations! are! added.! Sequential! estimation! of! parameters! in! a! model! provides! good! insight! into! building! the! model! and! determining! uncertainty! in! parameters.! For!example,!if!parameters!come!to!a!constant!value!after!a!certain!reasonable!time! then!the!experiment!can!be!stopped,!as!further!data!will!not!improve!the!parameter! estimate.! Prior! information! of! parameters! can! be! used! in! sequential! estimation! towards! estimation! of! parameters! for! a! particular! experiment.! The! advantage! of! sequential!over!OLS!is!that!more!insight!is!given!in!the!estimation!process,!because! parameters!are!updated!with!the!addition!of!each!datum.!!The!quality!of!the!model! for! a! given! data! set! is! judged! by! how! well! each! parameter! approaches! a! constant! before!the!end!of!the!experiment.! ! There!were!only!two!studies!found!in!the!food!literature!where!simultaneous! sequential!estimation!based!on!Gauss!minimization!was!used,!one!for!estimation!of! both! thermal! conductivity! and! volumetric! specific! heat! (Mohamed! 2009),! and! one! for!estimation!of!thermal!diffusivity!(Mohamed!2010).!!The!sequential!procedure!in! this! work! was! developed! using! the! matrix! inversion! lemma! (Beck! and! Arnold! 97 1977b,! p.! 277)! based! on! the! Gauss! minimization! method,! requiring! prior! information.!The!mathematical!form!of!nonJlinear!sequential!estimation!is!derived! from! maximum! a! posteriori! (MAP)! estimation.! ! The! minimization! function! in! the! Gauss!method!can!be!expressed!as:! T T ˆ ˆ S = ⎡Y − Y ( β ) ⎤ W ⎡Y − Y ( β ) ⎤ + ⎡ µ − β ⎤ U ⎡ µ − β ⎤ ! ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ! (4.22)! ˆ Where,!Y!is!the!experimental!response!variable!and! Y is!predicted!response,! μ! is! prior! information! of! parameter! vector! β,! W! is! inverse! of! covariance! matrix! of! errors! and! U! is! inverse! of! covariance! matrix! of! parameters.! The! extremum! of! the! function!given!by!equation!(4.22)!can!be!evaluated!by!differentiating!it!with!respect! to!β.'The!expression!can!be!given!as:! T ˆ ˆ ∇ β S = −2 ⎡∇ β Y ( β ) ⎤ W ⎡Y − Y ( β ) ⎤ − 2 ⎡ I ⎤U ⎡ µ − β ⎤ ! ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ! (4.23)! Eq.!(4.23)!can!be!set!to!zero,!and!β'solved!for!implicitly.!!Standard!statistical! assumptions! that! allow! the! use! of! sequential! estimation! are:! additive! errors,! zero! mean,!uncorrelated!errors,!normally!distributed!errors,!covariance!matrix!of!errors! is! completely! known,! no! errors! in! independent! variables! and! subjective! prior! information!of!parameters!are!known.!One!method!to!do!this!is!an!iterative!scheme! (Beck!and!Arnold!1977,!p.!277):! ! ! A i+1= Pi X T ! i+1 98 (4.24)! ! Δ i+1 = φ i+1+ X i+1 A i+1 ! (4.25)! ! K i+1= A i+1 Δ −1 ! i+1 (4.26)! ! ˆ e i+1= Yi+1 − Yi+1 ! (4.27)!! ( )⎤⎥⎦ ! b * = b *+ K i+1 ⎡ e i+1− X i+1 b *− b i+1 i i ⎢ ⎣ ! Pi+1 = Pi − K i+1 X i+1 Pi ! ! (4.28)! !!(4.29)! ! Where!b*i+1!is!the!updated!parameter!(px1,!p!is!the!number!of!parameters)!vector! at! time! step! i+1;! ! b*i! is! the! parameter! vector! at! the! previous! time! step! i;! ! b! is! the! parameter! vector! at! the! previous! iteration;! P! is! the! covariance! vector! matrix! of! parameters! (p' x' p),! X! is! the! sensitivity! coefficient! matrix! (nxp),! and! e! is! the! error! vector.!!The!scheme!is!started!by!providing!parameter!estimates,'computing!X!and! the!error!vector!e,!and!assuming!a!matrix!P.!!Because!we!did!not!have!accurate!prior! information!on!the!covariance!matrix,!we!set!P!as!a!diagonal!matrix!of!105.!Matrix!P,' X!and!e!are!functions!of!b!and!not!of!b*.!The!stopping!criteria!for!b!can!be!given!as! ! ! bk+1 − bk j j bk + δ1 j ! 99 <δ ! (4.30)! Where,!j!is!the!index!for!the!number!of!parameters.!The!magnitude!of δ could!be!in! the! order! of! 10J4.! Another! small! number! is! δ1 which! is! very! small! such! as! 10J8,! to! avoid!the!problem!when! bk !tends!to!zero.!! j 4.3'Results' 4.3.1'Instrument'calibration' The! calibration! curve! for! the! heater! is! presented! in! Figure! 4.6.! The! temperature! is! plotted! on! the! yJaxis! and! resistance! on! the! xJaxis.! A! linear! relation! was!found!between!the!resistance!and!temperature!as!given!by!Eq.!(4.31):!! ! ! T = 26.118R − 12609 ! ! 100 (4.31)! 120 o Temp erature, C 140 100 80 60 40 20 483 Experimental Predicted 484 485 486 487 Resistance (Ohm) 488 489 Figure!4.6!Calibration!curve!of!resistance!and!temperature!of!the!heating!element! ! ! ! After! the! verification! of! finite! element! code,! the! resistance! (converted! to! temperature)!data!collected!from!the!instrument!was!analyzed!using!the!sequential! estimation!procedure!(Dolan!and!others!2012;!Beck!and!Arnold!1977a).!Sequential! estimation! is! based! on! gauss! minimization! method! and! needs! prior! information! regarding!the!parameters!that!needs!to!be!estimated.!Sequential!procedure!updates! the!parameters!as!each!observation!is!added!to!the!estimation!procedure.!Analysis! software! was! developed! using! the! principles! of! sequential! estimation.! Once! the! temperature!data!was!entered!in!the!program,!the!software!outputs!the!estimated! 101 parameters! with! statistical! indices.! There! are! four! main! output! plots! from! the! software,! 1)! predicted! and! observed! temperature,! 2)! residuals,! 3)! sequential! estimation,!and!4)!scaled!sensitivity.!! ! The! instrument! was! calibrated! with! 95%! glycerol.! Simulated! heating! of! glycerol!at!two!different!locations!away!from!the!surface!of!the!heater!is!shown!in! Figure! 4.7.! However,! for! estimation! of! thermal! conductivity,! the! measured! temperature! using! the! resistance! of! the! heater! was! used.! The! power! input! of! the! heater!was!24!Watts.!The!thermal!conductivities!were!estimated!at!initial!and!final! temperature! of! the! heating! range.! The! estimated! value! of! thermal! conductivity! of! glycerol!at!19oC! was!0.268!W/moC!and!at!118.5oC!was!0.362!W/moC.!The!thermal! conductivity!of!glycerol!at!32oC!was!0.283!W/moC,!which!is!in!the!range!of!reported! values!in!literature!(Zhu!and!others!2007).!!! ! ! 102 130 25 120 20 100 90 15 80 70 10 60 Power (watts ) o Temper atur e, C 110 50 5 40 TC 1 TC2 30 20 0 20 40 60 80 0 100 Time (s ec) Figure!4.7!Simulated!heating!curves!of!glycerol!at!two!different!locations!for!a!given! power!input! ! 4.3.2'Instrument'experimental'result' Figure! 4.8! shows! the! experimental! temperature! and! predicted! temperature! versus!time!for!the!sweet!potato!puree.!Note!that!the!experimental!duration!is!less! than!40!seconds.!For!the!estimation!of!thermal!conductivities,!the!values!of!specific! heat!must!be!known.!The!value!of!glycerol!volumetric!heat!capacity!was!used!from! literature.!However,!the!values!of!temperatureJdependent!volumetric!heat!capacity! of!banana!puree!and!banana!oat!were!measured!using!a!DSC!(Differential!Scanning! Calorimeter,!Q2000,!TA!Instruments,!New Castle, DE).!!! 103 ! ! 120 o Temp erature, C 140 100 80 60 40 20 0 Experimental Predicted 10 20 Time (sec) 30 40 Figure! 4.8! Experimental! and! predicted! heating! profile! of! sweet! potato! puree! for! a! given!power!input!of!24!W.!! ! 104 0.6 o T(pred) - T(exp), C 0.8 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 0 10 20 Time (sec) 30 40 Figure!4.9!Residuals!of!sweet!potato!puree!for!a!given!power!input!of!24!W.! ! ! The!residuals!for!the!estimation!process!of!sweet!potato!puree!are!presented! in! Figure! 4.9.! The! residual! plot! does! not! seem! to! violate! any! standard! statistical! assumptions! (additive,! zero! mean,! uncorrelated,! constant! variance! and! normal! distribution! of! residuals).! It! looks! normally! distributed! and! there! is! no! apparent! pattern!or!signature!in!the!residuals.!The!estimated!values!of!thermal!conductivities! and! statistical! indices! are! reported! in! Table! 4.2.! The! estimated! value! of! thermal! conductivity! at! 25oC! was! 0.539! W/moC.! The! 95%! asymptotic! confidence! intervals! are!also!reported!in!Table!4.2.!! ! 105 ! A! commercially! available! unit! (KD2! PRO,! Decagon! Devices)! for! thermal! conductivity!measurement!was!used!to!compare!the!values!for!sweet!potato!puree.! The! measured! value! of! k1! at! 25oC! using! KD2! PRO! was! 0.529! W/moC! and! 0.546! W/moC!for!two!different!tests.!This!is!in!agreement!with!the!TPCell!measurements,! which! was! 0.539! W/moC.! The! composition! of! sweet! potato! puree! is! provided! in! Table! 4.1! (U.S. Department of Agriculture 2013).! Thermal! conductivity! at! 25oC,! predicted! with! composition! model! of! Choi! and! Okos,! was! 0.535! W/moC.! Higher! temperature! experiments! (140oC)! were! not! possible! with! KD2! PRO! as! it! was! not! possible! to! keep! the! oil! bath! temperature! in! equilibrium! with! the! sample! temperature.! However,! the! repeated! use! of! the! KD2! PRO! led! to! the! failure! of! the! sensor!without!any!reasonable!measurement!of!thermal!conductivity.!! ! Table! 4.1! Composition! of! sweet! potato! puree! from! USDA! nutrient! database! (U.S.! Department!of!Agriculture!2013)! Components! Percentage! Protein! 1.1! Fat!! 0.1! Carb! 13.2! Fiber! 1.5! Ash! 0! Water! 84.8! Ice! 0.00! 106 ! ! Table 4.2 Estimated values of thermal conductivities and statistical indices for sweet potato puree Material! Temp,!oC! K,!!W/moC! 25! 0.518! Sweet! Potato!1! 140! 0.585! 25! 0.532! Sweet! Potato!2! 140! 0.575! 25! 0.548! Sweet! Potato!3! 140! 0.568! 25! 0.539! Sweet! Potato!4! 140! 0.572! 25! 0.533! Sweet! Potato!5! 140! 0.574! 25! 0.522! Sweet! Potato!6! 140! 0.585! Std!error! Lower!CI! Upper!CI! C,!J/m3°C)! 4278660! 0.0019! 0.515! 0.521! 4858670! 0.0024! 0.581! 0.589! 4278660! 0.0019! 0.529! 0.535! 4858670! 0.0024! 0.571! 0.579! 4278660! 0.0019! 0.545! 0.552! 4858670! 0.0024! 0.563! 0.573! 4278660! 0.0019! 0.536! 0.542! 4858670! 0.0025! 0.568! 0.577! 4278660! 0.0018! 0.530! 0.537! 4858670! 0.0024! 0.569! 0.578! 4278660! 0.0018! 0.519! 0.525! 4858670! 0.0024! 0.581! 0.589! ! ! Table! 4.3! Estimated! values! of! thermal! conductivities! and! statistical! indices! for! several!food!materials! Material! Temp,!oC! K,!!W/moC! 25! 0.499! Banana! 140! 0.573! 25! 0.487! Banana! Rasp!Oat! 140! 0.566! Std!error! Lower!CI! Upper!CI! C,!J/m3°C)! 0.0027! 0.494! 0.505! 4190510! 0.0048! 0.563! 0.583! 4547100! 0.0026! 0.482! 0.492! 4431940! 0.0047! 0.557! 0.574! 5069170! ! ! 107 10 ˆ Xk1 ˆ Xk1 o SSC, C 0 −10 −20 −30 −40 0 10 20 30 Time (sec) 40 Figure!4.10!Scaled!sensitivity!coefficients!for!thermal!conductivities!at!T1!and!T2! ! ! ˆ ˆ The!scaled!sensitivity!coefficients,! X k and! X k are!shown!in!Figure!4.10!for! 1 2 ˆ k1!and!k2,!respectively.!For!initial!times,!the!value!of! X k is!larger!suggesting!that!k1! 1 ˆ can! be! estimated! with! good! accuracy! at! earlier! times.! The! value! of! X k is! larger! 2 towards! the! end! of! the! experiment,! in! this! case! after! 34! seconds.! The! k2' can! be! estimated!with!good!accuracy!only!if!the!experiment!was!performed!greater!than!>! 34! seconds.! Hence,! it! is! important! to! analyze! the! sensitivity! coefficients! to! get! insight!in!to!the!estimation!problem.!! ! 108 ! Figure! 4.11! shows! the! sequential! estimation! of! the! parameters.! It! is! important! to! note! that! the! values! of! k1! and! k2' comes! to! a! constant! value! after! 10! seconds.!This!is!a!good!sign!as!the!parameters!are!not!changing!towards!the!end!of! the! experimental! time! and! that! is! what! we! expect! from! the! sequential! results.! Sequential! results! provide! a! robust! tool! to! check! if! the! estimated! parameters! are! reliable.! If! the! value! of! parameter! does! not! come! to! a! constant! toward! the! end! of! experiment,!this!means!that!there!is!some!error!in!the!model!or!in!the!experiment.! To!see!the!constant!nature!of!parameters!towards!the!end,!the!sequential!plot!was! plotted!for!the!second!half!of!the!experiment!as!represented!in!Figure!4.12.!This!plot! reveals!any!trend!in!the!parameters!as!each!datum!is!added,!as!it!removes!the!large! values! of! parameters! at! the! beginning! of! the! experiment.! Hence,! sequential! parameter! estimation! can! help! to! diagnose! problems.! The! parameter! values! reported! in! Table! 4.2! and! Table! 4.3! are! the! values! at! the! end! of! the! sequential! estimation!procedure.!! ! 109 o Parameters, W/m C 20 15 10 k1 k2 5 0 −5 0 10 20 Time (sec) 30 40 Figure!4.11!Sequential!estimation!of!thermal!conductivities!at!T1!and!T2! ! ! 110 o Parameters, W/m C 0.6 0.58 0.56 k1 k2 0.54 0.52 0.5 0.48 20 25 30 Time (sec) 35 40 Figure!4.12!Sequential!estimation!of!thermal!conductivities!at!T1!and!T2!for!second! half!of!the!experimental!time! 4.4'Conclusions' ! The!dimensionless!derivation!of!scaled!sensitivity!coefficients!was!presented! and!implemented!as!an!intrinsic!verification!method!to!the!finite!element!code.!The! identity! given! by! scaled! sensitivity! relations! for! heat! transfer! problems! provides! a! method! for! checking! the! accuracy! of! a! computer! code! at! interior! and! boundary! points! and! at! any! time.! The! concept! is! quite! general! and! is! not! restricted! to! heat! conduction! or! linear! problems.! The! scaled! sensitivity! coefficients! can! also! provide! useful! insight! in! to! the! parameter! estimation! problem.! It! can! show! if! all! the! parameters!in!the!model!can!be!estimated!and!with!what!relative!accuracy.!! 