Heat transfer and mass transfer with heat generation in drops at high peclet number

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1 The Uivesity of Toledo The Uivesity of Toledo Diital Repositoy Theses ad Dissetatios 007 Heat tasfe ad mass tasfe with heat eeatio i dops at hih peclet umbe Adham Soucca The Uivesity of Toledo Follow this ad additioal woks at: Recommeded Citatio Soucca, Adham, "Heat tasfe ad mass tasfe with heat eeatio i dops at hih peclet umbe" (007. Theses ad Dissetatios. Pape 36. This Dissetatio is bouht to you fo fee ad ope access by The Uivesity of Toledo Diital Repositoy. It has bee accepted fo iclusio i Theses ad Dissetatios by a authoized admiistato of The Uivesity of Toledo Diital Repositoy. Fo moe ifomatio, please see the epositoy's About pae.

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4 A Abstact of Heat Tasfe ad Mass Tasfe with Heat Geeatio i Dops at Hih Peclet Numbe Adham Soucca Submitted as patial fulfillmet of the equiemets fo The Docto of Philosophy deee i Eieei The Uivesity of Toledo May 007 Koi ad Bik published a classic aalysis of taspot fom taslati doplets. Thei aalysis assumed that the bulk of the esistace to tasfe was i the doplet phase. It cosideed the limiti solutio as the Peclet umbe became vey lae. Thei wok has bee cited i may subsequet studies of doplet tasfe. Chapte Thee sectio 3. evisits thei solutio usi umeical techiques that wee ot the available. It was foud that oly the fist mode of thei solutio is accuate; hece, thei solutio is accuate at oly at lae times. I Chapte Thee Sectio 3., the tasiet heat tasfe fom a doplet with heat eeatio is ivestiated. It is assumed that the iii

5 bulk of the themal esistace esides i the doplet. Two cases wee discussed: low Peclet flows ad vey hih Peclet flows. As expected, it was foud that the tempeatue ise due to the heat eeatio was less fo hih Peclet flows. I additio, the tempeatue pofile espods moe quickly fo hih Peclet flows. This aalysis is also applicable to mass tasfe with a zeo-ode eactio. The system of diffeetial equatios that descibes this behavio was solved usi a semi-aalytical method ad a Matlab softwae compute poam. Key Wods: Spheical Doplet, Iteio Poblem, Hih Peclet Numbe, Ceepi Flow. iv

6 To my paets, my bothe ad my wife. v

7 ACKNOWLEDGMENTS I would like to expess my sicee atitude to my adviso, D. Doulas Olive, fo ivi me the iitial stimulus to wok o this poject, fo his suppot ad uidace thouh out my aduate studies. His costat ecouaemets, eeosity, costuctive commets ad ispii discussio have bee of immeasuable value to me. It s a hoo to be oe of his PhD studets. I would like to expess my deep atitude to the Mechaical, Idustial ad Maufactui Eieei depatmet fo its fiacial suppot, ad to D. A. Afjeh, D. M. Hefzy ad all the staff fo thei assistace dui my eseach. I also wat to exted my appeciatio to the membes of my dissetatio committee, D. Tey N, D. Cyil Masiulaiec, D. Soi Cioc ad D. Ezzatollah Salai. May thaks fo Pof. Bahe Haa ad fo M. Mac Iskada fo thei poofeadi my dissetatio. I also wish to thak my fied Paul La Fotaie fo his iitial help. Lastly, I would like to expess my deepest thaks to my beloved paets, my bothe ad my wife fo thei love, udestadi, ad cotiuous suppot. vi

8 Table of Cotets Pae Abstact Dedicatio Ackowledmets Table of cotets List of tables List of fiues Nomeclatue iii v vi vii x xi xii CHAPTER : INTRODUCTION. Motivatio. Theoetical Basis 6.3 Doplet Steam Lies 7.4 Poblem Fomulatio 0 CHAPTER : LITERATURE REVIEW. Heat ad Mass Tasfe fom Doplets 3. Vaiatio of Reyolds umbes 4.. Low Reyolds umbes 4.. Modeate Reyolds umbes 4..3 Hih Reyolds umbes 5 vii

9 .3 Vaiatio of elative esistace 5.3. Exteal Poblem 5.3. Iteal Poblem Cojuate Poblem 7.4 Heat Geeatio 9 CHAPTER 3: MATHEMATICAL ANALYSIS Itoductio Sectio 3. Aalysis of Heat Tasfe without Heat Geeatio 3.. Itoductio Poblem Fomulatio Solutio Pocedue 5 Sectio 3. Aalysis of Heat Tasfe fom a Doplet at Hih Peclet Numbes with Heat Geeatio 3.. Itoductio Poblem Fomulatio Solutio Pocedue Special Case : Peclet umbe a The steady state pat b The tasiet pat Special Case : Peclet umbe 37 viii

10 CHAPTER 4: RESULTS 4. Heat Tasfe fom a taslati Doplet at Hih Peclet Numbes Revisiti the Classic Solutio of Koi ad Bik Heat Tasfe fom a doplet at Hih Peclet Numbes with Heat Geeatio 55 CHAPTER 5: CONCLUSION AND FUTURE WORK 5. coclusio 6 5. Suestios fo futue wok Fial Thouhts 64 Refeeces 65 Appedix 75 Appedix 76 ix

11 List of Tables Table No. Title Pae 4- Coveece with Respect to Δ (without heat eeatio Coveece with Respect to Δ (with heat eeatio 55 x

12 List of Fiues Fiue No. Title Pae - Phase idetificatio of dops ad bubbles 4 - Spheical coodiate system 8-3 Shape eimes fo bubbles ad dops i uhideed avitatioal motio thouh liquids 9-4 Doplet Steam Lies 0-5 Steam lies ad thei othooal tajectoies i a vetical plae thouh the axis of a falli doplet 4- Fist two eie-fuctios, peset vs. Koi ad Bik Bulk Tempeatue Θ (τ Θ(, τ alo θ π /, Koi ad Bik vs. peset Nusselt Numbe, Koi ad Bik vs. Peset Θ(, τ alo θ π / Θ( alo θ π / at seveal times Θ as a fuctio of time ADI pedictios of bulk tempeatues. Copied fom Fayeweathe Illustatio of heat flux diectios: Hih Peclet umbe vs. low Peclet umbes Maximum Tempeatues as a fuctio of time 6 xi

13 Nomeclatue a Doplet adius. A Coefficiet, see Eq. (3.-7. B Coefficiet, see Eq. ( c j c p D jm Mola cocetatio Specific heat Mass diffusio coefficiet Δ ρ L Eo Eotvos umbe, Eo σ E & Rate of eey flow J j k max Nu Pe q& Mola diffusio flux vecto Themal coductivity Tucatio limit. Nusselt umbe o Shewood umbe Ua Peclet umbe, Pe. α Heat eeatio q Heat flux Fietia Re Reyolds umbe, ρ U L Re Fviscous μ Radial coodiate made dimesioless with the adius a. t Time T Tempeatue. U Fee Steam lies velocity. u Radial velocity v taetial velocity Geek Symbols γ ( Defied by Eq. (3.-9. T T Θ Dimesioless tempeatue without heat eeatio s. T T iit s xii

14 Θ Dimesioless tempeatue with heat eeatio ( T T k s q& a Θ Aveae o bulk tempeatue β ( Defied by Eq. (3.-0 Dimesioless spatial coodiate, ( si θ. ψ μ Dimesioless steam fuctio Dyamic viscosity. Ξ ( Eie-fuctio. ρ Desity α Themal diffusivity φ Azimuth coodiates α t τ Dimesioless time, τ. a δ Koecke delta fuctio θ Taetial coodiate. Ф Viscous dissipatio fuctio κ Ratio of dyamic viscosities μ ext / μ dop. 4 λ Eie-value Subscipts dop ext iit max s Doplet o dispesed phase. Cotiuous phase. With heat eeatio Iitial. Maximum Suface xiii

15 CHAPTER ONE INTRODUCTION. Motivatio I the eihteeth cetuy ad ealy ieteeth cetuies the study of heat tasfe was based o the caloic theoy which assets that heat is a mass-less, cololess, odoless, ad tasteless fluid substace which ca be poued fom oe body to aothe. Whe caloic was added to a body it aises its tempeatue ad vice vesa The mode physical udestadi of the atue of heat developed i the middle of the ieteeth cetuy whee heat was defied as the eey associated with the adom motio of atoms ad molecules. The huma body ejected heat to its suoudis ad the ate of this ejectio was elated to the huma comfot. This ate of heat tasfe is cotolled by adjusti clothi to the eviomet. A lot of home appliaces ae desied by usi heat tasfe piciples, (heati, ai coditioi, efieati, etc. Heat tasfe equipmet such as boiles, codeses, heates ad heat exchaes ae desied o the basis of heat tasfe aalysis, which i pactice ca be divided ito two majo oups:

16 - Rati poblems: which detemie the heat tasfe ate fo a existi system at a specific tempeatue diffeece - Sizi poblems: which detemie the size of the system i ode to tasfe heat at specific ate fo a specified tempeatue diffeece. A heat tasfe pocess ca be studied eithe expeimetally by taki measuemets, o aalytically by aalysis ad calculatios. Howeve the fist appoach deals with actual size systems with fewe expeimetal eos. Expeimetatio is expesive ad time cosumi. I cotast, the aalytical appoach has the advatae of bei fast ad elatively iexpesive. The wok doe i this study is aalytical i atue. Ay aalytical wok uses a set of assumptios to simplify the aalysis. The accuacy of the solutio depeds o the assumptios made i such aalysis. Mass tasfe is a simila pheomeo to heat tasfe. The extactio of a substace dissolved i fluid doplets, by a secod fluid suoudi the doplets, ad ot miscible with the fist oe, is a pocess of cosideable techical impotace. Bubbles, doplets ad paticles ae essetial i may atual pocesses like boili, femetatio, ai pollutio ad aifall, as well as ma elated activities such as idustial systems, uclea powe plats, etc Clouds ae assemblaes of small wate doplets which ude cetai cicumstaces coalesce, leadi to aifall. Oceas, seas, ad lakes cotai ai i dissolved fom kow as bubbles. Paticles play a pimay ole i all sots of spays. I idustial systems, such as chemical eactos, dops ad bubbles cay eactats ad poducts. I uclea powe plats, oe ecoutes bubbles i a boili wate eacto

17 3 ad dops i spay cooli compoets. I iteal combustio ad jet eies fuel is atomized. Eiees ad eseaches i may baches of eieei ad sciece ae faced with heat ad mass tasfe poblems. Whe cosidei heat o mass tasfe ea doplets it is ofte coveiet to coside the elative themal esistace i the doplet as compaed with the esistace i the suoudi fluid. If the themal esistace i the doplet is much hihe tha that of the suoudi fluid, the it is easoable to assume that the tempeatue at the doplet suface is equal to the ambiet tempeatue. Thus, heat tasfe is oly calculated i the doplet, ot i the ambiet fluid. This assumptio is associated with the so-called iteio poblem. Whe the themal esistace is pimaily i the suoudi phase, the tempeatue i the doplet is ofte assumed to be spatially uifom. This assumptio is associated with the so-called exteio poblem. Fially, if the themal esistace i the doplet is of the same ode of maitude as that of the ambiet fluid, the the heat tasfe must be calculated i both the doplet ad the ambiet fluid. This situatio is the so-called cojuate poblem. The wok of this study deals with the iteio poblem fo doplet heat tasfe. A doplet is a mass of liquid i a liquid o as medium, while a bubble is a mass of as i a exteal medium. The fluid paticle ad the suoudi medium ae sepaated by a well defied iteface. I some cases like soap bubble this iteface is a thi film. Moe complex situatios, (compoud dop cosist of pais of dops ad bubbles as show i fiue (-

18 4 (a liquid dop (b as bubble i a liquid (c Soap bubble (d Compoud dop thee itefaces (e Compoud dop two itefaces Gas Liquid Fiue - Phase idetificatio of dops ad bubbles: Ref [73] Applicatio of taspot studies elated to doplets icludes heat o mass tasfe fom bubbles isi i a liquid o fom dops movi i a secod fluid of diffeet popeties, combustio pocesses, ad chemical eactios ivolvi fluids. Dops, bubbles o compoud dops ae efeed to as fluid paticles which cosist of lae umbes of molecules so as to be cosideed a cotiuum.