111 ! Based! on! the! robust! principles! of! intrinsic! verification! and! insight! from!the!scaled!sensitivity!coefficients,!a!method!and!instrument!was!presented!to! estimate! temperatureJdependent! thermal! properties.! With! the! design! of! the! instrument! combined! with! the! model! capabilities,! this! instrument! is! capable! of! producing! results! in! significantly! less! time! as! compared! to! other! traditional! methods.!This!new!rapid!lineJsource!method!will!estimate!food!thermal!properties! in! one! experiment! under! conditions! approaching! an! actual! process! (experimental! run!time!less!than!1!min).!!The!previous!laborJintensive!requirement!to!run!multiple! tests!at!various!temperatures!over!many!hours!or!days!can!be!minimized!or!perhaps! eliminated.! ! This! new! device! will! allow! food! processors! to! obtain! rapid! thermal! property! information! to! be! used! to! design! processes! for! maximum! quality! while! maintaining!safety.!! ' 112 APPENDIX 113 Table!A.1!Data!recorded!from!TPCell!for!sweet!potato!test!1! ! Time!(sec)! Voltage!(Volt)! Current!(Amp)! 0! 105.7469! 0.2187! 0.2! 108.0289! 0.2233! 0.4! 108.0107! 0.2232! 0.6! 108.0207! 0.2232! 0.8! 108.0135! 0.2231! 1! 108.0262! 0.2231! 1.2! 108.0265! 0.2231! 1.4! 108.0019! 0.2230! 1.6! 107.9940! 0.2229! 1.8! 107.9907! 0.2229! 2! 107.9747! 0.2228! 2.2! 107.9648! 0.2228! 2.4! 107.9888! 0.2228! 2.6! 107.9921! 0.2228! 2.8! 107.9871! 0.2227! 3! 107.9955! 0.2227! 3.2! 108.0015! 0.2227! 3.4! 108.0032! 0.2227! 3.6! 107.9906! 0.2227! 3.8! 107.9892! 0.2226! 4! 108.0179! 0.2227! 4.2! 108.0121! 0.2226! 4.4! 107.9933! 0.2226! 4.6! 107.9849! 0.2225! 4.8! 107.9772! 0.2225! 5! 108.0057! 0.2225! 5.2! 108.0032! 0.2225! 5.4! 107.9927! 0.2225! 5.6! 107.9824! 0.2224! 5.8! 107.9764! 0.2224! 6! 107.9897! 0.2224! 6.2! 107.9928! 0.2224! 6.4! 107.9893! 0.2224! 6.6! 107.9774! 0.2223! 6.8! 107.9754! 0.2223! 7! 107.9914! 0.2223! 7.2! 108.0076! 0.2224! ! 114 Power!(Watt)! 23.1246! 24.1246! 24.1104! 24.1094! 24.1007! 24.1009! 24.0981! 24.0830! 24.0753! 24.0704! 24.0596! 24.0523! 24.0601! 24.0577! 24.0522! 24.0536! 24.0529! 24.0521! 24.0443! 24.0415! 24.0512! 24.0473! 24.0357! 24.0301! 24.0233! 24.0349! 24.0326! 24.0255! 24.0191! 24.0135! 24.0187! 24.0176! 24.0148! 24.0079! 24.0050! 24.0095! 24.0156! Table!A.1!(cont’d)! Time!(sec)! 7.6! 7.8! 8! 8.2! 8.4! 8.6! 8.8! 9! 9.2! 9.4! 9.6! 9.8! 10! 10.2! 10.4! 10.6! 10.8! 11! 11.2! 11.4! 11.6! 11.8! 12! 12.2! 12.4! 12.6! 12.8! 13! 13.2! 13.4! 13.6! 13.8! 14! 14.2! 14.4! 14.6! 14.8! ! Voltage!(Volt)! 107.9779! 107.9706! 107.9988! 108.0049! 107.9774! 107.9779! 107.9723! 108.0101! 108.0094! 107.9974! 107.9989! 107.9780! 108.0064! 108.0239! 108.0011! 107.9941! 107.9844! 107.9878! 107.9837! 107.9795! 107.9847! 107.9847! 108.0142! 108.0139! 108.0023! 108.0005! 107.9850! 108.0013! 108.0105! 108.0059! 108.0042! 107.9826! 108.0008! 108.0104! 107.9832! 107.9826! 107.9848! ! Current!(Amp)! 0.2223! 0.2222! 0.2223! 0.2223! 0.2222! 0.2222! 0.2222! 0.2222! 0.2222! 0.2222! 0.2222! 0.2221! 0.2222! 0.2222! 0.2221! 0.2221! 0.2221! 0.2221! 0.2220! 0.2220! 0.2220! 0.2220! 0.2221! 0.2221! 0.2220! 0.2220! 0.2220! 0.2220! 0.2220! 0.2220! 0.2220! 0.2219! 0.2219! 0.2220! 0.2219! 0.2219! 0.2219! 115 Power!(Watt)! 24.0000! 23.9948! 24.0066! 24.0067! 23.9943! 23.9927! 23.9871! 24.0035! 24.0027! 23.9956! 23.9954! 23.9834! 23.9954! 24.0023! 23.9898! 23.9858! 23.9810! 23.9813! 23.9771! 23.9754! 23.9771! 23.9744! 23.9864! 23.9856! 23.9802! 23.9769! 23.9692! 23.9750! 23.9783! 23.9754! 23.9742! 23.9633! 23.9701! 23.9738! 23.9606! 23.9584! 23.9594! Table!A.1!(cont’d)! Time!(sec)! 15! 15.2! 15.4! 15.6! 15.8! 16! 16.2! 16.4! 16.6! 16.8! 17! 17.2! 17.4! 17.6! 17.8! 18! 18.2! 18.4! 18.6! 18.8! 19! 19.2! 19.4! 19.6! 19.8! 20! 20.2! 20.4! 20.6! 20.8! 21! 21.2! 21.4! 21.6! 21.8! 22! 22.2! 22.4! Voltage!(Volt)! 108.0188! 108.0305! 108.0184! 107.9859! 107.9949! 108.0182! 108.0257! 108.0046! 108.0035! 107.9955! 108.0249! 108.0237! 108.0022! 108.0086! 108.0066! 108.0308! 108.0396! 108.0023! 108.0001! 107.9723! 108.0108! 107.9987! 107.9801! 107.9940! 107.9398! 107.9471! 107.9955! 107.9950! 108.0044! 107.9758! 108.0011! 108.0211! 108.0091! 108.0071! 107.9863! 108.0192! 108.0340! 108.0146! Current!(Amp)! 0.2220! 0.2220! 0.2219! 0.2219! 0.2219! 0.2219! 0.2219! 0.2219! 0.2218! 0.2218! 0.2219! 0.2219! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2217! 0.2218! 0.2217! 0.2217! 0.2217! 0.2216! 0.2216! 0.2217! 0.2217! 0.2217! 0.2216! 0.2217! 0.2217! 0.2217! 0.2217! 0.2216! 0.2217! 0.2217! 0.2217! 116 Power!(Watt)! 23.9751! 23.9774! 23.9715! 23.9574! 23.9592! 23.9698! 23.9712! 23.9612! 23.9597! 23.9553! 23.9677! 23.9658! 23.9558! 23.9579! 23.9568! 23.9658! 23.9685! 23.9520! 23.9502! 23.9367! 23.9528! 23.9470! 23.9375! 23.9438! 23.9191! 23.9206! 23.9420! 23.9410! 23.9446! 23.9312! 23.9411! 23.9494! 23.9438! 23.9413! 23.9317! 23.9464! 23.9516! 23.9427! Table!A.1!(cont’d)! Time!(sec)! 22.6! 22.8! 23! 23.2! 23.4! 23.6! 23.8! 24! 24.2! 24.4! 24.6! 24.8! 25! 25.2! 25.4! 25.6! 25.8! 26! 26.2! 26.4! 26.6! 26.8! 27! 27.2! 27.4! 27.6! 27.8! 28! 28.2! 28.4! 28.6! 28.8! 29! 29.2! 29.4! 29.6! 29.8! 30! Voltage!(Volt)! 108.0012! 107.9878! 108.0262! 108.0290! 108.0096! 108.0151! 107.9983! 108.0232! 108.0311! 108.0167! 108.0022! 107.9906! 108.0118! 108.0227! 108.0204! 108.0078! 108.0012! 108.0302! 108.0230! 108.0201! 108.0396! 108.0235! 108.0324! 108.0495! 108.1154! 108.1240! 108.1180! 108.1349! 108.1297! 108.0254! 108.0277! 108.0013! 108.0329! 108.0447! 108.0417! 108.0223! 108.0211! 108.0434! Current!(Amp)! 0.2216! 0.2216! 0.2217! 0.2217! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2215! 0.2216! 0.2216! 0.2216! 0.2215! 0.2215! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2217! 0.2217! 0.2217! 0.2218! 0.2217! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 117 Power!(Watt)! 23.9371! 23.9298! 23.9463! 23.9459! 23.9374! 23.9390! 23.9321! 23.9404! 23.9446! 23.9369! 23.9301! 23.9246! 23.9330! 23.9369! 23.9354! 23.9283! 23.9262! 23.9380! 23.9343! 23.9319! 23.9416! 23.9328! 23.9362! 23.9443! 23.9720! 23.9752! 23.9721! 23.9801! 23.9753! 23.9288! 23.9286! 23.9171! 23.9319! 23.9359! 23.9339! 23.9246! 23.9234! 23.9318! Table!A.1!(cont’d)! Time!(sec)! 30.2! 30.4! 30.6! 30.8! 31! 31.2! 31.4! 31.6! 31.8! 32! 32.2! 32.4! 32.6! 32.8! 33! 33.2! 33.4! 33.6! 33.8! 34! 34.2! 34.4! 34.6! 34.8! 35! 35.2! 35.4! 35.6! 35.8! 36! 36.2! 36.4! 36.6! 36.8! 37! 37.2! 37.4! 37.6! Voltage!(Volt)! 108.0567! 108.0387! 108.0352! 108.0284! 108.0274! 108.0407! 108.0187! 108.0148! 108.0121! 108.0442! 108.0608! 108.0327! 108.0387! 108.0301! 108.0380! 108.0355! 108.0238! 108.0088! 107.9785! 108.0031! 108.0192! 108.0099! 108.0040! 107.9874! 108.0027! 108.0366! 108.0267! 108.0085! 108.0019! 108.0442! 108.0641! 108.0571! 108.0550! 108.0470! 108.0607! 108.0690! 108.0469! 108.0331! Current!(Amp)! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2214! 0.2214! 0.2214! 0.2215! 0.2215! 0.2214! 0.2214! 0.2214! 0.2214! 0.2214! 0.2214! 0.2214! 0.2213! 0.2213! 0.2214! 0.2214! 0.2213! 0.2213! 0.2213! 0.2214! 0.2214! 0.2213! 0.2213! 0.2214! 0.2214! 0.2214! 0.2214! 0.2214! 0.2214! 0.2214! 0.2214! 0.2213! 118 Power!(Watt)! 23.9378! 23.9297! 23.9280! 23.9242! 23.9237! 23.9281! 23.9181! 23.9176! 23.9140! 23.9266! 23.9341! 23.9228! 23.9225! 23.9203! 23.9227! 23.9207! 23.9167! 23.9092! 23.8941! 23.9052! 23.9120! 23.9082! 23.9032! 23.8972! 23.9038! 23.9180! 23.9119! 23.9033! 23.9016! 23.9198! 23.9272! 23.9234! 23.9221! 23.9179! 23.9247! 23.9274! 23.9173! 23.9102! Table!A.1!(cont’d)! Time!(sec)! 37.8! 38! 38.2! 38.4! 38.6! 38.8! 39! 39.2! 39.4! 39.6! 39.8! 40! 40.2! 40.4! 40.6! 40.8! 41! 41.2! 41.4! Voltage!(Volt)! 108.0333! 108.0460! 108.0530! 108.0371! 108.0384! 108.0403! 108.0521! 108.0637! 108.0506! 108.0307! 108.0403! 108.0570! 108.0426! 108.0239! 108.0343! 108.0228! 108.0245! 108.0353! 108.0069! Current!(Amp)! 0.2213! 0.2213! 0.2214! 0.2213! 0.2213! 0.2213! 0.2213! 0.2214! 0.2213! 0.2213! 0.2213! 0.2213! 0.2213! 0.2213! 0.2213! 0.2212! 0.2213! 0.2213! 0.2212! 119 Power!(Watt)! 23.9108! 23.9155! 23.9179! 23.9100! 23.9106! 23.9118! 23.9157! 23.9218! 23.9142! 23.9054! 23.9090! 23.9164! 23.9098! 23.9013! 23.9049! 23.9000! 23.9005! 23.9047! 23.8904! Table!A.2!Data!recorded!from!TPCell!for!sweet!potato!test!2! ! Time!(sec)! Voltage!(Volt)! Current!(Amp)! 0! 103.4193! 0.2139! 0.2! 107.8192! 0.2229! 0.4! 107.8183! 0.2228! 0.6! 107.8332! 0.2228! 0.8! 107.8423! 0.2228! 1! 107.8203! 0.2227! 1.2! 107.8101! 0.2226! 1.4! 107.8150! 0.2226! 1.6! 107.7964! 0.2225! 1.8! 107.8233! 0.2225! 2! 107.8505! 0.2226! 2.2! 107.8238! 0.2225! 2.4! 107.8125! 0.2224! 2.6! 107.8326! 0.2224! 2.8! 107.8413! 0.2224! 3! 107.8334! 0.2224! 3.2! 107.8250! 0.2224! 3.4! 107.8134! 0.2223! 3.6! 107.8304! 0.2223! 3.8! 107.8396! 0.2223! 4! 107.8386! 0.2223! 4.2! 107.8212! 0.2222! 4.4! 107.8175! 0.2222! 4.6! 107.8419! 0.2222! 4.8! 107.8458! 0.2222! 5! 107.8493! 0.2222! 5.2! 107.8300! 0.2222! 5.4! 107.8158! 0.2221! 5.6! 107.8195! 0.2221! 5.8! 107.8248! 0.2221! 6! 107.8119! 0.2221! 6.2! 107.8027! 0.2220! 6.4! 107.8073! 0.2220! 6.6! 107.8012! 0.2220! 6.8! 107.8028! 0.2220! 7! 107.8134! 0.2220! 7.2! 107.8057! 0.2219! 7.4! 107.7882! 0.2219! 120 Power!(Watt)! 22.1174! 24.0322! 24.0249! 24.0255! 24.0241! 24.0102! 24.0020! 23.9989! 23.9878! 23.9957! 24.0056! 23.9892! 23.9820! 23.9869! 23.9882! 23.9823! 23.9751! 23.9677! 23.9742! 23.9751! 23.9724! 23.9625! 23.9593! 23.9664! 23.9675! 23.9663! 23.9552! 23.9468! 23.9463! 23.9474! 23.9403! 23.9338! 23.9345! 23.9303! 23.9301! 23.9328! 23.9270! 23.9180! Table!A.2!(cont’d)! Time!(sec)! 7.6! 7.8! 8! 8.2! 8.4! 8.6! 8.8! 9! 9.2! 9.4! 9.6! 9.8! 10! 10.2! 10.4! 10.6! 10.8! 11! 11.2! 11.4! 11.6! 11.8! 12! 12.2! 12.4! 12.6! 12.8! 13! 13.2! 13.4! 13.6! 13.8! 14! 14.2! 14.4! 14.6! 14.8! 15! Voltage!(Volt)! 107.7893! 107.8042! 107.8087! 107.7971! 107.7768! 107.7613! 107.7679! 107.8068! 107.8119! 107.8113! 107.8214! 107.8266! 107.8068! 107.8120! 107.7972! 107.8204! 107.8295! 107.8259! 107.8173! 107.8064! 107.8213! 107.8231! 107.8396! 107.8217! 107.8070! 107.8338! 107.8407! 107.8345! 107.8261! 107.8133! 107.8330! 107.8410! 107.8410! 107.8291! 107.8198! 107.8443! 107.8269! 107.8386! Current!(Amp)! 0.2219! 0.2219! 0.2219! 0.2219! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2217! 0.2218! 0.2218! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2216! 0.2217! 0.2217! 0.2217! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 121 Power!(Watt)! 23.9164! 23.9218! 23.9218! 23.9158! 23.9058! 23.8965! 23.8982! 23.9146! 23.9158! 23.9130! 23.9163! 23.9176! 23.9079! 23.9094! 23.9007! 23.9095! 23.9127! 23.9100! 23.9062! 23.8980! 23.9049! 23.9044! 23.9109! 23.9005! 23.8933! 23.9045! 23.9058! 23.9029! 23.8974! 23.8910! 23.8992! 23.9023! 23.9010! 23.8941! 23.8894! 23.8994! 23.8898! 23.8944! Table!A.2!(cont’d)! Time!(sec)! 15.2! 15.4! 15.6! 15.8! 16! 16.2! 16.4! 16.6! 16.8! 17! 17.2! 17.4! 17.6! 17.8! 18! 18.2! 18.4! 18.6! 18.8! 19! 19.2! 19.4! 19.6! 19.8! 20! 20.2! 20.4! 20.6! 20.8! 21! 21.2! 21.4! 21.6! 21.8! 22! 22.2! 22.4! 22.6! Voltage!(Volt)! 107.8386! 107.8302! 107.8272! 107.8291! 107.8359! 107.8457! 107.8473! 107.8873! 107.8687! 107.8754! 107.8713! 107.8529! 107.8920! 107.9898! 107.9844! 107.9694! 107.9666! 107.9831! 107.9881! 107.9793! 107.9854! 107.9673! 107.9765! 107.9757! 107.9683! 107.9532! 107.9620! 107.9585! 107.9811! 107.9582! 107.9493! 107.9532! 107.9728! 107.9874! 108.0021! 107.9963! 107.9652! 107.9595! Current!(Amp)! 0.2216! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2216! 0.2216! 0.2216! 0.2216! 0.2215! 0.2216! 0.2218! 0.2218! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2217! 0.2216! 0.2216! 0.2215! 122 Power!(Watt)! 23.8936! 23.8887! 23.8873! 23.8869! 23.8887! 23.8922! 23.8932! 23.9093! 23.8989! 23.9021! 23.9001! 23.8905! 23.9067! 23.9500! 23.9457! 23.9389! 23.9377! 23.9427! 23.9448! 23.9394! 23.9427! 23.9325! 23.9362! 23.9356! 23.9313! 23.9231! 23.9270! 23.9246! 23.9337! 23.9234! 23.9182! 23.9198! 23.9273! 23.9340! 23.9386! 23.9357! 23.9215! 23.9178! Table!A.2!(cont’d)! Time!(sec)! 22.8! 23! 23.2! 23.4! 23.6! 23.8! 24! 24.2! 24.4! 24.6! 24.8! 25! 25.2! 25.4! 25.6! 25.8! 26! 26.2! 26.4! 26.6! 26.8! 27! 27.2! 27.4! 27.6! 27.8! 28! 28.2! 28.4! 28.6! 28.8! 29! 29.2! 29.4! 29.6! 29.8! 30! 30.2! Voltage!(Volt)! 107.9622! 107.9619! 107.9494! 107.9511! 107.9608! 107.9635! 107.9668! 107.9614! 107.9574! 107.9640! 107.9862! 108.0071! 107.9833! 107.9743! 107.9633! 107.9890! 107.9925! 107.9783! 107.9761! 107.9916! 107.9951! 107.9962! 107.9874! 107.9752! 107.9887! 108.0027! 108.0204! 107.9956! 107.9965! 107.9766! 107.9996! 107.9855! 107.9877! 107.9828! 107.9849! 107.9882! 107.9931! 107.9825! Current!(Amp)! 0.2216! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2216! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2214! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2214! 0.2214! 0.2214! 0.2214! 0.2214! 0.2214! 0.2214! 0.2214! 0.2214! 123 Power!(Watt)! 23.9191! 23.9184! 23.9122! 23.9118! 23.9158! 23.9154! 23.9167! 23.9140! 23.9119! 