19 5 Moe tha five decades ao, Koi ad Bik s [66] wok ivestiated mass tasfe of a solute i a doplet. I paticula, they obtaied a semi-aalytic solutio fo the limiti case of the iteio poblem whee the Peclet umbe is vey lae ad the Reyolds umbe is vey low. Thei wok has bee extesively cited i the followi decades. Oe of the poblems with the wok of Koi ad Bik is that thei mathematical model oly allowed fo paabolic appoximatios to the cocetatio cotous. The itet of this wok is to ivestiate the accuacy of the Koi ad Bik solutio, especially at small times. I additio, this wok will exted thei model to iclude the effects of heat o mass eeatio.

20 6. Theoetical Basis The mathematical wok of Koi ad Bik was diected at mass tasfe. It is equally applicable to heat tasfe with the appopiate dimesioal covesio. As a esult of the similaity of modeli heat ad mass tasfe pocesses, the tems heat o mass tasfe ae sometimes used itechaeably. Althouh oly heat tasfe is modeled i the dissetatio, the esults ae equally applicable to mass tasfe See Appedix. Wheeve thee is tempeatue adiet sevi as a divi foce heat flow occus, this is kow as Fouie s law, ad is expessed as q k T (- whee q is the heat flux, k the themal coductivity ad T is the tempeatue adiet. Mass diffusio occus whee thee is a cocetatio adiet. A liea elatio kow as Fick s law ca descibe this ad could be mathematically stated as J j Djm c j j,,3... N whee J j is the mola diffusio flux vecto, D jm is the mass diffusio coefficiet ad cj is the mola cocetatio adiet. Applyi Fouie s law, Eq. (- ad the fist law of themodyamics to a cotolled volume a eey equatio may be developed ad expessed as:

21 7 T ρ c p + u T k T + q& + μ Φ (- t whee ρ is desity, c p is the specific heat, q& is the volumetic heat poductio ate, u is the adial velocity, μ is the dyamic viscosity ad Φ is the viscous dissipatio fuctio. Fo may heat tasfe applicatios, the themal coductivity ad desity ca be assumed to be costat. I additio i the cases of slow flow, the viscous dissipatio fuctio is eliible. Ude these coditios, the eey equatio is stated as: T q& + u T T + α t k (-3 whee α k / ρ c is the themal diffusivity. (-4 p I the absece of heat ad sik souces the eey equatio takes the fom of T + u T T α t (-5.3 Doplet Steam Lies Bubbles ad doplets i fee ise o fall ude the ifluece of avity ca be classified ito thee mai cateoies: Spheical Ellipsoidal Spheical-cap o Ellipsoidal- cap

22 8 The spheical coodiate system is applicable to spheical dops ad bubbles. Fo a illustatio of the spheical coodiates system see Fi. (- z φ θ y x Fiue - Spheical coodiate system whee φ is the azimuth coodiate, is the adial coodiate, ad θ the taetial coodiate. Due to the foce of suface tesio, small bubbles ad dops take a spheical shape. This meas that the itefacial tesio ad/o viscous foces ae much moe impotat tha the ietia foces. Fiue (-3 shows that at low Reyolds Numbes, spheical shapes fo fluid paticles occu.

23 9 Re F F ietia viscous ρ U L μ Δ Eo ρ L σ REYNOLDS NUMBER, Re EOTVOS NUMBER,Eo Fiue -3 Shape eimes fo bubbles ad dops i uhideed avitatioal motio thouh liquids. Adapted fom Fi. [Ref. ]

24 0 The steady steam fuctio fo the iteal flow field fo a spheical doplet at low Reyolds umbes is ive usi dimesioless coodiates by: 4 ( si θ Ua ψ (, θ. (-6 μ ext 4 + μ dop whee U ad a ae the doplet velocity ad adius, ad μ ext ad μ dop ae the dyamic viscosities of the cotiuous ad doplet phases. The adius,, is scaled by the doplet adius, a. See Fiue (-4 fo a illustatio of the steam lie ψ Fiue -4: Doplet Steam Lies.4 Poblem Fomulatio Koi ad Bik assumed that as the Peclet umbe became lae, the solute cocetatio cotous (o fo heat tasfe the tempeatue cotous became paallel with the steam fuctio cotous. See Equatio (-6

25 As stated, the tempeatue cotous ae assumed to be paallel with the steam fuctio cotous: lim T(, θ, φ, t T( ψ, t Pe (-7 Fo the iteio poblem, the bulk of the esistace to heat tasfe is assumed to be i the doplet phase. As such, the tempeatue at the suface, T s, is assumed to be costat. Followi the example of Koi ad Bik, Eq. (-7 may be estated i tems of the dimesioless spatial vaiable,, ad the dimesioless time,τ: lim Θ(, θ, φ, t Θ(, τ Pe lim Θ Pe (, θ, φ, t Θ (, τ Sectio 3. Sectio 3. (-8a (-8b α t whee: τ a whee 0. (-9 ad 4 ( si θ the adius is scaled by the doplet adius, a. See Fiue (-5 fo a illustatio of the positio of the fuctio Θ ad Θ ae the scaled tempeatues i the doplet such that: T T Θ s (without heat eeatio T T iit s Sectio 3. (-0 ( T T k Θ s (with heat eeatio Sectio 3. (- q& a whee T iit is the iitial doplet tempeatue ad T s is the suface tempeatue.

26 Fiue -5: Steam lies ad thei othooal tajectoies i a vetical plae thouh the axis of a falli doplet This will ivolve the solutio pocedue ad esults fo two diffeet foms of eey equatios: Sectio 3. will deal with Eq. (-5 i the absece of heat eeatio. Sectio 3. will deal with Eq. (-3 i the pesece of heat eeatio. Sectio 3. is a evisio of the classic wok of Koi ad Bik [66]. I that wok, oly two modes of eie-fuctios wee used. I additio, the eie-fuctios wee appoximated with quadatic polyomials. I the peset wok, up to 30 modes wee used with fiite-diffeece appoximatios fo the eie-fuctios. Sectio 3. is a ew study i the liht that o othe kow publicatio has ivestiated heat tasfe with heat eeatio iside doplets (iteio poblem at lae Peclet umbes.

27 3 CHAPTER TWO LITERATURE REVIEW. Heat ad Mass Tasfe fom Doplets The poblem of heat tasfe fom a sphee has bee the subject of seveal ivestiatios, stati with the oiial wok of Fouie [44] i 8. Fouie s pimay iteest was the cooli effect of the plaets; his wok was applied to iid sphees. The subject of heat ad mass tasfe fom doplets has bee ivestiated i may papes. Thee ae two excellet mooaphs covei the subject:- Bubbles, Dops, ad Paticle published i 978 [], ad Taspot Pheomea with Dops ad Bubbles published i 997 [73]. These mooaphs peset the theoy of heat ad mass tasfe ea doplets. Baue [36] ivestiated the ifluece of the distibutio coefficiet o the physical mass tasfe. Two spheical models wee cosideed; a iid sphee ad a fluid sphee with viscosity atio zeo (e.. a as bubble. Heat tasfe poblems fom doplets ae ofte classified as oe of the followi thee cases: exteal, iteal ad cojuate poblems. These poblems may be futhe cateoized by Reyolds umbe, Peclet umbe ad the ate of heat eeatio.

28 4. Vaiatio of Reyolds umbes.. Low Reyolds umbe Caslaw ad. Jaee [4] showed a solutio of the poblem of tasiet heat tasfe fom a sphee at ceepi flow (Re0. A aalytical solutio of the usteady heat tasfe fom a sphee at low Reyolds umbe ude steady velocity coditios was developed by Choudhouy ad Dake [6]. Fe ad Michaelides [88] have aalytically deived a expessio fo the heat tasfe fom a small sphee at low Peclet umbes assumi a Stokesia velocity distibutio aoud the sphee. Acivos ad Taylo [] used a asymptotic method ad applied thei study to a Stokesia velocity field which implies vey small Reyolds umbes; they discussed the effect of small but fiite Reyolds umbe of the iid sphee whe the Peclet Numbe is lae. They also deived a solutio fo the steady-state heat tasfe fom a sphee at small but fiite Peclet umbes. Abamzo ad Elata [4] computed the tasiet heat tasfe coefficiet fo iid sphees assumi a Stokesia velocity field implyi low Reyolds umbe outside the sphee. Haywood et al. [67] computed the taspot paametes of evapoati doplets... Modeate Reyolds umbe Chia et al. [] computed the tasiet heat tasfe fom evapoati doplets at vaious iitial tempeatues fo modeate Reyolds umbes. Abamzo & Elata [4] aalyzed the physical heat tasfe fom a sphee i Stokes flow at diffeet Reyolds umbes.

29 5..3 Hih Reyolds umbe A eview o umeical study o the tasiet heat tasfe fom a sphee at hih Reyolds umbes was made by Fe ad al. [6]. Fiedlade [69] was the fist to detemie asymptotically the Shewood umbe fo a small iid sphee ad fo lae Peclet umbes. The Shewood umbe is ouhly equivalet to the Nusselt umbe, but fo mass tasfe. Fiedlade applied the bouday laye theoy assumi a cocetatio pofile with coefficiets deived fom the bouday coditios..3 Vaiatio of elative esistace.3. Exteal Poblem Most of the theoetical wok is suitable to steady-state solutio fo the heat tasfe fom a isothemal sphee. The so-called exteal poblem is whe the tasfe esistace is assumed eliible iside the sphee as compaed to that of cotiuous phase (ambiet. A example of exteal poblem whee the esistace is pimay i the cotiuous phase ca be foud i a ai doplet descedi i the atmosphee whee the themal coductivity k 0/. Seveal authos ivestiated the exteal poblems whee the volumetic heat capacity atio was ifiite. With this assumptio the sphee emais at its iitial tempeatue. At Low Peclet umbes it was show that the Nusselt umbe appoaches.00. Bu [63] aalytically developed equatios fo the steady-state Nusselt umbes at low Reyolds umbes fo both fluid ad solid sphees.

30 6 Abamzo ad G.A. Fishbei [8] umeically solved the exteal poblem fo the steady-state eey poblem of with modeate Peclet umbes up to Pe 000. Thei esults suest that the bouday layes assumptios ae ot accuate fo Pe < Iteal poblem Whe the tasfe esistace is assumed eliible i the cotiuous phase compaed to that iside the sphee, the situatio is the called the iteio poblem. Accodi to Olive ad Chu, [7] thee is o equivalet steady-state situatio coespodi to a steady-state solutio fo a costat tempeatue sphee. At low Peclet umbes Newma, [] peseted a solutio fo diffusio of mass ito a sphee. Fom his wok it shows that the Nusselt umbe appoaches Johs ad Beckma [5] umeically iteated the eey equatio fo the doplet eio fo modeate Peclet umbes. They epoted oscillatios i the Nusselt umbes that wee due to the eciculatio of the fluid iside the doplet. Dwye, Kee ad Sades [38] obtaied simila esults usi a pomisi adaptive id scheme. Fo hih Peclet umbes, a covetioal bouday laye is appopiate oly at vey small times. May ivestiatios of the iteal poblem fo hih Peclet umbe icluded efeece to the classic wok of Koi ad Bik [66]. This classic wok ivestiated mass tasfe of a solute i a doplet. Thei wok obtaied a semi-aalytic solutio fo the limiti case of the iteio poblem whee the Peclet umbe is vey lae ad the Reyolds umbe is vey low. Koi ad Bik assumed that as the Peclet umbe becomes lae, the solute cocetatio cotous (o fo heat tasfe the tempeatue cotous became paallel with the steam fuctio cotous. The mathematical wok of Koi ad Bik was

31 7 diected to mass tasfe. This is equally applicable to heat tasfe with the appopiate dimesioal covesio. Due to the eciculatio of the fluid iside the doplet, Koi ad Bik [66] solved the eey equatio aalytically with the assumptio that the isothems ae paallel to the steam lies. Thei esults showed that as time iceases the Nusselt umbes appoaches 7.9. The wok of Koi ad Bik has bee cited i may subsequet woks as a limiti boud fo heat ad mass tasfe fom doplets at low Reyolds umbes ad hih Peclet umbes. Howeve, thei classic wok is a semi-aalytic appoximatio. The tial fuctios used wee oly quadatic fuctios. The Koi ad Bik solutio has bee show to be a accuate pedicto of heat ad mass tasfe ates at lae times. Howeve, it is ot clea if thei solutio was accuate at small times. Sice this classic aalysis of Koi ad Bik has bee fequetly cited, it seems easoable to evisit thei wok usi moe accuate umeical pocedues..3.3 Cojuate Poblem Whe the tasfe esistaces i both phases ae compaable, the cojuate poblem ivolves calculatios of the tempeatue field i both the cotiuous ad the dispesed phases. At low Peclet umbe Coope [] developed a aalytical solutio fo the tempeatue field fo vaious combiatios of themal popeties. His wok showed that fo all cojuate poblems, the Nusselt umbe vaishes fo lae time itevals with Pe 0.

32 8 Kleima ad Reed [54] aalyzed the cojuate tasfe fom a fluid i the pesece of chemical eactio. Latte, Jucu [3] aalyzed the cojuate heat ad mass tasfe betwee a iid sphee ad a ifiite medium i the pesece of a chemical eactio, whee the chemical eactio takes place iside the paticle. Jucu s wok was focused o themal coductivity, heat capacity, as well as diffusivity. Aalytical esults fo the cojuate mass tasfe poblems at hih Peclet umbes ae available by Levich et al. [78]. The fist umeical esults fo cojuate heat tasfe poblems wee published by Abamzo ad Bode [5]. Usteady cojuate heat tasfe fom a spheical doplet at low Reyolds umbes was ivestiated by Olive ad Chu [7]. They solved the eey equatio usi a implicit fiite diffeece method of Alteati Diectios Implicit (ADI, with a ae of 50 Pe < 000 fo Peclet umbes. They foud that the dimesioless tempeatue pofile asymptotically appoaches a steady-state value that is idepedet of the iitial pofile i the doplet. Fo modeate Reyolds umbes Olive ad Chu [8] solved the eey equatio usi (ADI fiite diffeece method with fluid motios iside ad outside the doplet. Chao [0] used bouday laye assumptios at hih Peclet umbes to estimate the heat tasfe ates fom sphees. Due to the elliptic atue of the iteio eio, such bouday laye solutio will be accuate oly at small times. Fo modeate Peclet umbes, Abamzo ad Bode [5] used a fiite diffeece method (ADI to iteate the eey equatio 0 Pe < 000. Thei wok which was caied at low Reyolds umbes povides a ood liteatue eview ito the taspot pocess i doplets ad solid sphees.

33 9 Recetly, seveal ivestiatios have bee made to chaacteize mass tasfe ea a bubble o doplet with chemical eactios. Fo example, Kliema ad Reed [54] ivestiated cojuate mass tasfe betwee a doplet ad a ambiet fluid with a fistode chemical eactio i the ambiet flow. Similaly, Jucu [9] ivestiated cojuate mass tasfe to a doplet with a secod-ode chemical eactio i the doplet. Accodi to Olive ad Chu [7], fo a fluid sphee at modeate Peclet umbes, the Nusselt umbes oscillates with decayi amplitude. The oscillatios ae due to the ciculatio of the fluid alteately supplyi hot ad cold fluid to the foe eio of the doplet whee most of the heat tasfe takes place. As time iceases the Nusselt umbe appoaches a steady value. Modeate Reyolds umbes wee used i the ae of 0 to 50. It was foud that by iceasi Reyolds umbe, the pedicted ate of heat tasfe is siificatly iceased fo fluid sphees as a esult of iceased fluid motios iside ad outside the doplet. It was also foud that the iceased velocities ea the itefacial suface of a dop ae a esult of a icease i the Reyolds umbe..4 Heat Geeatio Fewe eseaches have ivestiated the effects of distibuted heat (o mass souces o siks elated to heat ad mass tasfe ea doplet. Oe aim of the peset wok is to ivestiate the heat o mass tasfe fom a doplet with a uifom heat souce, q&. A heat souce may be ceated by a exothemic chemical o uclea eactio, o by a electo-maetic field. A heat sik may be ceated by a edothemic eactio. Recetly, seveal ivestiatios have bee made to chaacteize mass tasfe ea a bubble o doplet with chemical eactios; whee thee is a chemical eactio a ew

34 0 species is bei ceated, this will be aaloous to heat eeatio fo a mass tasfe poblem. The mass ad/o heat tasfe fom a sphee with uifom cocetatio ad/o tempeatue has bee aalyzed by Ruckestei et al. [9]. Ruckestei aalyzed the mass tasfe accompaied by a fist ode ievesible chemical eactio fom a sile compoet usi Duhamel s theoem. He deived aalytical expessios fo the Shewood umbe ad sphee aveae cocetatio. Sou ad Seas [83] umeically solved the same poblem assumi eliible diffusio i taetial diectio ad diffeet odes of chemical eactios. Kleima ad Reed [54] wee the fist to coside the chemical eactio elated to the cojuate poblem. They studied cojuate mass tasfe betwee a sile doplet (solid, liquid, o as ad a suoudi ambiet flow (liquid o as with exteal chemical eactio. Thei wok aalyzed the suoudi fluid flow with chemical eactio i the cotiuous phase. The asymptotic eime of mass tasfe was ivestiated fo lo times. Thei ivestiatio was iteal fo mass tasfe iside the paticle with Peclet umbe equals zeo ad exteal fo mass tasfe i the cotiuous phase fo Pe. Jucu [6] studied the cojuate heat ad mass tasfe betwee a iid paticle ad a ifiite covective cotiuous phase i the pesece of a o-isothemal chemical eactio. The exothemic ad edothemic chemical eactios wee aalyzed i two hydodyamic eimes: ceepi flow (Re 0 ad modeate Reyolds umbe. This wok assumed a Peclet umbe of 00 fo the cotiuous phase.

35 Jucu [9] ivestiated cojuate mass tasfe betwee a dop ad a suoudi fluid flow with secod ode ievesible chemical eactio iside the dop. The dispesed phase (the dop eactat was isoluble i the cotiuous phase. Two sphee models wee cosideed; a iid sphee ad a fluid sphee with iteal ciculatio. Fo each spheical model two hydodyamic eimes wee used, ceepi flow, ad modeate Reyolds umbes. Slow ad fast chemical eactios vayi fom 0-4 to 0 wee aalyzed at a modeate value of Pe 00. Oe of the most ecet papes is the wok of Jucu [8]. Jucu ivestiated cojuate heat/mass tasfe fom a cicula cylide with a iteal heat/mass souce i lamia coss flow at low Reyolds umbes. The heat/mass souce cosisted of a costat tempeatue/cocetatio wie imbedded i the cylide cete. Numeical ivestiatios wee caied out fo such a cylide with Reyolds umbes of ad 0 ad a Peclet umbe of 00. No wok was foud that ivestiated heat o mass tasfe fom a doplet at hih Peclet umbes with heat o mass eeatio i the doplet.

36 CHAPTER THREE MATHEMATICAL ANALYSIS Itoductio This chapte ivolves the mathematical aalysis of the taspot equatios ive by Eqs. (-3 ad (-5. Sectio 3. deals with taspot without heat eeatio Eq. (-5. Sectio 3. coces taspot with heat eeatio Eq. (-3. Usi the sepaatio of vaiables method, a evisio of the classic solutio of Koi ad Bik is pepaed ad the the esults ae examied ad compaed with the classical solutio. Heat eeatio is the added to the doplet at hih Peclet umbes. This chapte is divided ito two mai sectios:- Sectio 3.: Tasfe Fom a Doplet at Hih Peclet Numbes without heat eeatio. Sectio 3.: Tasfe Fom a Doplet at Hih Peclet Numbes with heat eeatio. which is futhe divided ito two subsectios :- i. Pue coductio Pe 0 ii. Vey hih Peclet umbe Pe

37 3 Sectio 3. Aalysis of Heat Tasfe without Heat Geeatio 3.. Itoductio The Koi ad Bik aalysis assumed that the bulk of the esistace to tasfe was i the doplet phase. It cosideed the limiti solutio as the Peclet umbe became vey lae. Thei wok has bee cited i may subsequet studies of doplet tasfe. This sectio evisits thei solutio usi umeical techiques that wee ot the available. The wok of Koi ad Bik has bee cited i may subsequet woks as a limiti boud fo heat ad mass tasfe fom doplets at low Reyolds umbes ad hih Peclet umbes. Howeve, this classic wok was a semi-aalytic appoximatio usi the Ritz method. The tial fuctios used wee oly quadatic fuctios. I additio, solutios fo oly two modes wee souht. The Koi ad Bik solutio was show to be a accuate pedicto of heat ad mass tasfe ates at lae times. Evidetly, it is ot clea if thei solutio is accuate at small times. Sice thei classic wok has ofte bee cited, it seems easoable to evisit thei wok usi moe accuate umeical pocedues that wee ot available fifty yeas ao. The pimay pupose is to ivestiate the miimum time at which the solutio of Koi ad Bik is accuate. I this sectio the same assumptios ae made eadi the physical paametes (as applied to heat tasfe ad ovei equatios as wee made by Koi ad Bik. Howeve, the solutio pocedue has bee modified to obtai a moe accuate solutio at small times.

38 4 3.. Poblem Fomulatio The steady steam fuctio fo the iteal flow field fo a spheical doplet at low Reyolds umbe is ive usi dimesioless coodiates by: 4 ( si θ Ua ψ (, θ. (3.- μ ext 4 + μ dop whee U ad a ae the doplet velocity ad adius, ad μ ext ad μ dop ae the dyamic viscosities of the cotiuous ad doplet phases. The adius is scaled by the doplet adius, a. As stated above, the tempeatue cotous ae assumed to be paallel with the steam fuctio cotous at all times lim T (, θ, φ, t T ( ψ, t Pe (3.- Θ is the scaled tempeatue i the doplet (see Eq.(-0 T Ts Θ T T iit s Whee T iit is the iitial doplet tempeatue ad T s is the suface tempeatue. Fo the iteio poblem, the bulk of the esistace to heat tasfe is assumed to be i the doplet phase. As such, the tempeatue at the suface, T s, is assumed to be costat. Followi the example of Koi ad Bik, Eq. (3.- may be estated i tems of the dimesioless spatial vaiable,, ad the dimesioless time,τ: lim Θ(, θ, φ, t Θ(, τ Pe (3.-3 α t whee: τ (3.-4 a

39 5 α is the themal diffusivity ad 4 ( si θ whee Solutio Pocedue Accodi to Eq. (8 of Koi ad Bik s pape (applied to heat tasfe, the diffeetial equatio fo heat taspot may be stated as: Θ β ( Θ γ (. ( τ The iitial ad bouday coditios imposed o Eq. (3.-5 ae: Θ(, τ 0 ( iitial coditio (3.-6 Θ( 0, τ 0 (alo the oute steam lie, ad (3.-7 Θ (, τ is fiite at the votex cete. (3.-8 The fuctios γ ( ad β ( ae ive by Eqs. (4 ad (5 of Koi ad Bik s pape as: + γ ( E K ( ( 4 3 ( 4 3 β ( K ( whee E(x ad K(x ae the complete elliptic iteals. Equatio (3.-5 is solved usi sepaatio-of-vaiables techiques whee Θ (, τ is assumed to be of the fom: 6λτ Θ (, τ A Ξ ( e. (3.-