23.9138! 23.9235! 23.9304! 23.9205! 23.9152! 23.9097! 23.9217! 23.9229! 23.9152! 23.9143! 23.9191! 23.9208! 23.9211! 23.9163! 23.9099! 23.9151! 23.9211! 23.9285! 23.9175! 23.9172! 23.9075! 23.9161! 23.9105! 23.9101! 23.9090! 23.9082! 23.9106! 23.9115! 23.9054! Table!A.2!(cont’d)! Time!(sec)! 30.4! 30.6! 30.8! 31! 31.2! 31.4! 31.6! 31.8! 32! 32.2! 32.4! 32.6! 32.8! 33! 33.2! 33.4! 33.6! 33.8! 34! 34.2! 34.4! 34.6! 34.8! 35! 35.2! 35.4! 35.6! 35.8! 36! 36.2! 36.4! 36.6! 36.8! 37! 37.2! 37.4! 37.6! 37.8! Voltage!(Volt)! 107.9753! 107.9696! 107.9897! 108.0115! 107.9781! 107.9660! 107.9658! 107.9905! 108.0003! 107.9829! 107.9728! 107.9746! 107.9986! 108.0079! 107.9922! 107.9843! 107.9798! 107.9868! 107.9961! 107.9804! 107.9874! 107.9825! 107.9986! 108.0097! 108.0023! 107.9948! 108.0057! 108.0079! 107.9989! 107.9576! 107.9724! 107.9612! 107.9679! 107.9516! 107.9199! 107.9537! 107.9544! 107.9616! Current!(Amp)! 0.2214! 0.2214! 0.2214! 0.2214! 0.2214! 0.2213! 0.2213! 0.2214! 0.2214! 0.2214! 0.2213! 0.2213! 0.2214! 0.2214! 0.2213! 0.2213! 0.2213! 0.2213! 0.2213! 0.2213! 0.2213! 0.2213! 0.2213! 0.2213! 0.2213! 0.2213! 0.2213! 0.2213! 0.2213! 0.2212! 0.2212! 0.2212! 0.2212! 0.2212! 0.2211! 0.2212! 0.2212! 0.2212! 124 Power!(Watt)! 23.9023! 23.8992! 23.9089! 23.9172! 23.9015! 23.8954! 23.8944! 23.9056! 23.9091! 23.9023! 23.8956! 23.8962! 23.9071! 23.9106! 23.9020! 23.9001! 23.8950! 23.8987! 23.9028! 23.8960! 23.8974! 23.8953! 23.9022! 23.9064! 23.9034! 23.8986! 23.9028! 23.9049! 23.8987! 23.8800! 23.8871! 23.8817! 23.8839! 23.8763! 23.8612! 23.8764! 23.8765! 23.8791! Table!A.2!(cont’d)! Time!(sec)! 38! 38.2! 38.4! 38.6! 38.8! 39! 39.2! 39.4! 39.6! 39.8! 40! 40.2! 40.4! 40.6! 40.8! 41! 41.2! 41.4! ! ! Voltage!(Volt)! 107.9622! 107.9811! 107.9959! 107.9917! 107.9874! 108.0006! 107.9800! 107.9691! 107.9652! 107.9937! 108.0028! 107.9851! 107.9772! 107.9792! 107.9891! 108.0035! 107.9729! 107.9851! Current!(Amp)! 0.2212! 0.2212! 0.2212! 0.2212! 0.2212! 0.2212! 0.2212! 0.2212! 0.2212! 0.2212! 0.2212! 0.2212! 0.2212! 0.2212! 0.2212! 0.2212! 0.2211! 0.2212! ! 125 Power!(Watt)! 23.8795! 23.8868! 23.8929! 23.8918! 23.8881! 23.8927! 23.8848! 23.8799! 23.8765! 23.8882! 23.8925! 23.8840! 23.8820! 23.8804! 23.8847! 23.8909! 23.8763! 23.8832! Table!A.3!Data!recorded!from!TPCell!for!sweet!potato!test!3! ! Time!(sec)! Voltage!(Volt)! Current!(Amp)! 0! 105.5313! 0.2182! 0.2! 108.0104! 0.2233! 0.4! 108.0524! 0.2233! 0.6! 108.0342! 0.2232! 0.8! 108.0353! 0.2232! 1! 108.0555! 0.2232! 1.2! 108.0059! 0.2230! 1.4! 108.0192! 0.2230! 1.6! 108.0333! 0.2230! 1.8! 108.0167! 0.2229! 2! 108.0235! 0.2229! 2.2! 108.0102! 0.2229! 2.4! 107.9830! 0.2228! 2.6! 107.9996! 0.2228! 2.8! 107.9754! 0.2227! 3! 107.9590! 0.2227! 3.2! 107.9637! 0.2226! 3.4! 107.9595! 0.2226! 3.6! 107.9964! 0.2227! 3.8! 108.0282! 0.2227! 4! 108.0229! 0.2227! 4.2! 108.0059! 0.2226! 4.4! 108.0241! 0.2226! 4.6! 107.9812! 0.2225! 4.8! 107.9581! 0.2225! 5! 107.9886! 0.2225! 5.2! 107.9837! 0.2225! 5.4! 108.0266! 0.2225! 5.6! 108.0127! 0.2225! 5.8! 108.0049! 0.2225! 6! 108.0187! 0.2225! 6.2! 107.9953! 0.2224! 6.4! 107.9986! 0.2224! 6.6! 107.9941! 0.2224! 6.8! 107.9864! 0.2223! 7! 108.0138! 0.2224! 7.2! 108.0184! 0.2224! 7.4! 108.0015! 0.2223! 126 Power!(Watt)! 23.0293! 24.1165! 24.1283! 24.1165! 24.1103! 24.1157! 24.0900! 24.0901! 24.0946! 24.0818! 24.0817! 24.0726! 24.0575! 24.0615! 24.0475! 24.0377! 24.0379! 24.0321! 24.0465! 24.0577! 24.0550! 24.0439! 24.0490! 24.0290! 24.0175! 24.0276! 24.0232! 24.0399! 24.0322! 24.0276! 24.0323! 24.0196! 24.0199! 24.0156! 24.0104! 24.0216! 24.0217! 24.0115! Table!A.3!(cont’d)! Time!(sec)! 7.6! 7.8! 8! 8.2! 8.4! 8.6! 8.8! 9! 9.2! 9.4! 9.6! 9.8! 10! 10.2! 10.4! 10.6! 10.8! 11! 11.2! 11.4! 11.6! 11.8! 12! 12.2! 12.4! 12.6! 12.8! 13! 13.2! 13.4! 13.6! 13.8! 14! 14.2! 14.4! 14.6! 14.8! 15! Voltage!(Volt)! 108.0312! 108.0187! 108.0131! 108.0230! 107.9970! 108.0170! 108.0294! 108.0329! 108.0307! 108.0257! 108.0187! 108.0255! 108.0208! 107.9974! 108.0236! 108.0206! 108.0124! 108.0242! 108.0072! 107.9968! 108.0217! 108.0097! 108.0281! 108.0269! 108.0026! 108.0178! 108.0070! 108.0056! 108.0337! 108.0162! 108.0539! 108.1617! 108.1430! 108.1452! 108.1491! 108.1434! 108.1476! 108.1475! Current!(Amp)! 0.2224! 0.2223! 0.2223! 0.2223! 0.2223! 0.2223! 0.2223! 0.2223! 0.2223! 0.2222! 0.2222! 0.2222! 0.2222! 0.2221! 0.2222! 0.2222! 0.2221! 0.2222! 0.2221! 0.2221! 0.2221! 0.2221! 0.2221! 0.2221! 0.2220! 0.2221! 0.2220! 0.2220! 0.2221! 0.2220! 0.2221! 0.2223! 0.2222! 0.2222! 0.2222! 0.2222! 0.2222! 0.2222! 127 Power!(Watt)! 24.0229! 24.0163! 24.0127! 24.0159! 24.0030! 24.0110! 24.0155! 24.0148! 24.0130! 24.0074! 24.0051! 24.0060! 24.0017! 23.9906! 24.0014! 23.9988! 23.9930! 23.9980! 23.9892! 23.9825! 23.9933! 23.9871! 23.9927! 23.9914! 23.9797! 23.9860! 23.9800! 23.9771! 23.9902! 23.9804! 23.9960! 24.0439! 24.0343! 24.0336! 24.0350! 24.0306! 24.0320! 24.0317! Table!A.3!(cont’d)! Time!(sec)! 15.2! 15.4! 15.6! 15.8! 16! 16.2! 16.4! 16.6! 16.8! 17! 17.2! 17.4! 17.6! 17.8! 18! 18.2! 18.4! 18.6! 18.8! 19! 19.2! 19.4! 19.6! 19.8! 20! 20.2! 20.4! 20.6! 20.8! 21! 21.2! 21.4! 21.6! 21.8! 22! 22.2! 22.4! 22.6! Voltage!(Volt)! 108.1300! 108.1407! 108.1129! 108.0451! 108.0599! 108.0254! 108.0297! 108.0563! 108.0456! 108.0372! 108.0290! 108.0120! 108.0241! 108.0025! 108.0123! 108.0256! 108.0232! 108.0608! 108.0710! 108.0568! 108.0691! 108.0494! 108.0620! 108.0626! 108.0561! 108.0539! 108.0662! 108.0663! 108.0684! 108.0659! 108.0640! 108.0809! 108.0950! 108.0712! 108.0822! 108.0614! 108.0469! 108.0440! Current!(Amp)! 0.2222! 0.2222! 0.2221! 0.2220! 0.2220! 0.2219! 0.2219! 0.2220! 0.2219! 0.2219! 0.2219! 0.2218! 0.2219! 0.2218! 0.2218! 0.2218! 0.2218! 0.2219! 0.2219! 0.2219! 0.2219! 0.2218! 0.2219! 0.2219! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2219! 0.2218! 0.2218! 0.2218! 0.2217! 0.2217! 128 Power!(Watt)! 24.0215! 24.0269! 24.0128! 23.9822! 23.9876! 23.9719! 23.9721! 23.9841! 23.9782! 23.9732! 23.9695! 23.9603! 23.9655! 23.9548! 23.9575! 23.9630! 23.9615! 23.9777! 23.9807! 23.9734! 23.9782! 23.9689! 23.9740! 23.9750! 23.9697! 23.9681! 23.9723! 23.9727! 23.9726! 23.9706! 23.9703! 23.9745! 23.9814! 23.9712! 23.9732! 23.9653! 23.9570! 23.9546! Table!A.3!(cont’d)! Time!(sec)! 22.8! 23! 23.2! 23.4! 23.6! 23.8! 24! 24.2! 24.4! 24.6! 24.8! 25! 25.2! 25.4! 25.6! 25.8! 26! 26.2! 26.4! 26.6! 26.8! 27! 27.2! 27.4! 27.6! 27.8! 28! 28.2! 28.4! 28.6! 28.8! 29! 29.2! 29.4! 29.6! 29.8! 30! 30.2! Voltage!(Volt)! 108.0391! 108.0300! 108.0408! 108.0147! 108.0605! 108.0553! 108.0373! 108.0473! 108.0503! 108.0753! 108.0917! 108.0471! 108.0688! 108.0882! 108.0817! 108.0939! 108.0842! 108.0778! 108.0875! 108.0848! 108.0667! 108.0645! 108.0533! 108.0587! 108.0722! 108.0657! 108.0747! 108.0659! 108.0448! 108.0753! 108.0720! 108.0799! 108.0638! 108.0581! 108.1031! 108.1003! 108.0667! 108.0779! Current!(Amp)! 0.2217! 0.2217! 0.2217! 0.2216! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2218! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2217! 0.2217! 0.2216! 0.2216! 129 Power!(Watt)! 23.9524! 23.9487! 23.9527! 23.9403! 23.9599! 23.9568! 23.9466! 23.9519! 23.9517! 23.9625! 23.9697! 23.9507! 23.9589! 23.9663! 23.9632! 23.9689! 23.9614! 23.9585! 23.9633! 23.9609! 23.9524! 23.9506! 23.9454! 23.9471! 23.9541! 23.9507! 23.9523! 23.9473! 23.9389! 23.9517! 23.9491! 23.9520! 23.9449! 23.9415! 23.9612! 23.9608! 23.9441! 23.9477! Table!A.3!(cont’d)! Time!(sec)! 30.4! 30.6! 30.8! 31! 31.2! 31.4! 31.6! 31.8! 32! 32.2! 32.4! 32.6! 32.8! 33! 33.2! 33.4! 33.6! 33.8! 34! 34.2! 34.4! 34.6! 34.8! 35! 35.2! 35.4! 35.6! 35.8! 36! 36.2! 36.4! 36.6! 36.8! 37! 37.2! 37.4! 37.6! 37.8! Voltage!(Volt)! 108.0714! 108.0734! 108.1098! 108.0513! 108.0523! 108.0832! 108.0743! 108.0706! 108.0902! 108.0616! 108.0768! 108.0953! 108.0721! 108.0859! 108.0934! 108.1683! 108.2153! 108.2077! 108.2130! 108.2191! 108.1997! 108.2045! 108.2254! 108.2322! 108.2078! 108.1208! 108.1300! 108.1607! 108.1247! 108.1356! 108.1179! 108.0953! 108.1092! 108.1074! 108.0927! 108.1064! 108.1174! 108.0981! Current!(Amp)! 0.2216! 0.2216! 0.2216! 0.2215! 0.2215! 0.2216! 0.2215! 0.2215! 0.2216! 0.2215! 0.2215! 0.2216! 0.2215! 0.2215! 0.2215! 0.2217! 0.2218! 0.2218! 0.2218! 0.2218! 0.2217! 0.2217! 0.2218! 0.2218! 0.2217! 0.2216! 0.2216! 0.2216! 0.2215! 0.2216! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 130 Power!(Watt)! 23.9438! 23.9456! 23.9605! 23.9342! 23.9355! 23.9478! 23.9434! 23.9397! 23.9492! 23.9361! 23.9415! 23.9494! 23.9391! 23.9450! 23.9458! 23.9799! 24.0013! 23.9964! 23.9986! 24.0002! 23.9925! 23.9928! 24.0021! 24.0042! 23.9933! 23.9544! 23.9573! 23.9714! 23.9542! 23.9598! 23.9509! 23.9406! 23.9460! 23.9451! 23.9379! 23.9436! 23.9484! 23.9389! Table!A.3!(cont’d)! Time!(sec)! 38! 38.2! 38.4! 38.6! 38.8! 39! 39.2! 39.4! 39.6! 39.8! 40! 40.2! 40.4! 40.6! 40.8! 41! 41.2! 41.4! 41.6! 41.8! ! ! Voltage!(Volt)! 108.1094! 108.0950! 108.1028! 108.1321! 108.1375! 108.1472! 108.1290! 108.1193! 108.1488! 108.1368! 108.1389! 108.1637! 108.1241! 108.1262! 108.1486! 108.1274! 108.1489! 108.1296! 108.1345! 108.1651! Current!(Amp)! 0.2215! 0.2214! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! ! 131 Power!(Watt)! 23.9439! 23.9369! 23.9401! 23.9523! 23.9559! 23.9586! 23.9507! 23.9448! 23.9578! 23.9525! 23.9532! 23.9629! 23.9449! 23.9453! 23.9551! 23.9452! 23.9541! 23.9467! 23.9470! 23.9602! Table!A.4!Data!recorded!from!TPCell!for!sweet!potato!test!4! ! Time!(sec)! Voltage!(Volt)! Current!(Amp)! 0! 105.6148! 0.2184! 0.2! 108.1338! 0.2235! 0.4! 108.1406! 0.2235! 0.6! 108.1549! 0.2235! 0.8! 108.1567! 0.2234! 1! 108.1367! 0.2233! 1.2! 108.1070! 0.2232! 1.4! 108.1278! 0.2232! 1.6! 108.1441! 0.2232! 1.8! 108.1425! 0.2232! 2! 108.1420! 0.2232! 2.2! 108.1823! 0.2232! 2.4! 108.2406! 0.2233! 2.6! 108.2462! 0.2233! 2.8! 108.2299! 0.2232! 3! 108.2359! 0.2232! 3.2! 108.2466! 0.2232! 3.4! 108.2239! 0.2232! 3.6! 108.2429! 0.2232! 3.8! 108.2379! 0.2232! 4! 108.2495! 0.2231! 4.2! 108.1970! 0.2230! 4.4! 108.1258! 0.2228! 4.6! 108.1269! 0.2228! 4.8! 108.1519! 0.2229! 5! 108.1350! 0.2228! 5.2! 108.1305! 0.2228! 5.4! 108.1182! 0.2227! 5.6! 108.1301! 0.2228! 5.8! 108.1241! 0.2227! 6! 108.1160! 0.2227! 6.2! 108.1276! 0.2227! 6.4! 108.1262! 0.2227! 6.6! 108.1349! 0.2227! 6.8! 108.1372! 0.2227! 7! 108.1282! 0.2226! 7.2! 108.1600! 0.2227! 7.4! 108.1444! 0.2226! 132 Power!(Watt)! 23.0666! 24.1725! 24.1700! 24.1700! 24.1650! 24.1522! 24.1345! 24.1393! 24.1423! 24.1384! 24.1338! 24.1497! 24.1716! 24.1728! 24.1615! 24.1615! 24.1630! 24.1522! 24.1564! 24.1534! 24.1555! 24.1301! 24.0949! 24.0933! 24.1023! 24.0925! 24.0882! 24.0822! 24.0864! 24.0802! 24.0746! 24.0779! 24.0759! 24.0776! 24.0774! 24.0700! 24.0850! 24.0749! Table!A.4!(cont’d)! Time!(sec)! 7.6! 7.8! 8! 8.2! 8.4! 8.6! 8.8! 9! 9.2! 9.4! 9.6! 9.8! 10! 10.2! 10.4! 10.6! 10.8! 11! 11.2! 11.4! 11.6! 11.8! 12! 12.2! 12.4! 12.6! 12.8! 13! 13.2! 13.4! 13.6! 13.8! 14! 14.2! 14.4! 14.6! 14.8! 15! Voltage!(Volt)! 108.1421! 108.1283! 108.1183! 108.1416! 108.1385! 108.1441! 108.1420! 108.1390! 108.1439! 108.1324! 108.1413! 108.1301! 108.1366! 108.1155! 108.1282! 108.1428! 108.1445! 108.1271! 108.1401! 108.1283! 108.1317! 108.1353! 108.1348! 108.1542! 108.1288! 108.1330! 108.1272! 108.1273! 108.1403! 108.1205! 108.1391! 108.1332! 108.1290! 108.1365! 108.1187! 108.1302! 108.1297! 108.1168! Current!(Amp)! 0.2226! 0.2226! 0.2225! 0.2226! 0.2225! 0.2225! 0.2225! 0.2225! 0.2225! 0.2225! 0.2225! 0.2224! 0.2224! 0.2224! 0.2224! 0.2224! 0.2224! 0.2224! 0.2224! 0.2223! 0.2223! 0.2223! 0.2223! 0.2224! 0.2223! 0.2223! 0.2223! 0.2223! 0.2223! 0.2222! 0.2223! 0.2222! 0.2222! 0.2222! 0.2222! 0.2222! 0.2222! 0.2221! 133 Power!(Watt)! 24.0729! 24.0660! 24.0598! 24.0684! 24.0646! 24.0664! 24.0640! 24.0615! 24.0633! 24.0559! 24.0589! 24.0529! 24.0537! 24.0437! 24.0475! 24.0532! 24.0526! 24.0429! 24.0484! 24.0401! 24.0416! 24.0423! 24.0405! 24.0487! 24.0366! 24.0370! 24.0332! 24.0330! 24.0368! 24.0267! 24.0352! 24.0307! 24.0292! 24.0304! 24.0217! 24.0253! 24.0243! 24.0177! Table!A.4!(cont’d)! Time!(sec)! 15.2! 15.4! 15.6! 15.8! 16! 16.2! 16.4! 16.6! 16.8! 17! 17.2! 17.4! 17.6! 17.8! 18! 18.2! 18.4! 18.6! 18.8! 19! 19.2! 19.4! 19.6! 19.8! 20! 20.2! 20.4! 20.6! 20.8! 21! 21.2! 21.4! 21.6! 21.8! 22! 22.2! 22.4! 22.6! Voltage!(Volt)! 108.1107! 108.1223! 108.1376! 108.1528! 108.1195! 108.1433! 108.1113! 108.1069! 108.1141! 108.0658! 108.0975! 108.0936! 108.1022! 108.0985! 108.0962! 108.1688! 108.1705! 108.1627! 108.1512! 108.1722! 108.1940! 108.1857! 108.2005! 108.2006! 108.1878! 108.1996! 108.1892! 108.2069! 108.2035! 108.1865! 108.1314! 108.1384! 108.1816! 108.2236! 108.3029! 108.2952! 108.2836! 108.2709! Current!(Amp)! 0.2221! 0.2221! 0.2222! 0.2222! 0.2221! 0.2221! 0.2221! 0.2221! 0.2221! 0.2220! 0.2220! 0.2220! 0.2220! 0.2220! 0.2220! 0.2221! 0.2221! 0.2221! 0.2221! 0.2221! 0.2221! 0.2221! 0.2221! 0.2221! 0.2221! 0.2221! 0.2221! 0.2221! 0.2221! 0.2221! 0.2219! 0.2219! 0.2220! 0.2221! 0.2223! 0.2222! 0.2222! 0.2222! 134 Power!(Watt)! 24.0129! 24.0183! 24.0239! 24.0301! 24.0135! 24.0234! 24.0086! 24.0069! 24.0076! 23.9853! 23.9992! 23.9971! 23.9997! 23.9974! 23.9969! 24.0268! 24.0273! 24.0227! 24.0163! 24.0251! 24.0345! 24.0296! 24.0359! 24.0348! 24.0273! 24.0332! 24.0281! 