40 6 whee Ξ ( is the eie-fuctio The coespodi odiay diffeetial equatio fo Ξ is the: d dξ γ ( + λ β ( Ξ ( 0. (3.- d d The bouday coditios imposed o Ξ ( ae: Ξ ( 0 0 (at the oute steam fuctio of the doplet, ad Ξ ( is fiite (at the doplet votex cete. Both of the fuctios γ ( ad β ( ae positive o the iteval (0,. I additio, lim γ ( 0. Ude these cicumstaces, Eq. (3.- is a Pope Stum- Pe Liouville like poblem whee the coespodi eie-fuctios, the iteval (0, with espect to the weihti fuctio β (. That is: whee δ m is the Koecke delta fuctio. 0 Oce the eie-values λ ad coespodi eie-fuctios Ξ ( ae ascetaied, the [ Ξ ( ] β ( d Ξ, ae othooal o Ξ m ( Ξ ( β ( d δ m (3.-3 The eie-values, λ, may be evaluated usi a eey balace aalysis (see Appedix 3 which equies that: λ eie-fuctios ae omalized so that 0 ( 8 ( Ξ Ξ d β d ( d 0 [ ( ] 0 Ξ β ( d (3.-5 0

41 7 A fiite-diffeece scheme usi Matlab has bee developed to fid the eievalues ad to umeically iteate Eqs. (3.-3, (3.-4, etc. See Appedix. The coefficiets, A may be obtaied usi the othooality of the eiefuctios, Ξ, coupled with the iitial coditio, Eq. (3.-6: A Ξ ( (3.-6 The ie poducts of both sides of Eq. (3.-6 ae take with Ξ ( m. Sice the eie- fuctios, Ξ, ae othoomal with espect to the weihti fuctio β (, the A Ξ ( β( d. ( The bulk o aveae tempeatue Θ (τ may be calculated by taki the weihted aveae of the tempeatue, with: o Θ( τ A ( β ( d e Ξ 0 A 6λ τ 3 6λ τ Θ ( τ A e ( Solvi Eq. (3.- ivolves fidi the eie-fuctios, Ξ (, the associated eie-values, λ, ad the coespodi coefficiets, A. Secod-ode fiite diffeece techiques wee used to solve Eq. (3.-. The values of λ wee iteatively adjusted util the chaacteistic equatio, Eq. (3.-4, was satisfied. Oce the eie-fuctios,

42 8 Ξ (, ad associated eie-values, λ, wee obtaied, the eie-fuctios wee the omalized. As a pactical matte, Eq. (3.- must be tucated with: max 6 λ τ Θ (, τ A Ξ ( e (3.-9 whee max is the tucatio limit, fo small times max has to be vey lae fo accuacy we used 30 modes which was Fially, the coefficiets A wee obtaied usi Eq. (3.-7. All iteatios wee pefomed usi the tapezoidal ule. The id fo iteatio was equally spaced i ad was idetical with the fiite-diffeece id spaci. The bulk o aveae tempeatue of the doplet may be detemied usi a tucated fom of Eq. (3.-8: max 3 6λ τ Θ ( τ A e ( Aothe measue of the mass tasfe is the Nusselt umbe (o Shewood umbe fo mass tasfe. Nusselt umbe may show as: Θ τ Nu. (3.- 3Θ Substitutio of Eq. (3.-0 ito Eq. (3.- yields: max 6λ τ 3 A λ e Nu (3.- 3 A e 6λ τ

43 9 Sectio 3. Aalysis of Heat Tasfe fom a Doplet at Hih Peclet Numbes with Heat Geeatio 3.. Itoductio This sectio ivestiates the heat o mass tasfe fom a doplet with a uifom heat souce, q&, Eq. (-3. A heat souce may be ceated by a exothemic chemical, uclea eactio, o by a electo-maetic field. 3.. Poblem Fomulatio Fo simplicity, futhe discussio of this poblem will emphasize the heat tasfe poblem. Coside a doplet i a ambiet fluid. The doplet is expeieci a uifomly distibuted ate of heat eeatio; q&. the ambiet fluid has o coespodi heat eeatio. This ivestiatio is limited to the so-called iteio poblem. Fo iteio poblems, all esistace to heat tasfe is assumed to eside i the doplet. As such, the tempeatue at the suface of the doplet, T s, is assumed to be the fee-steam tempeatue T s T ext. (3.- With this assumptio, oly the heat tasfe i the doplet phase eeds to be cosideed. Assumi symmety about the azimuth, the tempeatue pofile i the doplet will be a fuctio of two spatial coodiates, ad θ, as well as time, t, o T(,θ, t. A eey balace o a diffeetial volume of the sphee yields the followi patial diffeetial equatio:

44 30 T + T qa & siθ + siθ θ θ k Pe T u v T T + + θ τ (3.- whee is the adial coodiate scaled by the doplet adius a, u ad v ae the adial ad taetial velocities, scaled by the doplet velocity U. Pe is the Peclet umbe ad τ is the dimesioless time with Ua tα Pe ad τ, whee α is the themal diffusivity. α a Fially, the tempeatue may be made dimesioless with: Θ [ T (, θ, τ T ] k s (, θ, τ. (3.-3 qa & With this substitutio, Eq. (3.- becomes: Θ Θ Pe Θ v Θ Θ θ u θ θ si θ + θ (3.-4 si τ The iitial ad bouday coditios imposed o Eq. (3.-4 ae: Θ (, θ, τ 0 0 (Iitial coditio, (3.-4a Θ (, θ, τ 0 (Oute ede of the doplet, ad Θ θ Θ θ θ 0 θ π 0 (Azimuthally symmetic. Fo a spheical doplet at low Reyolds umbe ad without suface aets, the iteio steady steam fuctio is ive usi dimesioless coodiates by Ref [] as: 4 ( ψ (, θ si θ ψ (, θ. (3.-5 Ua 4( + κ Whee κ is the atio of dyamic viscosities, μ ext / μ dop. The scaled velocities ae the:

45 ( ψ cosθ u (, θ, ad (3.-6 siθ θ ( + κ ( + ψ siθ v (, θ (3.-7 siθ ( + κ Solutio Pocedue The solutio to Eq. (3.-4 has bee obtaied fo two limiti cases: fist, a aalytic solutio was obtaied fo pue diffusio with Pe 0; secod, a semi-aalytic solutio was obtaied fo limiti case of vey hih Peclet umbes Special Case : Pe 0. As the Peclet umbe appoaches zeo, the poblem loses its depedece o the taetial coodiate. Thus, the dimesioless tempeatue is a fuctio of ad τ oly. I this case, Eq. (3.-4 becomes: Θ Θ + (3.-8 τ o dθ Θ (3.-9 dτ The iitial ad bouday coditios imposed o Eq. (3.-9 ae: Θ (, τ 0 0 ( Iitial coditio Θ (, τ 0 (Bouday coditios Θ ( 0, τ fiite dθ d 0 0 Usi the stadad sepaatio-of-vaiable techique,

46 3 Θ (, τ K( M (, τ ( Substitute Eq. (3.-9 ito Eq. (3.-0 we obtai K (Steady-state pat (3.- ad M M τ (Tasiet pat ( a The steady-state pat K Bouday coditios: K ( 0 fiite d d dk d K ( 0 dk d C C dk d 3 K 6 C C ( + C fom BC [] 0 K ( C fom BC [] C 6 K( ( b The tasiet pat 6 M M, τ Iitial coditio: M (, τ 0 K, Bouday coditios: M (, τ 0, ad

47 33 M(0,τ is fiite. Applyi the oe-dimesioal laplacia i spheical coodiates esults i: d d dm d M, o τ M + dm d M τ. (3.-4 To solve fo M(,τ, let U (, τ M (, τ (3.-5 The iitial coditio ad bouday coditios fo U ae: Iitial coditios: U (, τ 0 ( 6 Bouday coditios: U (, τ 0 ad U ( 0, τ 0. The followi equatios covet Eq. (3.-4 ito a equivalet equatio fo U(,τ: U M + M (3.-6 U M U + the Fom Eq. (3.-6 M U U (3.-7 U M M M + +

48 34 M M U + (3.-8 Fom Eq. (3.-8 M U M (3.-9 Substitutio of Eq. (3.-7 ito Eq. (3.-9 esults i: U U U M (3.-0 τ τ M U U M τ τ (3.- Fially, with substitutio of Eqs. (3.-7, (3.-0 ad (3.- ito Eq. (3.-4 we obtai: τ U U (3.- Equatio (3.- may be solved usi sepaatio of vaiables with: ( (, ( τ τ V R U (3.-3 The bouday coditios o R( ae: 0 ( R, ad 0 0 ( R. Substitutio of Eq. (3.-3 ito the compoets of Eq. (3.- esults i the followi: d R d V U, ad (3.-4

49 35 U dv R τ dτ (3.-5 Substitute Eqs. (3.-4 ad (3.-5 ito Eq. (3.- d R V d VR " RV ' dv R dτ R " V& R V λ Thee exists a ifiite umbe of solutiosλ to the above diffeetial equatio. Hece, followi stadad sepaatio-of-vaiables techiques, let: R ( A cos( λ + B si( λ (3.-6 Usi the bouday coditio fo R at 0 ad with Eq. (3.-6 yields: R ( B si ( λ. The coespodi equatio fo V (τ is the V& V λ V λ τ e, whee λ π. (3.-7 Equatios (3.-6 ad (3.-7 may be substituted ito Eq. (3.-3: U (, λ τ τ B si ( λ e (3.-8 Usi the iitial coditio: U (, τ 0 ( 6 The esulti value fo U(,τ is the:

50 36 3 ( si (, ( e U τ λ λ λ τ. Hece, the value fo M(,τ is: 3 ( si (, ( e U M τ λ λ λ τ (3.-9 Sice, ( (, ( τ τ M K + Θ, the the tempeatue fo the special case whe Pe 0, ca be stated as: + Θ 3 si ( ( 6, ( e τ λ λ λ τ (3.-30 A paamete of iteest is the bulk o aveae tempeatue of the doplet, Θ, Θ Θ 0 0, ( d vol d vol τ ( vol π (3.-3 d d vol 4π (3.-33 π π π d vol d (3.-34 Thus, the bulk tempeatue is: π π π λ λ τ λ τ 3 4 (4 6 (4 si ( ( ( d d e + Θ

51 37 Iteatio yields the followi: 6 λ τ Θ e whee λ π 4 5 λ (3.-35 Aothe paamete of iteest is the maximum tempeatue i the doplet, Θ. Fo Pe 0, the maximum tempeatue will occu at the cete of the doplet. max To fid this value, the value of Eq. (3.-30 must be evaluated at 0. si λ lim 0 λ ( λ 3! λ Hece, Θ ( (, τ + max 6 π e λ τ (Pe 0 ( Special Case : Pe As the Peclet umbe becomes vey lae, the tempeatue cotous ae assumed to be paallel to the steam fuctio cotous [66]. With this assumptio, Eq. (3.-4 may be solved usi the pocedue descibed i Chapte 3 Sectio 3.. Specifically, the dimesioless tempeatue is assumed to be a fuctio of the steam fuctio, ψ, ad τ : lim Θ pe (, θ, τ Θ ( ψ, τ (3.-37 Followi the method poposed by Koi ad Bik, [66], Eq. (3.-37 may be estated i tems of the dimesioless spatial vaiable,, ad the dimesioless time,τ: lim Θ pe (, θ, τ Θ (, τ (3.-38 Whee: 6( + κ ψ 4 ( si θ. (3.-39

52 38 Applyi the Fist law of themodyamics to a cotolled volume: E & E& + E& E&. (3.-40 i out e st whee E & i is the ate of eey flow ito the cotolled volume, E & out is the ate of eey flow out of the cotolled volume, E & e is the ate of heat eeatio i the cotolled volume, ad E & st is the time-ate-of-chae of the eey stoed i the cotolled volume. The cotolled volume is assumed to be a diffeetial volume betwee two steamlie. Sice the tempeatue pofile is assumed to be paallel to the steam lies, the heat flux will be due to coductio oly. With this assumptio, the eey etei the cotolled volume may be foud usi Fouie s law of heat coductio: dt d & k π a siθ ds (3.-4 d ds E i ε Similaly, the eey exit the cotolled volume is ive as: dt d & k π a + Δ siθ ds (3.-4 d ds Eout +Δ Combii these yields: dt dt d & E& k ( ( π a siθ ds (3.-43 out +Δ i d d ds +Δ E Followi ef. [66], we obtai the equatio:

53 39 ds d 8 a siθ Δ (3.-44 a( dζ ds ζ ( cos θ Δ ds ϕ a siθ dϕ (3.-46 Whee Δ ( cos θ + ( si θ (3.-47 Thus, the et heat tasfe due to coductio ito the cotolled volume is: dt dt 8 siθ Δ & E& k ( ( π a siθ ds out +Δ (3.-48 i d d a +Δ E dt dt a( dζ & E& k ( ( 6π si θ out +Δ ( i dε +Δ d 4 cos θ E Aai, followi ef. [66] this may be witte as: T & E& γ ( 4 π a k (3.-50 i out E Whee fom Eq. (3-9 + γ ( K ( 4 3 E ( 4 3 The tem o the iht had side of Eq. (3.-40 may be evaluated as: de dt st T ρ c p dv (3.-5 t

54 40 de dt st T ρ c p S Sζ Sϕ (3.-5 t Substitutio of Eqs. (3.-44,(3.-45ad(3.-46 ito Eq. (3.-5 esults i: 3 dest T a ρ c β ( ε p ϕ (3.-53 dt t 8 Whee, ϕ π. Fom Eq. (3-0: β ( K. + + Equatio (3.-40 is the, de dt st 3 T a ρ c p β ( π (3.-54 t 4 The heat eeatio tem may be calculated as: & E q q& ΔV 3 a E& q& π β ( o, q 4 (3.-55 Substitute Eqs. (3.-50,(3.-54ad (3.-55 ito Eq. (3.-40 yields: E& i E& out de dt st E& q (3.-56

55 4 Θ Θ 6 ( ( τ β γ q t T c a a k T p & ρ π β π γ 4 ( 4 ( 3 Dividi by 4πak yields the patial diffeetial equatio: q t T c k a T p & ρ β γ 6 ( ( (3.-57 The above equatio i dimesioless fom is: (3.-58 The fuctios ( γ ad ( β as defied i Sectio 3. as Eqs. (3-9 ad (3-0 ( ( γ ( K E, (3.-59 Ad + + β ( K. Whee E(x ad K(x ae the complete elliptic iteals, [66]. The bouday coditios imposed o Eq. (3.-59 ae: 0 0, ( Θ τ (alo the exteio of the doplet, ad (3.-60, ( τ Θ is fiite, (at the votex cete. (3.-6 Ulike the pevious aalysis of Sectio 3., the iitial coditio imposed o Θ is: 0 0, ( Θ τ. (3.-6

56 4 Equatio (3.-59 is solved usi sepaatio-of-vaiables techiques whee F is the steady-state solutio, ad G tasiet solutio. Θ (, τ F ( + G(, τ (3.-63 The iitial ad bouday coditios imposed o Eq. (3.-63 ae: Θ(, τ 0 0, Θ( 0, τ 0 (alo the oute steam lie, ad Θ (, τ is fiite at the votex cete. Substitutio of Eq. (3.-63 ito (3.-58 yields the followi: ( F + G β ( ( F + G γ ( 6 τ F G β ( + F G γ ( ( ( + 6 τ τ γ F G β ( β ( G ( + γ ( + ( τ The diffeetial equatio fo F( is: F β ( γ ( ε 6 (3.-65 Ad fo G(,τ: γ G β ( G (. ( τ Equatio (3.-65 may be solved usi fiite-diffeece methods with the followi bouday coditios:

57 43 F( fiite, ad F(0 0. The solutio fo the tasiet G(,τ may be obtaied usi sepaatio of vaiables with: G(, τ A Ξ( ω( τ (3.-67 Followi stadad sepaatio-of-vaiable techiques, Eq. (3.-66 may be estated as: β ( Ξ dξ γ ( d 6 dω ω τ λ (3.-68 The odiay diffeetial equatio fo ω(τ is the: dω + 6 λ ω 0 dτ (3.-69 A solutio to this diffeetial equatio is ω 6λ ( e τ. Similaly, the odiay diffeetial equatio fo Ξ( is the: β ( Ξ dξ γ ( d λ, dξ o γ ( + λ ( Ξ 0 β d (3.-70 The eie-fuctio Ξ ( ad the eie-values λ have bee foud usi fiitediffeece techiques with Newto iteatios to fid the eie-values, taki ito accout the followi bouday coditios: Fom Eq. (3.-67 Ξ ( 0 F( at the oute steam fuctio of the doplet, ad Ξ ( is fiite at the doplet votex cete

58 44 Recall fom Eq. (3.-63 that: 6 λ τ A Ξ ( e (3.-7 G (, τ Θ This Θ (, τ is of the fom: (, τ G(, τ F( + 6λτ Θ (, τ F( + A Ξ ( e (3.-7 Note that the fuctio, F(, is the steady-state fuctio ad the tems i the summatio become vey small as time iceases, especially fo lae values of. As such, except fo vey small times, the summatio i Eq. (3.-7 may be tucated without a siificat loss of accuacy. The diffeetial equatio fo the steady-state compoet of Eq. (3.-7, F(, is: d df β ( γ (. (3.-73 d d 6 The bouday coditios imposed o F(, ae the: df d 0 6 (eey balace at the oute steam fuctio, F( 0 0, (alo the exteio of the doplet, ad F( is fiite. (at the votex cete, Equatio (3.-73 was solved iteatively usi secod-ode fiite diffeece techiques. The tasiet compoet of Eq. (3.-7 was solved i a mae simila to the solutio pocedue used i Chapte Thee Sectio 3. fo heat tasfe without heat eeatio. Usi stadad sepaatio-of-vaiables techiques ad substitutio of

59 45 Eq. (3.-7 ito Eq. (3.-73, the coespodi odiay diffeetial equatio fo the eie-fuctioξ is: d dξ γ ( + λ β( Ξ ( 0. (3.-74 d d The bouday coditios imposed o Ξ ( ae: Ξ ( 0 0 (at the oute steam fuctio of the doplet, ad Ξ ( is fiite (at the doplet votex cete. Equatio (3.-74 is solved usi secod-ode fiite diffeece techiques. With this solutio pocedue, Eq. (3.-74 is tasfomed ito a matix system of equatios. Both of the fuctios γ ( ad β ( ae positive o the iteval (0,. I additio, lim γ ( 0. 0 Ude these cicumstaces, Eq. (3.-74 is a pope Stum- Liouville like poblem whee the coespodi eie-fuctios, the iteval (0, with espect to the weihti fuctio β (. That is: whee δ m is the Koecke delta fuctio. equies that: 0 [ Ξ ( ] β ( d Ξ, ae othooal o Ξ m ( Ξ ( γ ( d δ m (3.-75 The eie-values, λ, may be evaluated usi a eey balace aalysis which λ 0 ( 8 ( Ξ Ξ d β d. ( d 0 0 Oce the eie-values ad coespodi eie-fuctios ae ascetaied, the eiefuctios ae omalized so that

60 46 0 [ ( ] Ξ β ( d. (3.-77 The coefficiets, A, may be obtaied usi the othooality of the eiefuctios, Ξ, coupled with the iitial coditio, Eq. (3.-6: 0 F( + A Ξ ( (3.-78 The ie poducts of both sides of Eq. (3.-78 ae take with Ξ (. Sice the eiefuctios, Ξ, ae othoomal with espect to the weihti fuctio β ( m Ξ ( ( β ( d A F. ( The solutio pocedue used to fid the eie-fuctios, Ξ, ad the eie-values, λ, was simila to the pocedue epoted i chapte thee. As such, these eie-fuctios ad eie-values will be the same as those cited i Sectio 3.. Howeve, the coefficiets, A, will be diffeet. The bulk o aveae tempeatue, Θ (τ, may be calculated by taki the weihted aveae of Θ(,τ, with: Θ 0 Θ (, τ (, τ d vol 0 d vol Whee 3 a vol β ( π d 4 the β ( d 8/3 0

61 47 Ξ + Θ τ λ β β τ ( ( ( ( 8 3 ( B e d A d F (3.-80 Defii, Ξ 0 ( ( β d B, the bulk tempeatue is the: + Θ τ λ β τ 0 6 ( ( 8 3 ( e B A d F (3.-8 All iteatios wee pefomed usi the tapezoidal ule usi the same ode poits as wee used fo the fiite diffeece scheme. As a pactical matte, the summatios used i Eq. (3.-7 ad subsequet equatios ae tucated such that: τ λ τ max 6 ( (, ( e A F Ξ + Θ, ad ( Θ τ λ β τ 0 6 max ( ( 8 3 ( e B A d F. (3.-83 To calculate, ( max τ Θ use Eq. (3.-77 whee ( ( 6, ( max + Ξ Θ τ λ τ F e A (3.-84

62 48 CHAPTER FOUR RESULTS 4. Heat Tasfe fom a Taslati Doplet at Hih Peclet Numbes: Revisiti the Classic Solutio of Koi ad Bik Whe usi umeical iteatio ad fiite-diffeece techiques, it is impotat to kow if the umeical id is small eouh to isue coveece. To illustate the coveece of the solutio with Δ, the values of A ad λ ae ive i Table (4-. As see i Table (4-, the coveece with espect to Δ is apid fo the fist few modes. Howeve, the coveece fo hihe-ode modes is slow. Nevetheless, except at vey small time itevals, the hihe ode tems ae ielevat. Fo example, with λ the thid mode is isiificat (elative to the fist mode fo τ > 0.0. Likewise, with λ 30,394.5 the thitieth tem is elatively isiificat fo τ > I Table (4- the fist two eie-fuctios ad coespodi eie-values may be compaed with those obtaied by Koi ad Bik. I the oiial wok of Koi ad Bik, max was chose at oly two.

63 49 Thei appoximate solutio was of the fom: Eie-fuctios ad Coefficiets pe Ref. [66]. A,,.3 λ.678 Ξ( ad A, 9.83, 0.73 λ Ξ( Table (4-: Coveece with Respect to Δ Δ 0.0 Δ Δ Δ Δ A.349 A.349 A.349 A.349 A.349 λ.6778 λ.6777 λ.6777 λ.6777 λ.6777 A A A A A λ λ λ λ λ A A A A A λ λ λ λ λ A A A A λ λ λ λ A A A λ 0,09. λ 0,054. λ 0,06.9 A A λ 30,354.6 λ 30,394.5 Fo compaiso puposes, plots of the poduct A i Ξ i ( fo the fist two eiefuctios ae illustated i Fi. (4-.