24.0350! 24.0324! 24.0236! 23.9989! 24.0004! 24.0212! 24.0378! 24.0728! 24.0684! 24.0630! 24.0565! Table!A.4!(cont’d)! Time!(sec)! 22.8! 23! 23.2! 23.4! 23.6! 23.8! 24! 24.2! 24.4! 24.6! 24.8! 25! 25.2! 25.4! 25.6! 25.8! 26! 26.2! 26.4! 26.6! 26.8! 27! 27.2! 27.4! 27.6! 27.8! 28! 28.2! 28.4! 28.6! 28.8! 29! 29.2! 29.4! 29.6! 29.8! 30! 30.2! Voltage!(Volt)! 108.2789! 108.2546! 108.2776! 108.2621! 108.2554! 108.1949! 108.1307! 108.1403! 108.1283! 108.1292! 108.1312! 108.1071! 108.1335! 108.1263! 108.1310! 108.1454! 108.1251! 108.1278! 108.1214! 108.1148! 108.1234! 108.1186! 108.1320! 108.1334! 108.1262! 108.1264! 108.1157! 108.1271! 108.1238! 108.1106! 108.1198! 108.1141! 108.1324! 108.1335! 108.1294! 108.1212! 108.1055! 108.1116! Current!(Amp)! 0.2222! 0.2221! 0.2222! 0.2221! 0.2221! 0.2220! 0.2219! 0.2219! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2216! 0.2217! 135 Power!(Watt)! 24.0580! 24.0481! 24.0579! 24.0491! 24.0460! 24.0186! 23.9891! 23.9926! 23.9871! 23.9875! 23.9870! 23.9756! 23.9875! 23.9820! 23.9849! 23.9911! 23.9809! 23.9800! 23.9780! 23.9759! 23.9784! 23.9758! 23.9794! 23.9798! 23.9768! 23.9766! 23.9718! 23.9751! 23.9736! 23.9665! 23.9696! 23.9671! 23.9746! 23.9738! 23.9727! 23.9688! 23.9610! 23.9638! Table!A.4!(cont’d)! Time!(sec)! 30.4! 30.6! 30.8! 31! 31.2! 31.4! 31.6! 31.8! 32! 32.2! 32.4! 32.6! 32.8! 33! 33.2! 33.4! 33.6! 33.8! 34! 34.2! 34.4! 34.6! 34.8! 35! 35.2! 35.4! 35.6! 35.8! 36! 36.2! 36.4! 36.6! 36.8! 37! 37.2! 37.4! 37.6! 37.8! Voltage!(Volt)! 108.1091! 108.1102! 108.1221! 108.1104! 108.1190! 108.1173! 108.1183! 108.1115! 108.0975! 108.1176! 108.1163! 108.1178! 108.1291! 108.1036! 108.1260! 108.1386! 108.1310! 108.0843! 108.0687! 108.1067! 108.1489! 108.1598! 108.1490! 108.1331! 108.1375! 108.1510! 108.1372! 108.1491! 108.1542! 108.1426! 108.1569! 108.1481! 108.1445! 108.1373! 108.1207! 108.1304! 108.1808! 108.1970! Current!(Amp)! 0.2216! 0.2216! 0.2217! 0.2216! 0.2217! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2215! 0.2215! 0.2215! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2215! 0.2215! 0.2216! 0.2217! 136 Power!(Watt)! 23.9605! 23.9614! 23.9665! 23.9617! 23.9648! 23.9619! 23.9624! 23.9585! 23.9516! 23.9610! 23.9595! 23.9600! 23.9644! 23.9523! 23.9624! 23.9675! 23.9619! 23.9424! 23.9347! 23.9508! 23.9694! 23.9723! 23.9674! 23.9614! 23.9627! 23.9678! 23.9621! 23.9665! 23.9677! 23.9613! 23.9676! 23.9634! 23.9615! 23.9588! 23.9504! 23.9530! 23.9759! 23.9838! Table!A.4!(cont’d)! Time!(sec)! 38! 38.2! 38.4! 38.6! 38.8! 39! 39.2! 39.4! 39.6! 39.8! 40! 40.2! 40.4! 40.6! 40.8! 41! ! ! Voltage!(Volt)! 108.1862! 108.1923! 108.1852! 108.1946! 108.1860! 108.2081! 108.2222! 108.2257! 108.2574! 108.2588! 108.2348! 108.2363! 108.2508! 108.2416! 108.2480! 108.2397! Current!(Amp)! 0.2216! 0.2217! 0.2216! 0.2216! 0.2216! 0.2217! 0.2217! 0.2217! 0.2217! 0.2218! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! ! 137 Power!(Watt)! 23.9782! 23.9813! 23.9763! 23.9800! 23.9764! 23.9852! 23.9923! 23.9915! 24.0048! 24.0066! 23.9947! 23.9951! 24.0019! 23.9963! 23.9994! 23.9943! Table!A.5!Data!recorded!from!TPCell!for!sweet!potato!test!5! ! Time!(sec)! Voltage!(Volt)! Current!(Amp)! 0! 105.4658! 0.2181! 0.2! 108.0402! 0.2233! 0.4! 108.0369! 0.2233! 0.6! 108.0612! 0.2233! 0.8! 108.0469! 0.2232! 1! 108.0563! 0.2232! 1.2! 108.0517! 0.2231! 1.4! 108.0411! 0.2231! 1.6! 108.0460! 0.2231! 1.8! 108.0512! 0.2230! 2! 108.0558! 0.2230! 2.2! 108.0398! 0.2229! 2.4! 108.0299! 0.2229! 2.6! 107.9888! 0.2228! 2.8! 107.9874! 0.2227! 3! 108.0118! 0.2228! 3.2! 108.0289! 0.2228! 3.4! 108.0171! 0.2227! 3.6! 108.0163! 0.2227! 3.8! 108.0123! 0.2227! 4! 108.0367! 0.2227! 4.2! 108.0571! 0.2227! 4.4! 108.0701! 0.2227! 4.6! 108.0626! 0.2227! 4.8! 108.0638! 0.2227! 5! 108.0578! 0.2227! 5.2! 108.0456! 0.2226! 5.4! 108.0422! 0.2226! 5.6! 108.0491! 0.2226! 5.8! 108.0606! 0.2226! 6! 108.0588! 0.2226! 6.2! 108.0569! 0.2225! 6.4! 108.0442! 0.2225! 6.6! 108.0711! 0.2225! 6.8! 108.2056! 0.2228! 7! 108.2155! 0.2228! 7.2! 108.2210! 0.2228! 7.4! 108.2112! 0.2228! 138 Power!(Watt)! 23.0017! 24.1301! 24.1214! 24.1278! 24.1149! 24.1151! 24.1088! 24.0994! 24.1000! 24.0972! 24.0956! 24.0856! 24.0793! 24.0571! 24.0532! 24.0613! 24.0664! 24.0587! 24.0556! 24.0511! 24.0611! 24.0666! 24.0697! 24.0657! 24.0635! 24.0599! 24.0518! 24.0490! 24.0483! 24.0513! 24.0492! 24.0469! 24.0398! 24.0504! 24.1074! 24.1101! 24.1109! 24.1057! Table!A.5!(cont’d)! Time!(sec)! 7.6! 7.8! 8! 8.2! 8.4! 8.6! 8.8! 9! 9.2! 9.4! 9.6! 9.8! 10! 10.2! 10.4! 10.6! 10.8! 11! 11.2! 11.4! 11.6! 11.8! 12! 12.2! 12.4! 12.6! 12.8! 13! 13.2! 13.4! 13.6! 13.8! 14! 14.2! 14.4! 14.6! 14.8! 15! Voltage!(Volt)! 108.2000! 108.2028! 108.2140! 108.2000! 108.1947! 108.2005! 108.2037! 108.2061! 108.2120! 108.2183! 108.2183! 108.2156! 108.2236! 108.2209! 108.2134! 108.2225! 108.2331! 108.2318! 108.1991! 108.1880! 108.1911! 108.1868! 108.2350! 108.2435! 108.2485! 108.2492! 108.2588! 108.2677! 108.2464! 108.2518! 108.2475! 108.2608! 108.2643! 108.2576! 108.2517! 108.2234! 108.2332! 108.2528! Current!(Amp)! 0.2227! 0.2227! 0.2227! 0.2227! 0.2227! 0.2227! 0.2227! 0.2226! 0.2226! 0.2226! 0.2226! 0.2226! 0.2226! 0.2226! 0.2226! 0.2226! 0.2226! 0.2226! 0.2225! 0.2225! 0.2225! 0.2224! 0.2225! 0.2225! 0.2225! 0.2225! 0.2225! 0.2226! 0.2225! 0.2225! 0.2225! 0.2225! 0.2225! 0.2225! 0.2224! 0.2224! 0.2224! 0.2224! 139 Power!(Watt)! 24.0990! 24.0979! 24.1023! 24.0923! 24.0905! 24.0924! 24.0919! 24.0913! 24.0929! 24.0943! 24.0917! 24.0911! 24.0919! 24.0907! 24.0860! 24.0888! 24.0909! 24.0874! 24.0741! 24.0681! 24.0680! 24.0647! 24.0857! 24.0878! 24.0895! 24.0885! 24.0917! 24.0953! 24.0836! 24.0856! 24.0831! 24.0872! 24.0880! 24.0846! 24.0797! 24.0667! 24.0700! 24.0762! Table!A.5!(cont’d)! Time!(sec)! 15.2! 15.4! 15.6! 15.8! 16! 16.2! 16.4! 16.6! 16.8! 17! 17.2! 17.4! 17.6! 17.8! 18! 18.2! 18.4! 18.6! 18.8! 19! 19.2! 19.4! 19.6! 19.8! 20! 20.2! 20.4! 20.6! 20.8! 21! 21.2! 21.4! 21.6! 21.8! 22! 22.2! 22.4! 22.6! Voltage!(Volt)! 108.2742! 108.2932! 108.2718! 108.2814! 108.2587! 108.2633! 108.2700! 108.2792! 108.2901! 108.2993! 108.3009! 108.2721! 108.2535! 108.2625! 108.2685! 108.2589! 108.2623! 108.2494! 108.2377! 108.2507! 108.2264! 108.2124! 108.1967! 108.2022! 108.2242! 108.2222! 108.2001! 108.2053! 108.2179! 108.2178! 108.2263! 108.2347! 108.2407! 108.2211! 108.2255! 108.2245! 108.2324! 108.2281! Current!(Amp)! 0.2225! 0.2225! 0.2224! 0.2224! 0.2224! 0.2224! 0.2224! 0.2224! 0.2224! 0.2224! 0.2224! 0.2224! 0.2223! 0.2223! 0.2223! 0.2223! 0.2223! 0.2223! 0.2222! 0.2223! 0.2222! 0.2222! 0.2221! 0.2221! 0.2222! 0.2222! 0.2221! 0.2221! 0.2221! 0.2221! 0.2221! 0.2221! 0.2222! 0.2221! 0.2221! 0.2221! 0.2221! 0.2221! 140 Power!(Watt)! 24.0864! 24.0926! 24.0850! 24.0870! 24.0758! 24.0770! 24.0803! 24.0816! 24.0864! 24.0899! 24.0892! 24.0745! 24.0666! 24.0704! 24.0717! 24.0659! 24.0668! 24.0596! 24.0551! 24.0600! 24.0488! 24.0411! 24.0329! 24.0338! 24.0431! 24.0421! 24.0317! 24.0332! 24.0388! 24.0392! 24.0409! 24.0430! 24.0461! 24.0372! 24.0375! 24.0365! 24.0392! 24.0366! Table!A.5!(cont’d)! Time!(sec)! 22.8! 23! 23.2! 23.4! 23.6! 23.8! 24! 24.2! 24.4! 24.6! 24.8! 25! 25.2! 25.4! 25.6! 25.8! 26! 26.2! 26.4! 26.6! 26.8! 27! 27.2! 27.4! 27.6! 27.8! 28! 28.2! 28.4! 28.6! 28.8! 29! 29.2! 29.4! 29.6! 29.8! 30! 30.2! Voltage!(Volt)! 108.1973! 108.1936! 108.2121! 108.2981! 108.2929! 108.2975! 108.2818! 108.2540! 108.1784! 108.1736! 108.1618! 108.1760! 108.1794! 108.1702! 108.1683! 108.1444! 108.1616! 108.1701! 108.1824! 108.1689! 108.1576! 108.1712! 108.1684! 108.1768! 108.1717! 108.1589! 108.1412! 108.1485! 108.1507! 108.1571! 108.1738! 108.1595! 108.1688! 108.1717! 108.1619! 108.1615! 108.1555! 108.1584! Current!(Amp)! 0.2220! 0.2220! 0.2220! 0.2222! 0.2222! 0.2222! 0.2222! 0.2221! 0.2219! 0.2219! 0.2219! 0.2219! 0.2219! 0.2219! 0.2219! 0.2218! 0.2219! 0.2219! 0.2219! 0.2219! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2217! 0.2217! 141 Power!(Watt)! 24.0218! 24.0208! 24.0274! 24.0649! 24.0620! 24.0639! 24.0555! 24.0422! 24.0083! 24.0063! 24.0007! 24.0056! 24.0061! 24.0024! 23.9998! 23.9894! 23.9972! 23.9994! 24.0042! 23.9979! 23.9919! 23.9973! 23.9948! 23.9990! 23.9958! 23.9893! 23.9822! 23.9850! 23.9838! 23.9870! 23.9953! 23.9875! 23.9908! 23.9914! 23.9872! 23.9862! 23.9829! 23.9831! Table!A.5!(cont’d)! Time!(sec)! 30.4! 30.6! 30.8! 31! 31.2! 31.4! 31.6! 31.8! 32! 32.2! 32.4! 32.6! 32.8! 33! 33.2! 33.4! 33.6! 33.8! 34! 34.2! 34.4! 34.6! 34.8! 35! 35.2! 35.4! 35.6! 35.8! 36! 36.2! 36.4! 36.6! 36.8! 37! 37.2! 37.4! 37.6! 37.8! Voltage!(Volt)! 108.1507! 108.1472! 108.1575! 108.1587! 108.1606! 108.1628! 108.1611! 108.1624! 108.1465! 108.1491! 108.1488! 108.1604! 108.1649! 108.1468! 108.1488! 108.1087! 108.0912! 108.1581! 108.1659! 108.1496! 108.1316! 108.1396! 108.1788! 108.1867! 108.2189! 108.2233! 108.2237! 108.2342! 108.2376! 108.2198! 108.2302! 108.2449! 108.1957! 108.1636! 108.2127! 108.2308! 108.2115! 108.2001! Current!(Amp)! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2216! 0.2216! 0.2215! 0.2216! 0.2217! 0.2216! 0.2216! 0.2216! 0.2217! 0.2217! 0.2217! 0.2218! 0.2218! 0.2218! 0.2218! 0.2217! 0.2218! 0.2218! 0.2217! 0.2216! 0.2217! 0.2217! 0.2217! 0.2217! 142 Power!(Watt)! 23.9799! 23.9771! 23.9814! 23.9817! 23.9821! 23.9836! 23.9806! 23.9803! 23.9750! 23.9736! 23.9744! 23.9783! 23.9795! 23.9712! 23.9710! 23.9530! 23.9456! 23.9731! 23.9777! 23.9689! 23.9599! 23.9654! 23.9813! 23.9848! 23.9967! 23.9985! 23.9991! 24.0031! 24.0056! 23.9968! 24.0007! 24.0065! 23.9834! 23.9691! 23.9920! 23.9982! 23.9894! 23.9835! Table!A.5!(cont’d)! Time!(sec)! 38! 38.2! 38.4! 38.6! 38.8! 39! 39.2! 39.4! 39.6! 39.8! 40! 40.2! 40.4! 40.6! 40.8! 41! 41.2! 41.4! 41.6! ! ! Voltage!(Volt)! 108.1966! 108.1928! 108.1958! 108.2251! 108.2145! 108.2089! 108.2346! 108.2227! 108.2299! 108.2196! 108.2182! 108.2152! 108.2217! 108.1966! 108.2022! 108.1996! 108.2135! 108.1955! 108.2073! Current!(Amp)! 0.2216! 0.2216! 0.2216! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2216! 0.2217! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! 0.2216! ! 143 Power!(Watt)! 23.9810! 23.9795! 23.9805! 23.9933! 23.9874! 23.9852! 23.9962! 23.9897! 23.9921! 23.9879! 23.9866! 23.9857! 23.9881! 23.9777! 23.9788! 23.9761! 23.9823! 23.9741! 23.9777! Table!A.6!Data!recorded!from!TPCell!for!sweet!potato!test!6! ! Time!(sec)! Voltage!(Volt)! Current!(Amp)! 0! 92.7069! 0.1917! 0.2! 107.8255! 0.2229! 0.4! 107.8238! 0.2229! 0.6! 107.8218! 0.2228! 0.8! 107.8279! 0.2228! 1! 107.8352! 0.2227! 1.2! 107.8424! 0.2227! 1.4! 107.8799! 0.2228! 1.6! 107.8697! 0.2227! 1.8! 107.8796! 0.2227! 2! 107.8877! 0.2227! 2.2! 107.8831! 0.2226! 2.4! 107.8716! 0.2226! 2.6! 107.8788! 0.2226! 2.8! 107.8919! 0.2226! 3! 107.8908! 0.2225! 3.2! 107.8916! 0.2225! 3.4! 107.8614! 0.2224! 3.6! 107.8737! 0.2224! 3.8! 107.8780! 0.2224! 4! 107.8810! 0.2224! 4.2! 107.8887! 0.2224! 4.4! 107.8868! 0.2224! 4.6! 107.8915! 0.2224! 4.8! 107.8956! 0.2223! 5! 107.8839! 0.2223! 5.2! 107.8904! 0.2223! 5.4! 107.8770! 0.2223! 5.6! 107.8785! 0.2222! 5.8! 107.8846! 0.2222! 6! 107.8853! 0.2222! 6.2! 107.8895! 0.2222! 6.4! 107.8817! 0.2222! 6.6! 107.8731! 0.2221! 6.8! 107.8856! 0.2222! 7! 107.8777! 0.2221! 7.2! 107.8724! 0.2221! 7.4! 107.8622! 0.2221! 144 Power!(Watt)! 17.7745! 24.0377! 24.0295! 24.0245! 24.0228! 24.0199! 24.0180! 24.0306! 24.0221! 24.0246! 24.0233! 24.0183! 24.0108! 24.0097! 24.0138! 24.0102! 24.0076! 23.9914! 23.9950! 23.9932! 23.9922! 23.9937! 23.9902! 23.9907! 23.9901! 23.9836! 23.9838! 23.9766! 23.9753! 23.9767! 23.9745! 23.9753! 23.9688! 23.9639! 23.9672! 23.9632! 23.9580! 23.9517! Table!A.6!(cont’d)! Time!(sec)! 7.6! 7.8! 8! 8.2! 8.4! 8.6! 8.8! 9! 9.2! 9.4! 9.6! 9.8! 10! 10.2! 10.4! 10.6! 10.8! 11! 11.2! 11.4! 11.6! 11.8! 12! 12.2! 12.4! 12.6! 12.8! 13! 13.2! 13.4! 13.6! 13.8! 14! 14.2! 14.4! 14.6! 14.8! 15! Voltage!(Volt)! 107.8612! 107.8737! 107.8715! 107.8687! 107.8427! 107.8425! 107.8513! 107.8591! 107.8622! 107.8653! 107.8708! 107.8698! 107.8705! 107.8837! 107.8697! 107.8627! 107.8752! 107.8723! 107.8765! 107.8744! 107.8752! 107.8312! 107.8414! 107.8737! 107.8788! 107.8860! 107.8869! 107.8832! 107.8793! 107.8604! 107.8656! 107.8415! 107.8481! 107.8522! 107.8556! 107.8413! 107.8549! 107.8604! Current!(Amp)! 0.2220! 0.2221! 0.2220! 0.2220! 0.2220! 0.2219! 0.2219! 0.2220! 0.2219! 0.2219! 0.2219! 0.2219! 0.2219! 0.2219! 0.2219! 0.2219! 0.2219! 0.2219! 0.2218! 0.2218! 0.2218! 0.2217! 0.2217! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2218! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2217! 0.2216! 0.2216! 0.2216! 145 Power!(Watt)! 23.9502! 23.9548! 23.9518! 23.9488! 23.9367! 23.9346! 23.9370! 23.9396! 23.9392! 23.9391! 23.9413! 23.9393! 23.9376! 23.9423! 23.9358! 23.9299! 23.9358! 23.9319! 23.9319! 23.9317! 23.9296! 23.9100! 23.9130! 23.9261! 23.9266! 23.9297! 23.9285! 23.9255! 23.9238! 23.9143! 23.9152! 23.9040! 23.9057! 23.9063! 23.9067! 23.8992! 23.9058! 23.9064! Table!A.6!(cont’d)! Time!(sec)! 15.2! 15.4! 15.6! 15.8! 16! 16.2! 16.4! 16.6! 16.8! 17! 17.2! 17.4! 17.6! 17.8! 18! 18.2! 18.4! 18.6! 18.8! 19! 19.2! 19.4! 19.6! 19.8! 20! 20.2! 20.4! 20.6! 20.8! 21! 21.2! 21.4! 21.6! 21.8! 22! 22.2! 22.4! 22.6! Voltage!(Volt)! 107.8433! 107.8246! 107.8242! 107.8435! 