64 Koi & Bik Peset Α Ξ ( Α Ξ ( Fi. Fiue : Fist 4- Two Fist Eiefuctios, two eie-fuctios, Peset peset vs. Koi vs. Koi & Bikad Bik Fo the fist eie-fuctio, the esults of this study poduces a ood match with those of Koi ad Bik. Howeve, fo the secod mode, the match is ot ood. The same is tue fo the coespodi eie-values. Fo Koi ad Bik s solutio the eie-fuctios wee limited to quadatic polyomials. I additio, umeical iteatio was (appaetly pefomed usi a value of Δ 0.. Thus, it is ot supisi that the pedictios of Koi ad Bik ae bette fo the fist mode tha they ae fo the secod mode. Hece, the solutio of Koi ad Bik is accuate oly fo the fist mode. As such, this classic solutio will be accuate oly whe the secod mode is isiificat. That is, it will be accuate whe: exp ( 6λ τ <<

65 5 O λ τ 0. 3 > Sice λ 8. 60, it follows that the Koi ad Bik solutio should be accuate fo τ > Koi & Bik + Koi & Bik Peset Peset Results Θ(τ τ Fiue Fi. 4- : Bulk Tempeatue, Θ (τθ(τ The bulk tempeatue pedicted by the Koi ad Bik model is the: Θ 6.85τ 57. τ (.3 + ( 0.73 e 3 ( τ e (Koi ad Bik (4.- The bulk tempeatue, Θ (τ, pedicted by the peset model Eq. (3.-0 is compaed with the bulk tempeatue pedicted by the Koi ad Bik model o Fi. (4-. Notice that fo τ > the bulk tempeatue pedicted by Koi ad Bik is ealy idetical to that pedicted by the peset model. The ood fit illustated o Fi. (4- fo τ > seems to cotadict the ealie statemet that the Koi ad Bik solutio was oly accuate fo τ > 0.03; howeve, the accuate bulk tempeatue pedictios of Koi ad Bik fo τ > do ot imply that thei model is accuate with espect to tempeatue pofiles at such low times.

66 5 + Koi & Bik Peset Θ(,τ τ τ Fiue 4-3 Fi. 3: Θ(,τ alo Θ(, θ τ alo π/, Koi θ π /, & B ik vs. P eset Koi ad Bik vs. peset The tempeatue pofiles pedicted by Koi ad Bik ae ive by: o, Θ(, τ.3 Θ(, τ ( τ ( τ ( e ( e 6.85τ 57.3τ ( ( e e (Koi ad Bik These pofiles ae compaed with the peset model Eq. (3.-9 o Fi. (4-3. The tempeatue pofiles ae plotted o Fi.(4-3 fo τ ad τ The details of the tempeatue pofile pedicted by the Koi ad Bik solutio ae ot vey accuate at τ Yet the tempeatue pofile pedicted by thei model is quite accuate at τ 0.03

67 53 Usi Eq. (3.- the Nusselt umbe based o the Koi ad Bik solutio is: 6.85τ 57.3τ ( 46.78e e 6.85τ 57.3τ ( e e Nu (Koi ad Bik ( Peset Solutio Nu Koi & Bik 0 Nu τ Fiue Fi. 4: 4-4 Nusselt Numbe, Koi & ad Bik Bik v. vs. Peset The Nusselt umbes, as pedicted by the peset model Eq. (3.- ad the Koi ad Bik model ae plotted o Fi. (4-4 as fuctio of τ. The Nusselt umbe based o these two models diffes substatially at low times. Howeve, fo τ > 0.03 they covee. Fo τ > 0.05, both the Koi ad Bik as well as the peset solutio covee to a steady value fo the Nusselt umbe of Nu lim Nu (4.-3 τ

68 54 Iteestily, both of the mooaphs metioed above eoeously state that the steadystate value fo the Nusselt umbe, accodi to Koi ad Bik, is The steadystate value fo the Nusselt umbe of 7.90 was also epoted by Olive ad DeWitt [4] Fially, the tempeatue pofiles pedicted i chapte Thee Sectio 3. ae plotted at vaious times o Fi. (4-5 fo τ Note that the cete of the votex, at, is udistubed util τ τ Θ(,τ τ τ 0.0 τ τ Fiue 4-5 Θ(, τ alo θ π / Fi. 5: Θ(,τ alo θ π/

69 55 4. Heat Tasfe fom a doplet at Hih Peclet Numbes with Heat Geeatio To illustate the coveece of the solutio with Δ, the values of A ad λ ae ive i Table (4-. As displayed i Table (4-, the coveece with espect to max ad Δ is apid fo the fist few modes. O the othe had, the coveece fo hihe-ode modes is slowe. The coefficiets, A, become iceasily small as iceases. Ispectio of Eq. (3.-8 demostates that, except at vey small times, the hihe ode tems ae ielevat due to the expoetial opeato. Table (4-: Coveece with Respect to Δ ad max Δ 0.0 max 5 Δ max 0 Δ max 30 A A A λ.6778 λ.6777 λ.6777 Δ max 30 A λ.6777 A A A A λ λ λ λ A A A A λ λ λ λ A A A λ λ λ A A λ λ A A λ λ

70 56 Fo pactici eiees, it is ofte valuable that simple, yet easoably accuate, appoximatios be epoted fo impotat paametes. The steady-state tempeatue pofile value may be well appoximated by the elatio: limθ (, τ F( 8 (4.- Substitutio of Eq. (7 ito yields: limθ (, θ, τ ( si θ (4.- limθ (, τ F( 8 Equatio (4.- may be exteded to fiite times by icludi a appoximatio of Eq. (3.-7 that is tucated at the fist mode while isui that the iitial coditio, Eq. (3.-4a is satisfied: Θ (, θ, τ 9 τ ( θ ( λ 4 6 si e. Fom Table (4-, the value of λ is Hece, τ > τ ( si θ ( e Θ (, θ, τ ( Ispectio of Fiue (4-6 idicates that Eq. (4.- is easoably accuate fo

71 Θ(, τ Steady-state Steady-State SoSolutio Eq. (37 Eq. (4 Eq. (3.-8 Eq. (4.-3 τ 0. τ τ Fiue 4-6 Θ( alo θ π / at seveal times Fiue (4-6 shows that the vey simple atue of Eq.(4.-3 eflects well the tempeatue distibutio except fo low times. This is coveiet sice Eq. (4.-3 is based o oly oe mode. A appoximatio fo the bulk tempeatue may be obtaied fom Eq. (4.-3 by applyi Eq. (3.-33: 6.8τ ( e 8 Θ ( τ ( This appoximatio is compaed with the pedictios of Eq. (3.-83 i Fi. (4-7. As may be see, Eq.(4.-4 pedicts the bulk tempeatue well, except at vey low times.

72 pue coductio Pe Θ Θ pue coductio Pe Pe Fiue 4-7 Θ as a fuctio of time. τ Fiue (4-7 shows a compaiso fo bulk tempeatue betwee pue coductio whee Pe zeo ad at vey hih Peclet umbes. It shows that as Pe the tempeatue pofile becomes steady much moe apidly tha is the case fo Pe 0. This is due to the fact that the maximum tempeatue of the doplet is at the cete of the votex at hih Peclet umbe, while it is at the doplet cete fo low Peclet umbes. Vey ecetly, Cal Fayeweathe [3] used a ADI method to umeically pedict the bulk tempeatue fo the same poblem ivestiated i this sectio. Fayeweathe plotted the bulk tempeatue as a fuctio of time ad Peclet umbe. His pedictios ae illustated i Fiue (4-8. This aph is a sto evidece to pedict the validity of my wok as limiti cases.

73 Fiue 4-8 ADI pedictios of bulk tempeatues. Copied fom Fayeweathe 59

74 60 As illustated i Fi. (4.9 the heat tasfe emaates fom the votex cete fo hih Peclet umbes, ad fom the doplet cete fo low Peclet umbes. Pe Pe 0 Fiue 4-9 Illustatio of heat flux diectios: Hih Peclet umbe vs. Low Peclet umbes As the Peclet umbe appoaches ifiity, the maximum tempeatue moves to the cete of the votex which is close to the doplet suface as show i Fi. (4-9.This is compaed with the case whe Pe 0, whee the maximum tempeatue is at the cete of the doplet. Notice i Fi.(4-9 that the cete of the votex is close to the doplet suface tha is the cete of the doplet. So we would expect lowe tempeatues at hih Peclet umbes tha at low Peclet umbes, ive the same value of q&.

75 pue coductio Pe Θ max Pe Fiue 4-0 Maximum Tempeatues τ Fiue (4-0 shows the maximum tempeatue i the doplet. The steady-state values fo the maximum tempeatue ae fo pue coductio, ad fo hih Peclet Numbes.

76 6 CHAPTER FIVE CONCLUSION AND FUTURE WORK 5. Coclusio The classic solutio of Koi ad Bik fo tasfe fom a doplet with hih Peclet umbe has bee evisited usi a solutio pocedue that icluded up to 30 modes. Plots of the bulk tempeatue, the spatially distibuted tempeatue, ad the Nusselt umbe wee obtaied. Fially, the Nusselt umbe fo lae times was show to be 7.90, athe tha the peviously (ad eoeously epoted value of It was foud that the Koi ad Bik model accuately pedicts the bulk tempeatue of the doplet fo τ > 0.003; howeve, the Koi ad Bik solutio did ot accuately model the details of the tempeatue pofile ad the Nusselt umbe util τ > As such, the peset model is moe accuate fo shot times, (τ < I additio, the tasfe fom a doplet with a distibuted heat eeatio has bee pedicted fo two special cases: at vey low Peclet umbes, ad at vey hih Peclet umbes. I both cases, all of the esistace to heat tasfe was assumed to be i the doplet (iteio poblem. A compaiso has bee made betwee these two cases to calculate the bulk ad the maximum tempeatues which ae poits of iteests. As may be expected, the doplet tempeatue aises less fo hih Peclet umbes tha it does fo low Peclet umbes. Similaly, the doplet tempeatue pofile espods

77 63 moe apidly at hih Peclet umbes tha it does at low Peclet umbes. A simple equatio was poposed that pedicts well the bulk tempeatue fo hih Peclet umbes: Θ ( τ τ ( e The above aalyses peseted the fist aalysis, kow to the autho, of heat tasfe fom a doplet with heat eeatio at hih Peclet umbes. 5. Suestios fo futue wok This wok had at least fou substatial limitatios: It was limited to the iteal poblem, whee the taspot esistaces wee assumed to be i the doplet phase; Oly limiti Peclet umbes wee ivestiated (Pe 0 ad Pe ; 3 The peset ivestiatio was theoetical without expeimetal veificatio; 4 The flow field was limited to ceepi flow (Re <. As such, it is ecommeded that futhe ivestiatios expad o this aalysis as suested i the followi paaaphs. Fo may heat tasfe poblems the pimay themal esistace is ot just i the doplet. Ude these cicumstaces, the peset iteio model will ude pedict the actual tempeatues. This wok was limited to aalyses whee the Peclet umbe was zeo o ifiity. I most cases, the Peclet umbe will take some fiite, itemediate value. Howeve, this wok ivestiated two limiti values fo the iteio poblem with heat eeatio, (e..

78 64 Pe 0 ad Pe. It is expected that the tempeatue fo itemediate Peclet umbes will be betwee these two limiti cases. Futue aalysis fo fiite Peclet umbes will ivolve at least a additioal spatial dimesio, Θ ( T,, θ. A ADI method o umeical aalysis ca be used to solve this kid of equatios. As this study was puely theoetical, a expeimetal study is ecessay to cofim the esults of the peset pedictios. It is well kow that theoetical ivestiatios aloe do ot always pedict well the physics of may pactical situatios. Fially, this ivestiatio assumed a flow field associated with vey low Reyolds umbes (e.. Re <<. May doplets will have modeate Reyolds umbes. This aalysis may ot be valid ude these coditios. 5.3 Fial Thouhts It is hoped that this wok will ispie othes to cotiue ivestiatios ito heat ad mass tasfe i dops with heat eeatio. This ivestiatio has demostated that the classic solutio of Koi ad Bik was oly valid at low times. I additio, this wok peseted the fist aalysis of heat tasfe i hih Peclet umbe doplets with heat eeatio fo the iteio poblem. This wok should fid elevace i heat ad mass tasfe opeatios associated with doplets.