107.8314! 107.8507! 107.8405! 107.8546! 107.8594! 107.8563! 107.8603! 107.8355! 107.8462! 107.8349! 107.8255! 107.8447! 107.8324! 107.8460! 107.8698! 107.8497! 107.8431! 107.8428! 107.8544! 107.8493! 107.8355! 107.8522! 107.8435! 107.8342! 107.8426! 107.8548! 107.8617! 107.8169! 107.8394! 107.8453! 107.8435! 107.8510! 107.8369! 107.8564! Current!(Amp)! 0.2216! 0.2215! 0.2215! 0.2216! 0.2215! 0.2216! 0.2215! 0.2216! 0.2216! 0.2216! 0.2215! 0.2215! 0.2215! 0.2215! 0.2214! 0.2215! 0.2215! 0.2215! 0.2215! 0.2215! 0.2214! 0.2214! 0.2215! 0.2214! 0.2214! 0.2214! 0.2214! 0.2214! 0.2214! 0.2214! 0.2214! 0.2213! 0.2214! 0.2214! 0.2213! 0.2214! 0.2213! 0.2214! 146 Power!(Watt)! 23.8983! 23.8880! 23.8879! 23.8946! 23.8889! 23.8949! 23.8909! 23.8967! 23.8974! 23.8958! 23.8959! 23.8842! 23.8878! 23.8822! 23.8770! 23.8858! 23.8795! 23.8842! 23.8948! 23.8851! 23.8815! 23.8795! 23.8846! 23.8813! 23.8746! 23.8816! 23.8770! 23.8709! 23.8748! 23.8798! 23.8811! 23.8603! 23.8704! 23.8720! 23.8702! 23.8735! 23.8667! 23.8756! Table!A.6!(cont’d)! Time!(sec)! 22.8! 23! 23.2! 23.4! 23.6! 23.8! 24! 24.2! 24.4! 24.6! 24.8! 25! 25.2! 25.4! 25.6! 25.8! 26! 26.2! 26.4! 26.6! 26.8! 27! 27.2! 27.4! 27.6! 27.8! 28! 28.2! 28.4! 28.6! 28.8! 29! 29.2! 29.4! 29.6! 29.8! 30! 30.2! Voltage!(Volt)! 107.8719! 107.8613! 107.8759! 107.8507! 107.8569! 107.8707! 107.8486! 107.8764! 107.8616! 107.8736! 107.8836! 107.8598! 107.8783! 107.8649! 107.8589! 107.8845! 107.8673! 107.8743! 107.8686! 107.8626! 107.8716! 107.8609! 107.8750! 107.8506! 107.8577! 107.8591! 107.8476! 107.8348! 107.8358! 107.8439! 107.8630! 107.8470! 107.8396! 107.8478! 107.8696! 107.8768! 107.8744! 107.8932! Current!(Amp)! 0.2214! 0.2213! 0.2214! 0.2213! 0.2213! 0.2214! 0.2213! 0.2213! 0.2213! 0.2213! 0.2213! 0.2213! 0.2213! 0.2213! 0.2213! 0.2213! 0.2213! 0.2213! 0.2213! 0.2212! 0.2213! 0.2212! 0.2213! 0.2212! 0.2212! 0.2212! 0.2212! 0.2211! 0.2212! 0.2212! 0.2212! 0.2212! 0.2211! 0.2212! 0.2212! 0.2212! 0.2212! 0.2212! 147 Power!(Watt)! 23.8811! 23.8750! 23.8805! 23.8691! 23.8720! 23.8775! 23.8662! 23.8773! 23.8711! 23.8768! 23.8795! 23.8680! 23.8766! 23.8708! 23.8653! 23.8771! 23.8689! 23.8723! 23.8684! 23.8645! 23.8688! 23.8627! 23.8692! 23.8573! 23.8588! 23.8607! 23.8543! 23.8473! 23.8480! 23.8514! 23.8594! 23.8522! 23.8470! 23.8505! 23.8593! 23.8618! 23.8599! 23.8684! Table!A.6!(cont’d)! Time!(sec)! 30.4! 30.6! 30.8! 31! 31.2! 31.4! 31.6! 31.8! 32! 32.2! 32.4! 32.6! 32.8! 33! 33.2! 33.4! 33.6! 33.8! 34! 34.2! 34.4! 34.6! 34.8! 35! 35.2! 35.4! 35.6! 35.8! 36! 36.2! 36.4! 36.6! 36.8! 37! 37.2! 37.4! 37.6! 37.8! Voltage!(Volt)! 107.8575! 107.8395! 107.8739! 107.8611! 107.8840! 107.8655! 107.8742! 107.8755! 107.8517! 107.8783! 107.8448! 107.8380! 107.8470! 107.8607! 107.8729! 107.8443! 107.8372! 107.8609! 107.8739! 107.8739! 107.8547! 107.8609! 107.8743! 107.8548! 107.8710! 107.9024! 107.8774! 107.8886! 107.8713! 107.8742! 107.8815! 107.8763! 107.8851! 107.8829! 107.8793! 107.8682! 107.8730! 107.8869! Current!(Amp)! 0.2211! 0.2211! 0.2212! 0.2211! 0.2212! 0.2211! 0.2212! 0.2212! 0.2211! 0.2211! 0.2211! 0.2211! 0.2211! 0.2211! 0.2211! 0.2210! 0.2210! 0.2211! 0.2211! 0.2211! 0.2211! 0.2211! 0.2211! 0.2210! 0.2211! 0.2211! 0.2211! 0.2211! 0.2211! 0.2211! 0.2211! 0.2211! 0.2211! 0.2210! 0.2210! 0.2210! 0.2210! 0.2210! 148 Power!(Watt)! 23.8525! 23.8441! 23.8585! 23.8523! 23.8619! 23.8533! 23.8572! 23.8576! 23.8466! 23.8564! 23.8424! 23.8376! 23.8421! 23.8461! 23.8523! 23.8389! 23.8353! 23.8457! 23.8505! 23.8514! 23.8424! 23.8442! 23.8493! 23.8403! 23.8474! 23.8596! 23.8495! 23.8527! 23.8455! 23.8463! 23.8481! 23.8465! 23.8491! 23.8473! 23.8454! 23.8407! 23.8422! 23.8483! Table!A.6!(cont’d)! Time!(sec)! 38! 38.2! 38.4! 38.6! 38.8! 39! 39.2! 39.4! 39.6! 39.8! 40! 40.2! 40.4! 40.6! 40.8! 41! 41.2! 41.4! 41.6! 41.8! ! ! Voltage!(Volt)! 107.8681! 107.8684! 107.8512! 107.8829! 107.8949! 107.8902! 107.8725! 107.8878! 107.8867! 107.8906! 107.8814! 107.8830! 107.8137! 107.6953! 107.7478! 107.7984! 107.8438! 107.8550! 107.8462! 107.8688! Current!(Amp)! 0.2210! 0.2210! 0.2210! 0.2210! 0.2210! 0.2210! 0.2210! 0.2210! 0.2210! 0.2210! 0.2210! 0.2210! 0.2209! 0.2206! 0.2207! 0.2208! 0.2209! 0.2209! 0.2209! 0.2209! ! 149 Power!(Watt)! 23.8386! 23.8391! 23.8316! 23.8451! 23.8495! 23.8471! 23.8394! 23.8458! 23.8441! 23.8456! 23.8407! 23.8425! 23.8108! 23.7582! 23.7816! 23.8030! 23.8219! 23.8269! 23.8227! 23.8307! REFERENCES 150 REFERENCES' Beck! JV.! 1970.! Nonlinear! estimation! applied! to! the! nonlinear! inverse! heat! conduction! problem.! International! Journal! of! Heat! and! Mass! Transfer! 13(4):703J16.! Beck!JV,!Arnold!KJ.!1977a.!Parameter!Estimation. New!York:!Wiley.! Beck! JV,! Arnold! KJ.! 1977b.! Parameter! estimation! in! engineering! and! science. New! York:!Wiley.! Betta! G,! Rinaldi! M,! Barbanti! D,! Massini! R.! 2009.! A! quick! method! for! thermal! diffusivity! estimation:! Application! to! several! foods.! Journal! of! Food! Engineering!91(1):34J41.! Bristow! KL,! Kluitenberg! GJ,! Horton! R.! 1994a.! Measurement! of! Soil! Thermal! Properties! with! a! DualJProbe! HeatJPulse! Technique.! Soil! Sci.! Soc.! Am.! J.! 58(5):1288J94.! Bristow! KL,! Kluitenberg! GJ,! Horton! R.! 1994b.! Measurement! of! soil! thermal! properties! with! a! dualJprobe! heatJpulse! technique.! Soil! science! society! of! America!Journal!58(5):1288J94.! COMSOL.! 2012.! COMSOL! Multiphysics.' 42a! ed:! COMSOL! Inc.,! Burlington,! Massachusetts,!United!States.! de! Monte! F,! Beck! JV,! Amos! DE.! 2008.! Diffusion! of! thermal! disturbances! in! twoJ dimensional! Cartesian! transient! heat! conduction.! International! Journal! of! Heat!and!Mass!Transfer!51(25J26):5931J41.! Denys!S,!Hendrickx!ME.!1999.!Measurement!of!the!thermal!conductivity!of!foods!at! high!pressure.!J!Food!Sci!64(4):709J13.! Dolan! K,! Valdramidis! V,! Mishra! D.! 2012.! Parameter! estimation! for! dynamic! microbial!inactivation;!which!model,!which!precision?!Food!Control.! Gratzek! JP,! Toledo! RT.! 1993.! Solid! Food! ThermalJConductivity! Determination! at! HighJTemperatures.!J!Food!Sci!58(4):908J13.! 151 Halliday!PJ,!Parker!R,!Smith!AC,!Steer!DC.!1995.!The!thermal!conductivity!of!maize! grits!and!potato!granules.!J!Food!Eng!26(3):273J88.! Mariani! VC,! Do! Amarante! ÁCC,! Dos! Santos! Coelho! L.! 2009.! Estimation! of! apparent! thermal!conductivity!of!carrot!purée!during!freezing!using!inverse!problem.! International!Journal!of!Food!Science!&!Technology!44(7):1292J303.! Mishra!DK,!Dolan!KD,!Yang!L.!2008.!Confidence!intervals!for!modeling!anthocyanin! retention! in! grape! pomace! during! nonisothermal! heating.! J! Food! Sci! 73(1):E9JE15.! Mishra! DK,! Dolan! KD,! Yang! L.! 2011.! Bootstrap! confidence! intervals! for! the! kinetic! parameters!of!degradation!of!anthocyanins!in!grape!pomace.!Journal!of!Food! Process!Engineering!34(4):1220J33.! Mohamed! IO.! 2009.! Simultaneous! estimation! of! thermal! conductivity! and! volumetric! heat! capacity! for! solid! foods! using! sequential! parameter! estimation!technique.!Food!Res!Int!42(2):231J6.! Mohamed! IO.! 2010.! Development! of! a! simple! and! robust! inverse! method! for! determination!of!thermal!diffusivity!of!solid!foods.!J!Food!Eng!101(1):1J7.! Monteau!JJY.!2008.!Estimation!of!thermal!conductivity!of!sandwich!bread!using!an! inverse!method.!J!Food!Eng!85(1):132J40.! Murakami! EG,! Sweat! VE,! Sastry! SK,! Kolbe! E,! Hayakawa! K,! Datta! A.! 1996.! Recommended! design! parameters! for! thermal! conductivity! probes! for! nonfrozen!food!materials.!J!Food!Eng!27(2):109J23.! Nahor!HB,!Scheerlinck!N,!Verniest!R,!De!Baerdemaeker!J,!Nicolai!BM.!2001.!Optimal! experimental! design! for! the! parameter! estimation! of! conduction! heated! foods.!J!Food!Eng!48(2):109J19.! Roy!CJ.!2005.!Review!of!code!and!solution!verification!procedures!for!computational! simulation.!Journal!of!Computational!Physics!205(1):131J56.! Salari!K,!Knupp!P.!2000.!Code!Verification!by!the!Method!of!Manufactured!Solutions.! Other!Information:!PBD:!1!Jun!2000.!p.!Medium:!P;!Size:!124!pages.! Scheerlinck!N,!Berhane!NH,!Moles!CG,!Banga!JR,!Nicolai!BM.!2008.!Optimal!dynamic! heat! generation! profiles! for! simultaneous! estimation! of! thermal! food! 152 properties! using! a! hotwire! probe:! Computation,! implementation! and! validation.!Journal!of!Food!Engineering!84(2):297J306.! ShariatyJNiassar!M,!Hozawa!M,!Tsukada!T.!2000.!Development!of!probe!for!thermal! conductivity! measurement! of! food! materials! under! heated! and! pressurized! conditions.!J!Food!Eng!43(3):133J9.! USDA!National!Nutrient!Database!for!Standard!Reference,!Release!26.!Nutrient!Data! Laboratory!Home!Page,!http://www.ars.usda.gov/ba/bhnrc/ndl.!2013.! Yang! CJy.! 1999.! Estimation! of! the! temperatureJdependent! thermal! conductivity! in! inverse! heat! conduction! problems.! Applied! Mathematical! Modelling! 23(6):469J78.! Yang! CJy.! 2000a.! Determination! of! the! temperature! dependent! thermophysical! properties! from! temperature! responses! measured! at! medium's! boundaries.! International!Journal!of!Heat!and!Mass!Transfer!43(7):1261J70.! Zhu! S,! Ramaswamy! HS,! Marcotte! M,! Chen! C,! Shao! Y,! Le! Bail! A.! 2007.! Evaluation! of! thermal! properties! of! food! materials! at! high! pressures! using! a! dualJneedle! lineJheatJsource!method.!J!Food!Sci!72(2):E49JE56.! Zueco! J,! Alhama! F,! González! Fernández! CF.! 2004.! Inverse! determination! of! the! specific!heat!of!foods.!J!Food!Eng!64(3):347J53.! ! ! 153 Chapter'5 ' Nutritional'Values'Of'The'Food'Material'And'Comparison'Of'Degradation'In' Aseptic'And'Conventional'Thermal'Processing' ! Abstract' ! Thermal! degradation! of! ascorbic! acid! and! thiamin! was! studied! in! different! processing!conditions.!A!sample!food!matrix,!sweet!potato!puree,!was!chosen!for!the! study!and!was!fortified!with!ascorbic!acid!and!thiamin.!The!objective!of!this!study! was!to!develop!a!methodology!to!determine!the!product!quality!using!mathematical! modeling!technique!coupled!with!optimal!experimental!design!based!on!a!chemical! agent’s!kinetic!behavior.!Experiments!for!retort!processing!were!performed!using!a! water! immersion! still! retort.! Several! timeJtemperature! combinations,! for! a! particular! lethality! value! were! considered! for! retort! and! aseptic! processing.! The! time! and! temperature! combination! in! the! aseptic! system! was! obtained! by! varying! the! flow! rate! of! the! product! and! temperature! setting! of! the! coiled! heater.! TemperatureJdependent! thermal! properties! were! used! to! predict! temperature! inside! the! product.! The! timeJtemperature! history! of! the! product! was! then! used! in! the! kinetic! model! to! estimate! the! kinetic! parameters! by! minimizing! the! sum! of! squares!of!measured!and!predicted!nutrient!degradation.!A!robust!model!may!help! to! optimize! the! food! processing! by! simulating! the! experimental! conditions! on! computer!rather!than!doing!numerous!experiments!in!lab!or!pilot!plant.!For!retort! processing,! different! processing! times! at! 121.1oC! were! used.! Quantification! of! vitamin! C! and! thiamin! was! performed! with! high! performance! liquid! 154 chromatography!system.!Degradation!kinetics!of!the!nutrients!were!analyzed!using! temperatureJdependent! thermal! properties! of! the! sweet! potato! puree.! Kinetic! parameters! for! the! nutrient! retention! were! estimated! to! compute! the! nutrient! retention!in!retort!and!aseptic!processing!conditions.!The!reaction!rate!for!ascorbic! acid!in!aseptic!processing!and!retort!processing!was!0.0073!minJ1!and!0.0114!minJ1! at!a!reference!temperature!of!127oC,!respectively.!The!activation!energy!for!ascorbic! acid!in!aseptic!processing!and!retort!processing!was!26.62!KJ/gJmol!and!3.43!KJ/gJ mol,! respectively.! The! kinetic! parameter! of! thiamin! could! not! be! estimated! due! to! insufficient!degradation!in!aseptic!as!well!as!in!retort!processing.!! 5.1'Introduction' The! most! important! and! widely! used! food! preservation! method! is! thermal! processing.! Quality! of! processed! food! is! considered! a! key! factor! from! a! nutritional! point! of! view.! However,! due! to! microbiological! safety! reasons! quality! is! often! compromised.!OverJheating!and!underJheating!are!two!major!concerns!for!the!food! industry.!Due!to!food!safety!concerns!and!limitations!of!the!proper!control!systems,! the!food!industry!has!a!tendency!to!overJprocessing!the!products.!To!ensure!a!safe! product,!the!most!heatJresistant!pathogenic!or!spoilage!organism!that!will!grow!at! expected!storage!temperatures!must!be!taken!into!account.!This!organism!might!be! Clostridium'botulinum!in!the!lowJacid!food.!The!thermal!process!is!designed!in!such! a!way!that!probability!of!survival!of!C.'botulinum!in!lowJacid!food!is!no!higher!than! one! in! 1012! cans.! Validation! of! the! thermal! process! is! normally! done! with! 155 thermocouples!inserted!into!the!coldest!spot!of!the!package!containing!the!food.!The! timeJtemperature! history! of! the! product! can! be! used! to! calculate! the! lethality! received!by!the!product!using!standard!calculation!methods.!If!the!thermal!process! is! more! conservative! in! nature,! then! it! tends! to! overJprocess! the! product.! OverJ processing!of!the!food!product!leads!to!the!loss!of!vital!nutrients!and!lower!overall! quality!of!the!product,!whereas!underJheating!the!product!leads!to!product!spoilage! due! to! microbial! growth! and! poses! health! concern! for! the! consumers.! Hence,! it! is! important! to! optimize! the! processing! conditions! to! achieve! the! highest! nutritional! retention!and!keep!the!product!safe!from!microorganism!growth!at!the!same!time.!! The! effect! of! thermal! processing! on! nutrient! degradation! can! be! defined! by! the!kinetic!parameters!in!the!mathematical!model.!The!timeJtemperature!history!of! the!product!can!be!used!to!model!the!process!and!estimate!the!kinetic!parameters!in! the!model.!Thermal!parameters,!such!as!thermal!conductivity!and!heat!capacity,!are! important!for!simulating!the!timeJtemperature!history!of!the!product.!Even!though! the! thermal! processes! are! dynamic! and! cover! a! large! temperature! range! (20oC! –! 140oC),! the! thermal! properties! are! often! considered! constant! in! modeling! the! kinetic!parameters.!Thermal!properties!of!food!vary!considerably!with!temperature! (Dolan! and! Mishra! 2013).! ! The! rate! of! quality! change! varies! exponentially! with! temperature.! Hence,! a! thermal! process! designed! to! be! run! at! 140oC,! may! have! significant! impact! on! quality! if! the! thermal! properties! used! in! the! model! are! measured!at!room!temperature!(20oC)(Gratzek!and!Toledo!1993).!! 156 Food! processing! systems! are! dynamic! in! nature,! meaning! the! processing! temperature! will! be! varying! with! time.! Hence,! performing! experiments! at! isothermal!temperature!is!usually!a!simplified!way!of!generating!data!in!lab!settings.! However,!isothermal!experiments!do!not!truly!replicate!the!dynamic!behavior,!and! the! kinetic! parameters! obtained! from! isothermal! experiments! may! be! misleading! (Levieux! and! others! 2007;! Banga! and! others! 2003).! The! nonisothermal! method! of! estimating! the! kinetic! parameters! has! advantages! over! the! isothermal! method! (Banga!and!others!2003).!The!nonisothermal!method!simulates!the!actual!thermal! process!rather!than!doing!experiments!that!are!different!than!the!actual!process!and! isothermal!in!nature!(Margarida!C.!Vieira!2001;!Dolan!and!others!2007;!Banga!and! others!2003;!Cohen!and!others!1994;!Cohen!and!Saguy!1985).!Estimation!of!kinetic! parameters! has! been! studied! in! literature! using! the! nonisothermal! method! by! performing! experiments! in! a! retort! (Mishra! and! others! 2008;! Dolan! and! others! 2007;!Nasri!and!others!1993).!However,!there!is!little!research!done!on!evaluation! of!kinetic!parameters!in!aseptic!system!experiments.!!Cohen!and!Saguy!(1985)!and! Cohen!and!others!(1994)!proposed!a!method!to!estimate!the!kinetic!parameters!for! continuous! thermal! processing! of! grapefruit! juice.! Paired! Equivalent! Isothermal! Exposures! ! (PEIE)! method! was! used! to! evaluate! the! kinetic! parameters! in! continuous! flow! system! (Margarida! C.! Vieira! 2001).! A! review! on! nutritional! comparison! was! carried! out! for! Vitamin! B,! C! and! phenolic! compounds! for! fresh,! frozen! and! canned! fruits! and! vegetables! (Rickman! and! others! 2007).! In! another! study,!optimal!experimental!design!was!used!for!estimating!the!kinetic!parameters! 157 in! the! Arrhenius! model! with! a! linearly! increasing! temperature! profile! (Cunha! and! Oliveira!2000).!! Retention! of! thiamin! was! measured! for! various! retort! temperatures.! The! losses! were! 7.4! J! 55.2%! for! a! temperature! range! of! 104! –! 121.1oC,! respectively! (Ariahu!and!Ogunsua!1999).!However,!kinetic!parameter!estimation!was!not!done!in! the!paper.!Kinetics!of!degradation!of!vitamin!C!in!peas!sealed!and!retorted!in!large! cans!was!studied!(M.!A.!Rao!1981).!Degradation!kinetics!of!vitamin!C!in!orange!juice! was! studied! under! conventional,! ohmic! and! microwave! heating! processes.! Isothermal! experiments! were! performed! and! it! was! found! that! ohmic! heating! retained!maximum!amount!of!vitamin!C!(Vikram!and!others!2005).!There!are!very! few!studies!in!literature!that!have!been!done!on!the!optimal!experimental!design!for! kinetic! evaluation! of! food! in! aseptic! processing.! Modeling! of! the! kinetics! based! on! temperatureJdependent! thermal! properties! also! was! not! found! in! the! literature! surveyed.!Application!of!the!optimal!design!to!obtain!the!best!times!for!experiment! for! accurate! estimation! of! parameters! is! very! limited! in! food! engineering! area.! DJ optimal! design! was! used! for! the! kinetic! parameter! estimation! of! thermal! degradation!kinetics!of!ascorbic!acid!at!low!water!contents!(Frías!and!others!1998).! ! Very! few! studies! are! available! comparing! the! nutritional! degradation/retention! of! retorted! and! aseptically! processed! foods.! A! comprehensive! study! of! optimum! nutrient! retention! comparison! in! aseptic! and! conventional!processing!is!limited!in!the!literature.!The!objective!of!this!study!is!to! compare! the! heat! stability! of! heat! liable! compounds! in! aseptic! and! conventional! 158 processing! and! to! compare! the! degradation! kinetics! in! those! processes.! The! nutrients! selected! for! this! study! were! ascorbic! acid! and! thiamin! mixed! in! a! food! matrix!of!sweet!potato!puree.!! 5.2'Materials'and'Methods'' 5.2.1'Sample'preparation' ' Sweet! potato! puree! (Yamco! LLC,! NC)! was! used! for! aseptic! and! retort! experiments.! Nutritional! fortification! of! vitamin! C! and! thiamin! was! done! with! premixed!fortified!samples!(fortitech®,!NY).!Premix!was!mixed!adequately!with!the! sweet!potato!puree!in!a!food!mixer.!A!4!oz!glass!jar!was!used!for!retort!processing.! Also,!for!the!aseptic!processing,!4!oz!jars!were!filled!with!the!product.!The!samples! were!stored!at!refrigerated!temperature!before!analysis.! ! ! 5.2.2'Aseptic'and'retort'trials' Several!timeJtemperature!combinations,!for!a!particular!Fo!value,!were!used! for! retort! and! aseptic! processing.! Experiments! for! the! retort! processing! were! performed!using!a!vertical!water!immersion!still!retort,!as!shown!in!a!schematic!in! Figure! 5.1.! The! retort! temperature! profile! was! selected! to! represent! commercial! processing! food! processing! for! a! minimum! F0! value! of! 6! min.! Other! timeJ temperature! combinations! were! designed! to! be! lower! than! F0! =6! min.! Jars! filled! with!a!mixed!sample!of!sweet!potato!and!vitamin!premix!were!put!on!the!rack!and! lowered!in!the!retort.!The!test!was!started!as!soon!as!the!retort!was!filled!with!cold! water!and!door!was!locked.!After!the!preJselected!test!condition!was!achieved,!the! 159 retort! was! cooled! down! below! 25oC! and! samples! were! collected! and! stored! at! refrigerated!condition!(2oC)!until!further!analysis.! !! The! experimental! runs! for! the! aseptic! process! were! performed! on! Microthermics®! equipment! as! shown! in! Figure! 5.2.! The! equipment! consists! of! a! positive! displacement! pump,! two! heaters,! a! hold! tube! and! two! coolers.! A! steam! controller!controls!the!heater’s!hot!water!set!point!of!the!aseptic!system.!Flow!rate! can! be! adjusted! with! the! positive! displacement! pump.! Test! conditions! were! achieved! with! the! flow! rate! setting! and! the! heater! temperature! set! points.! A! data! recorder!(NI!9213,!National!Instruments)!was!attached!to!the!equipment!to!record! experimental! temperature! at! the! end! of! each! unit.! Temperature! profiles! were! selected! to! represent! commercial! aseptic! processing.! The! cooled! product! after! the! coolers!was!filled!in!4Joz!glass!jars!and!stored!at!refrigerated!condition!(2oC)!until! further!analysis.! 160 F E D E B A.#Steam B.#Water C.#Drain#Overflow D.#Vents,#Bleeders E.#Air F.#Safety#Valves,##### Pressure&Relief& Valves ! F C D C B D Manual&Valves: !!!!!!!!!!!!! Close ! ! Gate A Figure!5.1!Vertical!still!water!immersion!retort! ! 161 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! Flow% Control' Valve ! ! ! ! Hot$ Water& Temperature Generato r Pressure Chill% Water Steam& Controls ! ! ! Product( Outlet ! Flow% Control' Valve Back% Pressure& Valve ! ! Figure!5.2!Microthermics!equipment!for!aseptic!processing! ! ! Product( Instrumentation Cooler ! ! ! Pump Hold%Tubes Heater ! Dual% Product( Inlet Preheater Jumper' Panel 162! 5.2.3%Analytical%methods% Quantification* of* vitamin* C* and* thiamin* was* performed* with* a* high* performance* liquid* chromatography* system,* using* a* rapid* detection* method* for* vitamin* C* * (Furusawa* 2001).* The* method* uses* 10* g* of* sample* and* 70* ml* of* metaC phosphoric* acid.* The* sample* was* then* left* for* 5* min.* The* HPLC* was* fitted* with* a* column* (waters* carbohydrate,* 125* Å,* 10* μm,* 3.9* ** 300* mm)* and* a* guard* column* (Bondpak*AZ/Corasil,*37C50*μm)*at*ambient*temperature.*The*injection*volume*was* 20* μl.* The* mobil* phase* was* 80:20* ACN:phosphate* buffer,* flow* rate* of* 2.5* ml/min* with*a*UV@*254*nm*detector.*The*run*time*was*approximately*10*minutes.* Thiamin*was*also*quantified*with*HPLC*method*(Pinto*and*others*2002).*The* injection*volume*was*50*μl*through*a*column*(Spherisorb*ODS*5*μm,*4.6*X*250*nm* with*a*flow*rate*of*1.5*ml/minute.*The*run*time*was*approximately*10*minutes.*** 5.3%Mathematical%model% 5.3.1%Sterilization%value% The*general*method*for*the*lethal*rate*(Bigelow*and*others*1920)*forms*the* basis*for*modern*thermal*process*calculations.*In*order*to*calculate*the*process*time* for*a*product,*thermal*death*time*(F)*should*be*known*at*all*temperatures*to*which* the*product*has*been*exposed.*The*equation*for*the*thermal*death*time*as*a*function* of*temperature*can*be*written*in*the*form*of*F*notation*as,* * 163 * Fr F ( 1 z )(T-Tr ) * =10 (5.1)* Thermal*death*time*Fr*is*calculated*at*a*reference*temperature*Tr.*The*ratio*Fr/F*is* called*lethal*rate*and*is*expressed*as*L*(Ball*1923).** * ( 1 )(T-Tr ) * L=10 z (5.2)* L*is*calculated*at*each*temperature*T*and*lethal*rate*can*be*plotted*against.*The*total* area*under*the*lethal*rate*curve*can*be*obtained*by*integrating*the*curve*over*the* timeCtemperature*history*and*is*referred*as*lethality*(F)*(Patashnik*1953).*F*can*be* expressed*as:* * t ( 1 z )(T-Tr )dt * F = ∫ 10 (5.3)* 0 The*process*lethality*calculated*based*on*equation*(5.3)*must*match*the*anticipated* lethality*for*the*specific*product*in*order*to*determine*the*process*time.* 5.3.2%Thermal%degradation%kinetics% Thermal* destruction* kinetics* is* important* to* characterize* the* thermal* process* for* a* specific* product.* Kinetics* can* be* defined* as* the* study* of* rate* of* reaction,* which* varies* with* several* factors,* such* as* moisture,* pH,* temperature,* concentration*and*other*processing*factors.*The*reaction*rate*equation*is*given*as:! 164 − * dC = kC n * dt (5.4)* The*relationship*between*k*and*temperature*is*generally*modeled*by*the*Arrhenius* Equation:! * k = kr e − Ea ⎛ 1 1 ⎞ − R ⎜ T Tr ⎟ ⎝ ⎠ * (5.5)* The*use*of*a*reference*temperature*Tr*ensures*that*the*correlation*between*kr*and* Ea* is* not* 1.0* or* C1.0.* Tr* can* arbitrarily* be* set* to* an* average* value* of* temperature* range*used*in*experiments*(Van*Boekel*1996).*Alternatively,*the*value*of*Tr*can*be* optimized* by* using* an* inverse* problem* to* get* a* best* estimate* of* Tr* (Schwaab* and* Pinto*2007).! Most* reactions* in* food,* such* as* nutrient* retention,* quality* factors* and* microorganism* destruction* follow* firstCorder* reaction* kinetics.* Hence,* for* a* first* order*(n*=*1)*reaction,*Eq.*(5.4)*can*be*written*as:* * −k ψ C Co = e r * (5.6)* Where*Ψ*is*the*time*temperature*history*and*is*the*integrated*value*of*temperature* T(t)*over*the*entire*time*domain.** 165 − Ea ⎛ 1 1⎞ ⎜ T (t) − T ⎟ t R ⎝ r ⎠ dt * ψ = ∫0 e g * (5.7)* Retention*can*be*calculated*for*any*product*by*using*Eq.*(5.6)*provided*that* the*kinetic*parameters*for*particular*compound*(kr*and*Ea),*and*timeCtemperature* history* are* known.* TimeCtemperature* history* was* calculated* with* COMSOL®* (COMSOL* 2012)* and* MATLAB®* (MATLAB* 2012)* using* temperatureCdependent* thermal*properties.*TPCell*measured*temperatureCdependent*thermal*conductivity,* while* heat* capacity* was* measure* by* a* Differential* Scanning* Calorimeter.* Microthermics*experiments*data*were*analyzed*using*Eq.*(5.6)*and(5.7).* For*kinetic*analysis*of*retort*data,*the*Arrhenius*model*was*considered.**Rate* constant* and* activation* energy* were* estimated* using* the* mathematical* model* (Mishra* and* others* 2008)* for* degradation* in* a* can.* The* mass* average* retention* of* nutrients*can*be*described*by*Eq.*(5.8).* * ⎛ ⎜ ⎝ ⌠⌠ ⎮⎮ ⎮⎮ ⎞ C =2 ⎮⎮ e Co ⎟ ⎠ pred ⎮⎮ ⎮⎮ ⎮⎮ ⌡⌡ ⌠t ⎮ -k ⎮ e r⎮ ⎮ ⌡0 -Ea ⎛ 1 1⎞ - ⎟ Rg ⎜ T (r, z ',t) Tr ⎠ ⎝ dt r drdz * (5.8)* Sensitivity* coefficients* were* calculated* for* all* the* parameters* in* the* model* (Beck*1977).*The*scaled*sensitivity*coefficient*is*defined*as:* 166 ∂C ˆ ** βi = βi ∂β i * (5.9)* These* scaled* sensitivity* coefficients* will* provide* better* insight* in* deciding* which* parameters* are* more* sensitive* to* variation* in* processing* conditions.