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82 68 [8] Gh. Jucu, Cojuate heat/mass tasfe fom a cicula cylide with a iteal heat/mass souce i lamia coss-flow at low Reyolds umbes, It. J. Heat Mass Tasfe 48: 49-44, 005. [9] Gh. Jucu, Cojuate mass tasfe to a spheical dop accompaied by a secod ode chemical eactio iside the dop, It. J. Heat Mass Tas., vol. 45: , 00. [30] Gh. Jucu, Cojuate usteady heat tasfe fom a sphee i Stokes flow, Chem. E. Sci. 5: , 977. [3] Gh. Jucu, R. Mihail, The effect of diffusivities atio o cojuate mass tasfe fom a doplet, It. J. Heat Mass Tasfe 30: 3-6, 987. [3] Gh. Jucu, The ifluece of the cotiuous phase Pe umbes o themal wake pheomeo, J. Heat mass tasfe 34: 03-08, 998. [33] Gh. Jucu, The ifluece of the physical popeties atios o the cojuate heat tasfe fom a dop, Heat Mass Tasfe 35:5-57, 999. [34] Gh. Jucu, Usteady cojuate heat tasfe fo a sile paticle ad i multi paticle systems at low Reyolds umbes, It. J. Heat Mass Tasfe 4: , 998. [35] Gh. Jucu, Usteady heat ad/o mass tasfe fom a fluid sphee i ceepi flow, It. J. Heat Mass Tasfe 44: 39-46, 00. [36] H. Baue, Usteady state mass tasfe thouh the iteface of spheical paticles, It. J. Heat Mass Tasfe : , 978.

83 69 [37] H. Watada, A. E. Hamielec ad A.I. Johso, A theoetical study of mass tasfe with chemical eactio i dops, Ca. J. Chem. E. 48: 55-6, 970. [38] H.A. Dwye, R.J. Kee ad B.R. Sades, Adaptive id method fo poblems i fluid mechaics ad heat tasfe, AIAA jl 8: 05-, 980. [39] H.Bee, Foced covectio heat ad mass tasfe at small Peclet umbes fom a paticle of abitay shape, Chem. E. Sci 8: 09-5, 963. [40] H.D. Nuye, S. Paik, J.N.C. Chu, Usteady cojuate heat tasfe associated with a taslati doplet: a diect umeical simulatio, Heat tasfe A 4: 6-80, 993. [4] H.F. Baue & W. Eidel, Maaoi covectio i a spheical liquid system. Acta Astoautica, 5: 75-90, 987. [4] H.S. Caslaw, J.C. Jaee, Coductio of heat i solids, Oxfod Uivesity Pess, Oxfod, 947. [43] Hadamad, J., Mouvemet pemaet let d ue sphee liquide et visqueuse das u liquide visqueux, C.R. Acad. Sci, vol.5: , 9. [44] J. Fouie, Theoie aalytique de la Chaleu, Pais 8. [45] J.H. Hake ad J. Ahmadzadeh, The effect of electic fields o mass tasfe fom falli dops. It. J. Heat Mass Tasfe, 7: 9-5, 974. [46] J.L. Adeso, Doplet iteactios i themocapillay motio. It. J. Multiphase Flow, :83-84, 985.

84 70 [47] J.N. Chu & D.L.R. Olive, Tasiet heat tasfe i a fluid sphee taslati i a electic field. ASME J. Heat Tasfe, : 84-9, 990. [48] J.N. Chu, The motio of paticles iside a doplet. ASME J. Heat Tasfe, 04: , 98. [49] K. D. Bato & R.S. Subamaia, The miatio of liquid dops i a vetical tempeatue adiet. J. Colloid Iteface Sci., 33:-, 989. [50] K. Schueel, R. Haesel, E. Schlichti, W. Halwachs, Reactive extactios, It. Chem. E. 8: , 988. [5] L.E. Johs, J. ad R.B. Beckma, Mechaism of dispesed phase mass tasfe i viscous, sile dop extactio systems, A.I.Ch.E. : 0-6, 966. [5] L.S. Cha & J.C. Be, Fluid flow ad tasfe behavio of a dop taslati i a electic field at itemediate Reyolds umbes. Itl. J. Heat ad Mass Tasfe, 6: , 98. [53] L.S. Cha T.E. Chaleso & J.C. Be, Heat ad mass tasfe to a taslati dop i a electic field. It. J. Heat Mass tasfe, 5:03-030, 98. [54] Leoid S. Kleima ad X. B. Reed, Usteady Cojuate Mass Tasfe betwee a Sile Doplet ad a Ambiet Flow with Exteal Chemical Reactio, Id. E. Chem. Res., vol. 35: , 996. [55] M. Hähel, V. Delitzsch, & H. Eckelma. The motio of doplets i a vetical tempeatue adiet. Phys. Fluids A, : , 989. [56] M.D. Va Dyke, A model of seies tucatio applied to some poblem i fluid mechaics, Stafod Uivesity Repot SUDAER 47, 965.

85 7 [57] N. Koopliv ad E.M. Spaow, Tasiet heat tasfe betwee a movi sphee ad a fluid, 4 th It. Heat Tasfe Cof., III, FC7.4, Pais-Vesailles,970. [58] N.O. You, J.S. Goldstei, & M.J. Block, The motio of bubbles i a vetical tempeatue adiet. J. Fluid Mech., vol. 6: , 959. [59] P.Aamalai, N.Shaka, R. Cole, & R.S. Subamaia, Bubble miatio iside a liquid dop i a space laboatoy. Appl. Sci. Res., 38:79-86, 98. [60] P.J. Ataki, Tasiet pocesses i a iid sluy doplet dui liquid vapoizatio ad combustio. Comb. Sci. Techo., 46:3-35, 986. [6] P.J. Bailes & J.D. Thoto, Electically aumeted liquid-liquid extactio i a compoet system.. Multi doplet studies. I poc. It. Solvet. Extactio Cof., vol. :0-07, Lyos, 974. [6] P.N. Choudhouy, D.G. Dake, Usteady heat tasfe fom a sphee i a low Reyolds umbe flow, Quat. J. Mech. ad Appl. Math 4: 3-36, 97. [63] P.O. Bu, heat o mass tasfe fom sile sphees i a low Reyolds umbe flow, It. j. E Sci. 0: 87-8, 98. [64] P.S. Ayyaswamy, Diect cotact tasfe pocesses with movi liquid doplets. I advaces i Heat Tasfe, vol. 6, Academic Pess, New Yok, 995. [65] P.S. Ayyaswamy, Mathematical method i diect-cotact tasfe studies with doplets. I Aual Review of Heat Tasfe, vol.7, Beell House, New Yok, 996. [66] R. Koi ad J.C. Bik, O the Theoy of Extactio fom Falli Doplets, Appl. Sci. Res. Se. A, :4-54, 950.

86 7 [67] R.J. Haywood, N. Nafzie, ad M. Resibulut, A detailed examiatio of as ad liquid phase tasiet pocesses i covective doplet evapoatio, J. Heat Tasfe : , 989. [68] S.C.R. Deis, J.D.A. Walke ad J.D. Hudso, Heat tasfe fom a sphee at low Reyolds umbe, J. Fluid Mech.60: 73-83, 973. [69] S.K. Fiedlade, Mass ad heat tasfe to sile sphees ad cylides ad low Reyolds umbe, AIChE J. 3: 43-48, 957. [70] S.K. Giffiths & F.A. Moiso J., Low Peclet umbe heat ad mass tasfe fom a dop i a electic field, J. Heat Tasfe, 0: , 979. [7] S.K. Giffiths & F.A. Moiso J., The taspot fom a dop i a alteati electic field, It. J. Heat Mass Tasfe, 6: 77-76, 983. [7] S.L. Soo, Multiphase Fluid dyamics, Sciece Oess Beiji, 990. [73] S.S. Sadhal, P.S. Ayyaswamy, ad J.N. Chu, Taspot Pheomea with Dops ad Bubbles by Spie, Beli, 997. [74] See Eqs ad of M. Abamowitz ad I. Steu, Hadbook of Mathematical Fuctios, Natioal Bueau of Stadads, 964. [75] T.D. Taylo, ad A. Acivos, O the Defomatio ad Da of a Falli Viscous Dop at low Reyolds Numbe, J. Fluid Mech., Vol. 8: [76] V. Ya. Rivkid, G.M. Ryski, Flow stuctue i motio of a spheical dop i a fluid medium at itemediate Reyolds umbes, Fluid Dy.: 5-, 976.

87 73 [77] V.G. Levich, Physicochemical Hydodyamics, Petice Hall, Elewood Cliffs, NJ, 96. [78] V.G. Levich, V.S. Kylov, V.S. Vootili, O the theoy of o steady diffusio fom a movi doplet (Russia. Dokl. Akad. Nauk, 6 (3: , 965. [79] W. Halwacks, K. Schueel, Ivestiatio of eactive extactio o sile doplets, Chem. E. Sci. 38: , 983. [80] W.A. Siiao, Fluid Dyamics ad Taspot of Doplets ad spays, Cambide Uivesity Pess, Cambide, 999. [8] W.C. Che ad R. Pfeffe, Local ad ove-all mass tasfe ates aoud solid sphees with fist ode homoeous chemical eactio, Id. E. Fudam. 9(: 0-07, 970. [8] W.F Ames, Numeical Methods fo patial Diffeetial Equatios, Academic Pess, New Yok,977. [83] W.Y. Sou, J.T. Seas, Effect of eactio ode ad covectio as bubbles i a as-liquid eacti system, Chem. E. Sci. 6: [84] Y.H. Pao, Usteady mass tasfe with chemical eactio, Chem. E. Sci. 9: 694, 964. [85] Y.T. Shah, Mass tasfe fom a sile movi spheical bubble i the pesece of a complex chemical eactio, Ca. J. Chem. E. 50:74-79, 97. [86] Z.G. Fe, E.E. Michaelides ad E. Efstathios, A umeical study o the tasiet heat tasfe fom a sphee at hih Reyolds ad Peclet umbes, It. J. Heat Mass Tasfe 43: 9-9, 000.

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89 75 Appedix Applicatio to Mass Tasfe This aalysis may be applied to mass tasfe. Fo a explaatio of the aaloy betwee heat tasfe ad mass tasfe see pp. 0-4 of efeece []. Coside a paet species A that poduces the dauhte species, thouh a chemical o uclea eactio, at a costat volumetic ate, R &. Futhe, assume that the dauhte species is stable without futhe eactio o decay. A diffeetial mass balace of the dauhte species with a coespodi eeatio tem yields the followi diffeetial equatio: c + c θ + θ θ si θ si Ra & D Pe Sc c v c c u + + θ τ (A. Whee c is the dauhte species cocetatio level, D is the mass diffusivity, R & is the volumetic ate of poductio of the dauhte species, Sc is the Schmidt umbe, ad τ is the dimesioless time: μ Sc D ρ dop D t ad τ. (A. a Fo mass tasfe, the vaiable Θ epesets the dimesioless species cocetatio with: ( c c D Θ s (A.3 R& a With these substitutios, ad the associated implied assumptios, the above aalysis is applicable to mass tasfe.