* Statistical* indices* were* calculated* to* ascertain* the* accuracy* of* the* estimated* parameters.* Sensitivity* coefficients* are* good* indicators* of* the* identifiability* of* the* several* parameters*occurring*in*the*model*(Beck*and*Arnold*1977c).* Alternatively,*the*scaled*sensitivity*coefficient*can*be*approximated*by*the*finite* difference*method*as:* * βi ∂C C((1+ δ )β i ) − C( β i ) ** ≈ ∂β i δ (5.10)* Where,*d*is*a*very*small*number*such*as*0.001.* 5.4%Results% * The*chromatogram*of*the*vitamin*C*analysis*is*presented*in*Figure*5.3.*The* ascorbic* acid* peak* occurs* at* 4.8* minutes.* The* peak* for* thiamin* occurs* around* 20* seconds,*as*shown*in*Figure*5.4.* 167 0.014 ! 0.012 ! 0.010 ! 0.008 ! 0.006 ! 0.004 ! 0.002 ! Ascorbic(Acid( AU 0.016! !!!!!!2.00$$$$$$$3.00$$$$$$$4.00$$$$$$5.00$$$$$$6.00$$$$$$7.00$$$$$$8.00$$$$$$9.00$$$$10.00 Minutes 220 ! 200 ! 160 ! 140 ! 120 ! 100 ! 80 ! 40 ! 20 ! Thiamine AU* Figure*5.3*Chromatogram*for*HPLC*analysis*of*vitamin*C*as*ascorbic*acid* !!0.00#####0.10#######0.20######0.30#####0.40######0.50######0.60######0.70######0.80#####0.90####1.00 Minutes* * Figure*5.4*Chromatogram*for*HPLC*analysis*of*thiamin* 168 5.4.1%Aseptic%Experiment% Viscosity*of*the*sweet*potato*puree*at*several*temperatures*was*determined* using*rheometer*(RS600,*Haake,*Thermo*Scientific).**Reynolds*number*is*calculated* using*Eq.*(5.11).* ⎛ n 2−n ⎞ D (u ) ρ ⎛ 4n ⎞ n ⎜ ⎟ ** N Re,PL = ⎜ 8n−1 K ⎟ ⎜ 3n + 1⎟ ⎝ ⎠ ⎝ ⎠ * * Where*volumetric*average*velocity*is*given*by,* u= * * (5.11)* V 4Q ** = A π D2 (5.12)* Laminar* flow* of* a* power* law* fluid* exists* in* the* tube* if* ( ) N Re,PL < N Re,PL * critical * ( N Re,PL )critical = 2100 + 875(1− n) ** 169 (5.13)* 3500 Exp Pred Viscosity (cP) 3000 o 22.76 oC 39.20 oC 59.17 oC 70.18 oC 80.13 oC 90.19 C 100.20 oC 110.23 oC 120.20 oC o 130.20 oC 138.18 C 66.56 oC o 22.94 C 2500 2000 1500 1000 500 0 0 100 200 300 400 500 Shear Rate (1/sec) Figure*5.5*Viscosity*of*sweet*potato*puree*measured*at*different*temperatures,*Q*=*2* lpm*and*D*=*0.43*in.* * Table*5.1*Viscosity*of*sweet*potato*puree*and*Reynolds*number* Temp*(oC)* K*(Pa.sn)* 22.76* 39.20* 59.17* 70.18* 80.13* 90.19* 100.20* 110.23* 120.20* 130.20* 138.18* 66.56* 22.94* 28.37* 19.87* 14.65* 10.59* 9.20* 8.74* 8.22* 7.62* 6.88* 5.75* 4.37* 9.63* 17.70* Viscosity,* cP*@* 100/s* 1009.74* 785.41* 634.28* 523.31* 456.91* 413.66* 383.05* 332.20* 281.78* 232.68* 183.00* 385.91* 637.21* n* 0.28* 0.30* 0.32* 0.35* 0.35* 0.34* 0.33* 0.32* 0.31* 0.30* 0.31* 0.30* 0.28* 170 Reynolds* Number* 9.70* 12.53* 15.60* 19.06* 21.83* 24.04* 25.94* 29.79* 35.00* 42.36* 53.96* 25.53* 15.37* Critical* Reynolds* Number* 2848.74* 2795.78* 2752.96* 2695.77* 2693.65* 2713.97* 2720.63* 2749.83* 2778.79* 2784.09* 2768.40* 2789.11* 2842.72* Reynolds* number* of* sweet* potato* puree* is* lower* than* that* of* the* critical* Reynolds* number* (Table* 5.1)* and* hence* the* flow* profile* in* aseptic* system* is* laminar.* For* aseptic* experiments,* the* timeCtemperature* data* for* experimental* design* is* provided* in* Table* 5.2,* along* with* the* vitamin* C* and* thiamin* data.* The* diameter* of* tubes* in* aseptic* system* is* 0.43’* and* a* hold* tube* of* 70* feet* length* was* used* for* all* experiments.* The* velocity* through* the* system* for* 1* lpm* flow* rate* was* 0.58* ft/sec* and* for* 2* lpm* was* 1.17* ft/sec.* The* simulated* and* experimental* time* temperature* profile* is* shown* in* Figure* 5.6.* The* average* value* of* the* unprocessed* sample*of*sweet*potato*vitamin*C*was*809*mg/100g.*The*simulated*profile*was*used* for*kinetic*parameter*estimation.*Sequential*estimation*results*are*shown*in*Table* 5.3.* The* estimated* value* of* rate* constant* was* 0.0114* minC1* and* the* activation* energy*was*12*KJ/mol.*The*standard*errors*for*rate*constant*and*activation*energy* were*0.0011*minC1*and*2.560*KJ/mol,*respectively.*The*reference*temperature*used* in*the*analysis*was*127oC.*The*kinetics*of*thiamin*could*not*be*performed,*as*there* was*not*enough*degradation*(table*5.2)*of*thiamin*even*at*higher*temperatures.* 171 Temp erature ( o C ) 140 TPred 120 TExp 100 80 60 40 20 0 0 50 100 150 time (sec) 200 250 Figure* 5.6* Experimental* and* predicted* temperature* profile* of* sweet* potato* puree* (Test*2)*as*it*goes*through*various*sections*of*aseptic*system.* * * * * 172 Table*5.2*TimeCtemperature*data*for*aseptic*experiment* Test* Raw* 1* 2* 3* 4* 5* 6* 7* 8* 9* 10* 11* Flow*Rate* (lpm)* 0* 0* 0* 1* 1* 1* 1* 1* 1* 1* 1* 1* 1* 1* 1* 1* 1* 1* 2* 2* 2* 2* 2* 2* 2* 2* 2* 2* 2* 2* 2* 2* 2* 2* 2* 2* IT** PreCHeater** Heater** Hold*Tube** Vit*C,** Thiamin,* mg/100g* mg/100g* (oC)* (oC)* (oC)* (oC)* C* C* C* C* 809* 0.356* C* C* C* C* 802* 0.342* C* C* C* C* 816* 0.374* 16.67* 91.94* 122.56* 121.94* 790* 0.220* 16.67* 91.94* 122.56* 121.94* 782* 0.204* 16.67* 91.94* 122.56* 121.94* 784* 0.283* 16.67* 90.94* 124.39* 123.72* 762* 0.274* 16.67* 90.94* 124.39* 123.72* 770* 0.165* 16.67* 90.94* 124.39* 123.72* 765* 0.211* 16.67* 90.56* 127.78* 127.22* 745* 0.270* 16.67* 90.56* 127.78* 127.22* 756* 0.294* 16.67* 90.56* 127.78* 127.22* 755* 0.218* 16.67* 91.11* 128.89* 128.28* 732* 0.267* 16.67* 91.11* 128.89* 128.28* 736* 0.281* 16.67* 91.11* 128.89* 128.28* 748* 0.222* 16.67* 93.44* 134.67* 134.06* 712* 0.276* 16.67* 93.44* 134.67* 134.06* 721* 0.284* 16.67* 93.44* 134.67* 134.06* 720* 0.208* 17.78* 90.56* 107.78* 107.22* 773* 0.315* 17.78* 90.56* 107.78* 107.22* 769* 0.239* 17.78* 90.56* 107.78* 107.22* 767* 0.204* 17.78* 90.94* 111.11* 110.56* 768* 0.198* 17.78* 90.94* 111.11* 110.56* 778* 0.293* 17.78* 90.94* 111.11* 110.56* 783* 0.256* 17.78* 90.11* 117.78* 116.94* 785* 0.316* 17.78* 90.11* 117.78* 116.94* 781* 0.249* 17.78* 90.11* 117.78* 116.94* 779* 0.281* 17.78* 89.44* 122.39* 121.94* 768* 0.204* 17.78* 89.44* 122.39* 121.94* 771* 0.224* 17.78* 89.44* 122.39* 121.94* 770* 0.234* 17.78* 88.06* 129.72* 129.11* 767* 0.243* 17.78* 88.06* 129.72* 129.11* 771* 0.235* 17.78* 88.06* 129.72* 129.11* 761* 0.239* 17.78* 87.11* 134.56* 133.44* 732* 0.232* 17.78* 87.11* 134.56* 133.44* 726* 0.212* 17.78* 87.11* 134.56* 133.44* 714* 0.222* 173 Table 5.2 (cont’d) Test* 12* IT** PreCHeater** Heater** Hold*Tube** Vit*C,** Thiamin,* Flow*Rate* (lpm)* mg/100g* mg/100g* (oC)* (oC)* (oC)* (oC)* 2* 17.78* 88.22* 140.22* 139.56* 686* 0.210* 2* 17.78* 88.22* 140.22* 139.56* 700* 0.183* 2* 17.78* 88.22* 140.22* 139.56* 702* 0.226* * * Table* 5.3* Parameter* estimates* and* statistical* indices* for* kinetic* parameters* of* vitamin*C*degradation*in*aseptic*system* kr,*minC1* Lower* Upper* Final* Standard* Relative* confidence* confidence* Estimates* error* error,*%* level* level* 0.01140* 0.00110* 9.61855* 0.00935* 0.01344* Ea,*J/mol* 26621.93* 2560.21* Parameters* 9.62* 21848.24* 31392.02* Tr,*oC* 127* 127* * Experimental*and*predicted*degradation*of*vitamin*C,*(C/C0)*in*sweet*potato* puree* is* shown* in* Figure* 5.7.* Since* there* were* two* flow* rates* and* several* timeC temperature* combinations,* the* selection* of* xCaxis* was* based* on* Eq.* (5.7).* The* maximum* degradation* was* about* 12%* at* the* hold* tube* temperature* of* 139.56* oC.* This* processing* temperature* was* selected* to* provide* a* F0* value* of* >6* min,* to* simulate*a*commercial*aseptic*process.** * 174 Vitamin C Retention 1.05 Experimental Predicted 1 0.95 0.9 0.85 0.8 0 2 Ψ 4 6 Figure* 5.7* Experimental* and* predicted* degradation* of* vitamin* C* in* aseptically* processed*sweet*potato*puree* 175 o T(exp) - T(pred), C 0.06 0.04 0.02 0 −0.02 −0.04 −0.06 0 2 Ψ 4 Figure*5.8*Residuals*of*vitamin*C*in*aseptic*processing* 176 6 Sequential Parameters 2.5 kr Ea 2 1.5 1 0.5 0 −0.5 0 2 Ψ, min 4 6 Figure*5.9*Sequential*parameter*estimates*of*vitamin*C*in*aseptic*processing* The* residuals* are* plotted* in* Figure* 5.8* and* the* sequential* parameters* estimates*are*plotted*in*Figure*5.9.*The*sequential*parameters*are*normalized*with* the*final*estimate*of*the*parameter.*This*is*because*of*the*different*scales*of**kr*as* compared*to*Ea.*The*sequentially*estimated*parameters*come*to*a*constant*towards* the*end*of*experiment.* 177 Scaled Sensitivity Coeff 0.05 kr 0 Ea −0.05 −0.1 −0.15 −0.2 −0.25 −0.3 −0.35 0 2 Ψ 4 6 Figure* 5.10* Scaled* sensitivity* coefficient* of* kr* and* Ea* in* the* kinetic* degradation* model*of*vitamin*C*in*sweet*potato*puree*processed*in*retort* Scaled*sensitivity*coefficients*are*shown*in*Figure*5.10.*It*can*be*inferred*that* the*magnitude*of*scaled*sensitivity*of*rate*constant*is*low*as*compared*to*the*total* scale,* which* is* 1* unit,* in* this* case.* This* suggests* that* there* would* be* difficulty* in* estimating* this* parameter.* However,* the* magnitude* of* scaled* sensitivity* of* activation*energy*is*comparatively*larger.* 5.4.2%Retort%Experiments% Retention*of*vitamin*C*and*thiamin*for*the*retort*trials*is*presented*in*Table* 5.4.* For* all* the* conditions,* the* retort* temperature* was* 121.67* oC.* The* time* presented* in* Table* 5.4* includes* the* comeCup* time* of* the* retort.* Simulation* of* the* 178 glass* jar* heated* in* the* retort* was* done* using* COMSOL®.* Figure* 5.11* provides* a* simulated*temperature*profile*of*sweet*potato*in*a*glass*jar.*Since*Gauss*points*were* chosen* for* integration* over* space* in* the* glass* jar,* a* simulated* timeCtemperature* profile*at*nine*difference*Gauss*points*(T1!0!T9)*is*shown*in*Figure*5.11.* 179 * Table*5.4*TimeCtemperature*and*vitamin*C*data*for*retort*experiments* Test* Unprocessed* 1* 2* 3* 4* 5* 6* 7* 8* 9* 10* Time*(min)* Retort** Temperature*(oC)* Vit*C** (mg/100g)* Thiamin* (mg/100g)* 0* 0* 0* 14* 14* 14* 18* 18* 18* 22* 22* 22* 26* 26* 26* 30* 30* 30* 34* 34* 34* 38* 38* 38* 46* 46* 46* 52* 52* 52* 52* 52* 52* C* C* C* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 121.67* 886* 908* 812* 857* 879* 884* 800* 796* 805* 780* 768* 764* 747* 742* 730* 742* 751* 724* 707* 727* 711* 704* 664* 676* 664* 652* 671* 642* 651* 648* 646* 649* 620* 0.275* 0.261* 0.26* 0.249* 0.239* 0.285* 0.342* 0.293* 0.31* 0.369* 0.35* 0.3* 0.284* 0.297* 0.274* 0.311* 0.265* 0.246* 0.24* 0.18* 0.283* 0.25* 0.294* 0.204* 0.142* 0.22* 0.264* 0.13* 0.151* 0.166* 0.264* 0.342* 0.374* 180 y ! ! ! 120 ! ! ! 119 ! ! ! 118 ! ! ! 117 ! ! ! 116 ! ! ! 115 z x Figure* 5.11* Simulated* temperature* profile* of* sweet* potato* puree* in* a* glass* jar* processed*in*retort* * * * 181 140 T1 o Temp erature, C 120 T2 100 T3 T4 80 T 5 T6 60 T7 40 T8 T9 20 0 0 1000 2000 Time (sec) 3000 Figure*5.12*Simulated*temperature*profile*of*sweet*potato*puree*at*gauss*points*in*a* glass*jar*processed*in*retort* * Table* 5.5* Parameter* estimates* and* statistical* indices* for* kinetic* parameters* of* vitamin*C*degradation*in*retort* Lower* Upper* Final* Standard* Relative* Parameters* confidence*confidence* Estimates* error* error,*%* level* level* kr,*minC1* Tr,*oC* 0.00660* 0.00029* 4.32443* 0.00615* 0.00706* 88* Ea,*J/*gCmol* 3430.14* 148.51* 4.33* 3188.71* 3664.94* 88* * The* kinetic* parameter* estimates* for* the* degradation* of* vitamin* C* in* sweet* potato*during*processing*in*retort*are*presented*in*Table*5.5.*The*estimated*value*of* 182 rate* constant* was* 0.0066* minC1* and* the* activation* energy* was* 3.43* KJ/gCmol.* The* standard* errors* for* rate* constant* and* activation* energy* were* 0.00029* minC1* and* 0.148*KJ/gCmol,*respectively.*The*reference*temperature*used*in*the*analysis*was*88* oC.* The* confidence* interval* of* parameters* is* also* presented* in* Table* 5.5.* The* experimental* and* predicted* degradation* are* presented* in* Figure* 5.13.* The* maximum*degradation*was*about*30%*and*that*was*at*the*longest*processing*time* of*40*min.*This*was*also*the*commercial*thermal*process*with*a*delivered*lethality* of*6*min*at*the*center*of*the*can.*Residuals*are*plotted*in*Figure*5.14*and*there*is*no* apparent*sign*of*correlation.*Sequentially*estimated*parameters*are*shown*in*Figure* 5.15,* the* parameters* come* to* a* constant* towards* the* end* of* experiment.* Scaled* sensitivity*coefficient,*as*shown*in*Figure*5.16,*is*large*for*kr*as*compared*to*Ea.* 183 Vitamin C Retention 1.05 Experimental Predicted 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0 10 20 Time (min) 30 40 Figure* 5.13* Experimental* and* predicted* degradation* of* vitamin* C* in* retort* processing* 184 o T(exp) - T(pred), C 0.06 0.04 0.02 0 −0.02 −0.04 −0.06 0 10 20 Time (min) 30 40 Figure*5.14*Residuals*of*vitamin*C*degradation*in*retort*experiment* 185 5 kr 4 Ea Parameters 3 2 1 0 −1 −2 −3 0 10 20 Time (min) 30 40 Figure*5.15*Normalized*sequential*parameter*estimates*of*vitamin*C*in*retort* 186 Vitamin C Retention 0.1 kr 0.05 Ea 0 −0.05 −0.1 −0.15 −0.2 −0.25 0 10 20 Time (min) 30 40 Figure* 5.16* Scaled* sensitivity* coefficient* of* kr* and* Ea* in* the* kinetic* degradation* model* * * Thiamin* was* not* sufficiently* degraded* to* perform* kinetic* analysis.* The* optimization*we*discussed*so*far*is*related*to*the*parameter*estimation*problem*in* the* model.* Parameters* in* the* kinetic* model* of* nutrient* kinetics* and* enzyme* inactivation* can* be* estimated* with* optimal* experiments.