90 76 Appedix Extactio fom falli doplets with ad without Heat Geeatio clc; clea; close path(path,'u:\phd\paul\adham'; hold o; fomat lo; max 30; % umbe of eievalues % mode ; step /000; % icemet fo xi debu 0; ew 0; if ew % debui % 0 o ew values. This eables us % to avoid compili each values aai % Computatio of p(xi ad q(xi x step : step : -step; fo k : leth(x xi x(k; m (-sqt(xi/(+sqt(xi; [K,E] ellipke(m; p(k /3*(sqt((+sqt(xi*((4-3*xi*E- (4*sqt(xi-3*xi*K; q(k /(sqt(+sqt(xi*k; xiv(k xi; ed p(leth(x+ 0; xi ; m (-sqt(xi/(+sqt(xi; [K,E] ellipke(m; q(leth(x+ /(sqt(+sqt(xi*k; p; q;

91 77 % Solve S (Steady State ite ; bc.05; dbc.0; while dbc > 0^-5 fo : leth(x f( -q(/6; if eqm(,+ p(/step^ + (p(+- 8/3/(*step^; else eqm(,- p(/step^ - (p(+-p(- /(*step^; if < leth(x eqm(,+ p(/step^ + (p(+-p(- /(*step^; ed ed eqm(, -*p(/step^; ed f( f( - bc * (p(/step^ + (p(+-p(- /(*step^; Siv(eqm*f'; if ite Sp( (4*S(-S(/(*step - /6; Sp( Sp(; bc bc + dbc; ite ; else Sp( (4*S(-S(/(*step - /6; if si(sp( si(sp( bc bc + dbc; else bc bc - dbc*9/0; dbc dbc / 0; ed ed ed S [S ; bc-dbc];

92 78 % Computatio of the (eievalues ad eiefuctios % The elatio betwee mu ad lambda is such that : % mu lambda^ f0; ite ; lambda ; % fist uess muv lambda^; % Loop ove the max modes fo mode : max delta 0.; muv lambda.^; mu muv(mode; while delta > 0^-5 fo : leth(x eqm(, -*p(/step^ + mu*q(; if eqm(,+ p(/step^ + (p(+- 8/3/(*step^; else eqm(,- p(/step^ - (p(+-p(- /(*step^; if < leth(x eqm(,+ p(/step^ + (p(+- p(-/(*step^; ed ed ed f( - (p(/step^ + (p(+-p(- /(*step^; path(path,'u:\phd\tidiaoal solutio'; diaoal dia(eqm; diaoal_uppe dia(eqm,; diaoal_lowe dia(eqm,-; [ L, U, U ] ti_facto ( diaoal_lowe, diaoal, diaoal_uppe ; Z ti_solve (L,U,U,f; Z[Z;]; m leth(z; deo (q * Z.^ - /*q(m*z(m^ * step; Z /sqt(deo * Z; % to omalize Z ume (q*z - /*q(m*z(m * step; if ite

93 79 mu; mu; ed F 8/3 * (-Z(+4*Z( / (*step*ume - F F; ite ; mu mu + delta; else F 8/3 * (-Z(+4*Z( / (*step*ume - if si(f si(f mu mu + delta; F; else mu mu - delta; delta delta / 0; ed ed ed ite ; ZZ(:,mode Z; A(mode ume; muv(mode mu; lambda(mode sqt(mu; lambda(mode+ lambda(mode + 0.; % S % ZZ % muv %***************************************** % Computatio of A (coefficiet m leth(s; fo : max A( 0; fo k : m A( A( - q(k*s(k*zz(k, * step; ed A( A( + /*q(m*s(m*zz(m, * step; ed if step /00 save koko00 elseif step /00 save koko00 elseif step /500

94 80 save koko500 elseif step /000 save koko000 else save koko ed else if step /00 load koko00 elseif step /00 load koko00 elseif step /500 load koko500 elseif step /000 load koko000 else load koko ed ed xi [0 ; x';]; /*(-*(-xi+.^(/.^(/; /*(+*(-xi+.^(/.^(/; fo i : leth( i(i (leth(+-i; ed [ ; i']; time [ [ : -.05 : 0.] ]; fo j : leth(time t time(j; fo k : leth(s T 0; fo : max T T + A(*ZZ(k,*exp(-6*muv(*t; ed tt(k T; ed T S' + tt; T [0 ; T']; fo i : leth(t Ti(i T(leth(T+-i; ed Temp(:,j [T;Ti']; plot(,temp(:,j,'k' % plot(,temp(:,j,'k',,temp(:,j,'k' ed

95 8 text(.5,0.008,'t0.0065','fotsize', text(.5,0.0,'t0.003','fotsize', text(.5,0.033,'t0.065','fotsize', text(.3,.045,'steady state','fotsize', xlabel('','fotsize',3 ylabel('\theta','fotsize',3 title('\theta S. State + Tasiet','fotsize',4 if step /00 text(.05,.055,'step /00' elseif step /00 text(.05,.055,'step /00' elseif step /500 text(.05,.055,'step /500' elseif step /000 text(.05,.055,'step /000' else text(.05,.055,'use step' ed id off >>> % To calculate bulk Tempeatue v/step % Peifiity % to calculate B (Z*q fo m :max; fo ::v; alpha(,mzz(,m*q(; ed ed fo m:max; sumalpha(m0 fo ::(v-; sumalpha(msumalpha(m+alpha(,m; ed ed fo m::max B(m(sumalpha(m+alpha(v,m/*step ed

96 8 % daw theta vesus time daw fo peifiity theta0 t[0:0.0:] fo m::0 thetatheta + 3/8*A(m*B(m*(exp(-6*lambda(m^*t- ed %plot(t,theta % daw theta vesus time pue coductio Pe0 theta0 t[0:0.0:] fo ::max lambda*pi theta theta+6/(lambda^4*exp(-lambda^*t ed theta/5-theta hold o plot(t, theta,t,theta,'-' >>> t[0:0.00:;] % To calculate maximum tempeatue %thetamaximum(peifiity %at the cete of votex j/step sum0; fo i:max sum sum +A(i*ZZ(j,i*exp(-muv(i*6*t; ed thetamax sum +S(j plot (t,thetamax hold o

97 83 %pue coductio (pe0 t[0:0.00:]; thetamax0 thetamax0 fo :max thetamax thetamax + (*(-^/(*pi^*exp(- (*pi^*t ed thetamaxthetamax + /6 %plot(t,thetamax plot(t,thetamax,t,thetamax hold o >>> % To plot a seies of costat xi % cuves pe Koi & Bik % path(path,'c:\documets ad % Settis\dolive\My Documets\ % matlab\reseach\koibik' % >>>>>>Uppe cicle xi[.0880,.045,.464,.8560];% values of xi ds0.0; hold off hold o fo j :leth(xi k xiplot(xi(j,ds, ed % plot uit cicle x(0;y(0; theta [0:pi/00:pi]; fo j :leth(theta; x(j+cos(theta(j; y(j+si(theta(j; ed x(j+0; y(j+0; plot(x,y; % >>>>>Lowe cicle xi[.0943,.440,.8560];% values of xi ds0.0; fo j :leth(xi

98 84 k xiplot(xi(j,ds,- ed % plot uit cicle x(0;y(0; theta [0:pi/00:pi]; fo j :leth(theta; x(j+cos(theta(j; y(j+si(theta(j; ed x(j+0; y(j+0; plot(x,-y; % axis([0,,0,] >>> %%% xi plot FUNCTION % To plot a cuve of costat xi fo a % Koi ad Bik Sphee % xi - value fo xi % ds - appoximate step size fo plots % facto: fo uppe cicle, % - fo lowe cicle fuctio jxiplot(xi,ds,facto; j ; Uj(jsqt((+sqt(-xi/; % uppe side of cicle thetauj(jpi/; zsi ; while zsi > 0; j j+; Uj(j-;thetathetaUj(j-; dtheta ds/sqt((^3-^/(- *^^*(cos(theta/si(theta^ + ^; d (^3-/(-*^*(cos(theta/si(theta*dtheta; 0 +d;theta0theta+dtheta; zetata (0*cos(theta0^4/(*0^-; [Uj(j,thetaUj(j]xizeta(xi,zetata,0,theta0; zsi si(zetata; ed j ; Lj(jsqt((-sqt(-xi/; % lowe side of cicle thetalj(jpi/; zsi -; while zsi < 0;

99 85 j j+; Lj(j-;thetathetaLj(j-; dtheta ds/sqt((^3-^/(- *^^*(cos(theta/si(theta^ + ^; d (^3-/(-*^*(cos(theta/si(theta*dtheta; 0 +d;theta0theta+dtheta; zetata (0*cos(theta0^4/(*0^-; [Lj(j,thetaLj(j]xizeta(xi,zetata,0,theta0; zsi si(zetata; ed xu Uj.*cos(thetaUj; yu facto*uj.*si(thetauj; xl Lj.*cos(thetaLj; yl facto*lj.*si(thetalj; plot(xu,yu,'k',xl,yl,'k',-xu,yu,'k',-xl,yl,'k' %%% xi zeta FUNCTION % Fid ad theta, ive xi & zeta fom Koi & Bik fuctio [,theta]xizeta(xita,zetata,0,theta0 t0;xcos(theta0; dr [;]; while sqt(dr'*dr > t^; 3t*; 4t*3; 5t*4; xi 4*(-4*(-x^; F xi-xita; zeta (t*x^4/(*-; F zeta-zetata; Jacobia [(8*t-6*3*(-x^, -8*(-4*x ;... 4*x^4*(5-3/(*-^, 4*4*x^3/(*t^- ]; dr -iv(jacobia*[f;f]; t t+dr(; x x+dr(; ed t; theta acos(x; >>>>

100 86 %%% Tisolve FUNCTION % fuctio x ti_solve ( L, U, U, b % % TRI_SOLVE solves a tidiaoal liea system factoed by TRI_FACTOR. % % Iput, eal L(N-, the subdiaoal elemets of the uit lowe % tiaula facto of A. % % Iput, eal U(N, U(N-, the diaoal ad supediaoal elemets % of the uppe tiaula facto of A. % % Iput, eal b(n, the iht had side. % % Output, eal x(n, the solutio of the liea system. fuctio Z ti_solve ( L, U, U, f NN leth ( U ; % Solve L * y b. y zeos(nn,; y( f(; fo i : NN y(i f(i - L(i- * y(i-; ed % % Solve U * Z y. % Z zeos(nn,; fo i NN : - : Z(i y(i / U(i; y(i- y(i- - U(i- * Z(i; ed Z( y( / U(; >>>> %%% Tifacto FUNCTION % fuctio [ L, U, U ] ti_facto ( A, A, A3 % % TRI_FACTOR computes the LU factos of a compact stoae tidiaoal matix. % % Discussio:

101 % % The aloithm assumes that Gaussia elimiatio ca be pefomed % without pivoti. % % The tidiaoal matix A is stoed i a compact fomat. % A(:N- cotais the elemets of the subdiaoal; % A(:N cotais the elemets of the diaoal; % A3(:N- cotais the elemets of the supediaoal. % % This stoae aaemet is suested by the followi pictues: % % Whee the eties of A et stoed i A, A, A3: % % A( A3( % A( A( A3( 0 0 % 0 A( A(3 A3(3 0 % 0 0 A(3 A(4 A3(4 % A(4 A(5 % % How the eties of A ae stoed i A, A, A3: % % A(, A(, A(, % A(3, A(, A(,3 % A(4,3 A(3,3 A(3,4 % A(5,4 A(4,4 A(4,5 % A(5,5 % % The output matices L ad U ae also stoed compactly: % % U( U( %L( U( U( 0 0 % 0 L( U(3 U(3 0 % 0 0 L( U(4 U(4 % L( U(5 % % ito the vectos L, U ad U: % % L(, U(, U(, % L(3, U(, U(,3 % L(4,3 U(3,3 U(3,4 % L(5,4 U(4,4 U(4,5 % U(5,5 % % Modified: % 87

102 88 % 30 Mach 000 % % Autho: % % Joh Bukadt % % Paametes: % % Iput, eal A(N-, A(N, A3(N-, the subdiaoal, diaoal, ad % supediaoal elemets of the matix A. % % Output, eal L(N-, the subdiaoal elemets of the uit lowe % tiaula facto of A. % % Output, eal U(N, U(N-, the diaoal ad supediaoal elemets % of the uppe tiaula facto of A. fuctio [ L, U, U ] ti_facto ( A, A, A3 L A; U A; U A3; leth ( U ; fo i : U(i U(i - U(i- * L(i- / U(i-; L(i- L(i- / U(i-; ed >>>