* This* type* of* optimization* leads* to* the* global* optimization.* The* global* optimization* is* based* on* different* processing*conditions*of*a*processing*system.*The*maximum*retention*of*a*nutrient* can* be* achieved* by* processing* the* product* in* such* a* condition* that* will* allow* the* proper* sterilization* of* the* product* and* well* as* proper* inactivation* of* the* enzyme.* Optimum*conditions*for*the*thermal*processing*of*soy*milk*were*determined*in*the* study*of*(Kwok*and*others*2002).*Optimality*was*achieved*by*using*the*degradation* 187 of* thiamin,* riboflavin,* color* and* flavor* and* the* inactivation* of* trypsin* inhibitor* activity*(TIA).*The*optimal*condition*in*this*study*was*found*to*be*a*single*step*UHT* process,* for* example,* 143oC/60* s,* with* satisfactory* inactivation* of* TIA,* color* and* flavor*in*acceptable*limit*and*retention*of*thiamin*between*90*and*93%.*In*another* study*(Manoj*M.*Nadkarni*1985),*optimal*nutrient*retention*was*determined*by*the* use* of* optimal* control* theory* for* the* conductionCheated* canned* food.* They* found* that* the* rapid* heating* and* rapid* cooling* rates* as* permitted* by* the* process* constraints*was*the*optimal*control,*and*provided*the*maximum*nutrient*retention* for* a* given* reduction* in* microbial* load.* Also,* it* was* recommended* that* only* one* heating*and*cooling*cycle*should*be*used*during*the*sterilization*process*instead*of* several* steps* of* heating* and* cooling.* Similar* results* were* obtained* for* the* conduction* heated* food* in* retortable* pouches* (Yoshimi* Terajima* 1996).* Response* surface*methodology*(RSM)*is*helpful*as*an*initial*study*to*observe*trends.**However* RSM* has* drawbacks* because* of* the* local* and* stationary* nature* of* the* algebraic* models.* ModelCbased* optimization* approach* has* been* developed* in* recent* years* that*accounts*for*the*timeCdependent*robustness*in*the*model*and*hence*has*great* power*to*improve*food*processing*techniques.** The* dynamic* temperature* profile* obtained* from* the* retort* and* the* aseptic* system* will* be* helpful* in* determining* the* maximum* nutrient* retention* in* the* product.* It* will* also* help* in* determining* the* type* of* temperature* profile* that* best* suits*the*maximum*retention.* % % 188 5.5%Conclusions% *Nutritional*studies*were*performed*to*compare*retention*of*vitamins*in*the* aseptic* system* with* conventional* retort* processing.* Vitamin* C* and* thiamin* were* selected* as* model* vitamins.* The* retention* in* the* aseptic* system* was* higher* as* compared* to* the* retort* processing.* The* retention* of* vitamin* C* for* the* commercial* process*(F0*=*6*min)*of*aseptic*system*was*85%*and*for*retort*it*was*70%.*So,*the* aseptic* process* was* higher* in* retention.* TemperatureCdependent* thermal* properties* were* used* in* the* modeling* of* kinetic* of* degradation* of* nutrients.* Modeling* of* the* kinetic* parameters* of* degradation* for* both* vitamins* showed* differences*in*the*kinetic*parameters.*For*the*retort*processing,*the*rate*constant*at* a* reference* temperature* of* 127* oC* was* 0.0073* minC1* and* for* aseptic* processing* it* was*0.0114*minC1.*This*difference*might*be*because*of*the*difference*in*temperature* history* for* both* the* processing* systems.* The* maximum* temperature* in* aseptic* processing*was*140*oC***and*for*retort*processing*was*121.67*oC.*The*degradation*of* thiamin* was* not* enough* to* estimate* the* kinetic* parameters* in* both* systems.* The* aseptic*process*is*better*for*the*retention*of*nutrients*such*as*vitamin*C.** * * 189 REFERENCES 190 REFERENCES% Ariahu* CC,* Ogunsua* OA.* 1999.* Effect* of* thermal* processing* on* thiamine* retention,* inCvitro* protein* digestibility,* essential* amino* acid* composition,* and* sensory* attributes*of*periwinkleCbased*low*acid*foods.*p.*121C34.* Ball*CO.*1923.*Thermal*process*time*for*canned*foods.*bull.*Natl.*Res.*Counsil*7,*Part* 1*(37),*76.* Banga* JR,* BalsaCCanto* E,* Moles* CG,* Alonso* AA.* 2003.* Improving* food* processing* using* modern* optimization* methods.* Trends* in* Food* Science* &* Technology* 14(4):131C44.* Beck*JV,*and*Arnold,*K.J.**.*1977.*Parameter*estimation*in*engineering*and*science:* John*Wiley*and*Sons,*NY.* Beck* JV,* Arnold* KJ.* 1977c.* Parameter* Estimation* in* Engineering* and* Science. New* York:*Wiley.* Bigelow* WC,* Bohart* GS,* Richardson* AC,* Ball* CO.* 1920.* Heat* penetration* in* processing*canned*foods.*National*Canners*Association.*Bull.*No.*16L.* Cohen* E,* Birk* Y,* Mannheim* CH,* Saguy* IS.* 1994.* Kinetic* Parameter* Estimation* for* Quality*Change*during*Continuous*Thermal*Processing*of*Grapefruit*Juice.*p.* 155C8.* Cohen*E,*Saguy*I.*1985.*Statistical*evaluation*of*Arrhenius*model*and*its*applications* in*predicting*of*food*quality*losses.*p.*273.* COMSOL.* 2012.* COMSOL* Multiphysics.! 42a* ed:* COMSOL* Inc.,* Burlington,* Massachusetts,*United*States.* Cunha* LM,* Oliveira* FAR.* 2000.* Optimal* experimental* design* for* estimating* the* kinetic*parameters*of*processes*described*by*the*firstCorder*Arrhenius*model* under* linearly* increasing* temperature* profiles.* Journal* of* Food* Engineering* 46(1):53C60.* Dolan*KD,*Mishra*DK.*2013.*Parameter*Estimation*in*Food*Science.*Annual*review*of* food*science*and*technology*4:401C22.* 191 Dolan* KD,* Yang* L,* Trampel* CP.* 2007.* Nonlinear* regression* technique* to* estimate* kinetic* parameters* and* confidence* intervals* in* unsteadyCstate* conductionC heated*foods.*Journal*of*Food*Engineering*80(2):581C93.* Frías*JM,*Oliveira*JC,*Cunha*LM,*Oliveira*FA.*1998.*Application*of*DCoptimal*design* for* determination* of* the* influence* of* water* content* on* the* thermal* degradation*kinetics*of*ascorbic*acid*at*low*water*contents.*Journal*of*Food* Engineering*38(1):69C85.* Furusawa* N.* 2001.* Rapid* highCperformance* liquid* chromatographic* identification/quantification* of* total* vitamin* C* in* fruit* drinks.* Food* Control* 12(1):27C9.* Gratzek* JP,* Toledo* RT.* 1993.* Solid* Food* ThermalCConductivity* Determination* at* HighCTemperatures.*J*Food*Sci*58(4):908C13.* Kwok* KCC,* Liang* HCH,* Niranjan* K.* 2002.* Optimizing* Conditions* for* Thermal* Processes*of*Soy*Milk.*p.*4834C8.* Levieux* D,* Geneix* N,* Levieux* A.* 2007.* InactivationCdenaturation* kinetics* of* bovine* milk* alkaline* phosphatase* during* mild* heating* as* determined* by* using* a* monoclonal* antibodyCbased* immunoassay.* Journal* of* Dairy* Research* 74(03):296C301.* M.* A.* Rao* CYLJKHJC.* 1981.* A* Kinetic* Study* of* the* Loss* of* Vitamin* C,* Color,* and* Firmness*During*Thermal*Processing*of*Canned*Peas.*p.*636C7.* Manoj* M.* Nadkarni* TAH.* 1985.* Optimal* Nutrient* Retention* during* the* Thermal* Processing* of* ConductionCHeated* Canned* Foods:* Application* of* the* Distributed*Minimum*Principle.*p.*1312C21.* Margarida* C.* Vieira* AATCLMS.* 2001.* Kinetic* Parameters* Estimation* for* Ascorbic* Acid* Degradation* in* Fruit* Nectar* Using* the* Partial* Equivalent* Isothermal* Exposures* (PEIE)* Method* under* NonCIsothermal* Continuous* Heating* Conditions.*p.*175C81.* MATLAB.*2012.*MATLAB*and*Statistics*Toolbox*Release.!2012b*ed:*The*MathWorks* Inc.,*Natick,*Massachusetts,*United*States.* 192 Mishra*DK,*Dolan*KD,*Yang*L.*2008.*Confidence*intervals*for*modeling*anthocyanin* retention* in* grape* pomace* during* nonisothermal* heating.* J* Food* Sci* 73(1):E9CE15.* Nasri*H,*Simpson*R,*Bouzas*J,*Torres*JA.*1993.*UnsteadyCstate*method*to*determine* kinetic*parameters*for*heat*inactivation*of*quality*factors:*ConductionCheated* foods.*Journal*of*Food*Engineering*19(3):291C301.* Patashnik* M.* 1953.* A* simplified* procedure* for* thermal* process* evaluation.* Fd* Technol.,*Champaign*7,*1.* Pinto* E,* Pedersén* M,* Snoeijs* P,* Van* Nieuwerburgh* L,* Colepicolo* P.* 2002.* Simultaneous* Detection* of* Thiamine* and* Its* Phosphate* Esters* from* Microalgae*by*HPLC.*Biochemical*and*Biophysical*Research*Communications* 291(2):344C8.* Rickman* JC,* Barrett* DM,* Bruhn* CM.* 2007.* Nutritional* comparison* of* fresh,* frozen,* and* canned* fruits* and* vegetables* II.* Vitamin* A* and* carotenoids,* vitamin* E,* minerals*and*fiber.*Journal*of*the*Science*of*Food*and*Agriculture*87(7).* Schwaab*M,*Pinto*JC.*2007.*Optimum*reference*temperature*for*reparameterization* of* the* Arrhenius* equation.* Part* 1:* Problems* involving* one* kinetic* constant.* Chemical*Engineering*Science*62(10):2750C64.* Van* Boekel* M.* 1996.* Statistical* Aspects* of* Kinetic* Modeling* for* Food* Science* Problems.*Journal*of*Food*Science*61(3):477C86.* Vikram* VB,* Ramesh* MN,* Prapulla* SG.* 2005.* Thermal* degradation* kinetics* of* nutrients* in* orange* juice* heated* by* electromagnetic* and* conventional* methods.*Journal*of*Food*Engineering*69(1):31C40.* Yoshimi*Terajima*YN.*1996.*Retort*Temperature*Profile*for*Optimum*Quality*during* ConductionCHeating*of*Foods*in*Retortable*Pouches.*p.*673C8.* * 193 Chapter%6 % Overall%Conclusions% * 6.1%Conclusions% * Based* on* the* dimensionless* derivation* of* the* scaled* sensitivity* coefficients,* an* identity,* intrinsic* sum,* was* developed.* Intrinsic* sum* was* used* to* verify* the* numerical*code*for*its*accuracy*to*provide*temperature*predictions.*It*was*also*used* to*get*insight*into*the*parameter*estimation*problem.*It*was*also*shown*that*if*the* sum*of*all*scaled*sensitivity*coefficients*is*equal*to*zero,*then*not*all*the*parameters* can* be* estimated* uniquely* and* simultaneously.* However,* if* the* sum* of* scaled* sensitivity*coefficients*is*not*equal*to*zero,*then*it*might*be*possible*to*estimate*all* parameters* uniquely* and* simultaneously.* In* the* case* of* heat* transfer* problems* where*heat*flux*is*a*boundary*condition,*it*was*shown*that*thermal*conductivity*and* specific* heat* can* be* estimated* simultaneously.* This* was* because* the* sum* of* all* scaled*sensitivity*coefficients*was*equal*to*the*temperature*rise*of*the*product.*This* was* the* fundamental* principal* behind* design* of* TPCell* instrument* for* measuring* temperatureCdependent* thermal* properties.* TPCell* was* designed* to* measure* thermal*properties*up*to*140oC*with*a*testing*time*of*less*than*a*minute.*TPCell*is* fully* programmable* to* provide* different* heating* rates* for* different* types* of* materials.* Several* food* products* were* tested* and* the* thermal* conductivity* was* reported.*The*thermal*conductivity*at*room*temperature*had*good*agreement*with* other* values* reported* in* literature.* The* temperatureCdependent* thermal* 194 conductivity* obtained* from* TPCell* was* used* to* model* the* kinetic* parameters* of* degradation*for*vitamin*C*and*thiamin.** * Nutrient* studies* were* performed* to* compare* the* degradation* of* vitamin* C* and* thiamin* in* aseptic* and* retort* processing* conditions.* The* aseptic* processing* system* had* a* better* retention* than* retort* processing* for* vitamin* C.* For* the* retort* processing,*the*rate*constant*at*a*reference*temperature*of*127oC*was*0.0073*minC1* and*for*aseptic*processing*it*was*0.0114*minC1.*This*difference*might*be*because*of* the* difference* in* temperature* for* both* the* systems.* The* maximum* temperature* in* aseptic* processing* was* 140oC* and* for* retort* processing* was* 121.67oC.* The* degradation*of*thiamin*was*not*enough*to*estimate*the*kinetic*parameters*in*both* systems.*The*aseptic*process*is*better*for*the*retention*of*nutrients*such*as*vitamin* C.* * The*food*industry*will*benefit*from*this*research*with*regard*to*the*thermal* properties*at*elevated*temperatures.*With*the*new*and*novel*processes*relying*on* faster*heating*and*faster*cooling,*the*majority*of*the*time*spent*by*food*products*is* at* higher* temperatures.* So,* the* temperatureCdependent* thermal* properties* would* provide*a*means*to*optimize*the*quality*of*the*food*product*while*keeping*the*food* safe.* Without* the* knowledge* of* accurate* thermal* properties* at* elevated* temperatures,*food*processors*tend*to*be*conservative*with*the*thermal*process*and* end*up*with*poor*quality*of*the*processed*food.** 195 6.2%Recommendations%for%future%work% Here* are* recommendations* for* future* work* related* to* current* research* and* to*make*advancements*with*TPCell.* 1. Investigate* the* effect* of* reactions* such* as,* endothermic* reactions,* glass* transition,* protein* denaturation* and* lipid* melting* on* thermal* conductivity*of*food*materials.* 2. *Perform* more* experimental* runs* with* TPCell* for* several* different* temperature* ranges,* such* as* 20oC* C* 80oC,* and* compare* the* results* with*the*complete*range*of*20oC*C*140oC.* 3. Perform*experiments*with*TPCell*on*solid*materials.* 4. Investigate* the* effect* of* temperature* profile* of* TPCell* heater* on* estimated* thermal* conductivity.* Does* the* temperature* history* effect* thermal*conductivity?* 5. Investigate*the*effect*of*oxygen,*carbon*dioxide*and*headspace*in*the* jar*on*degradation*of*nutrients.* % 196