A contribution towards the numerical study of bubble dynamics in nucleate boiling at local scale using a conservative level set method

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1 Thesis Pesented to the Univesity of Cegy Pontoise in patial fulfillment of the degee equiements fo the degee of Docto of Philosophy in Civil Engineeing by: Muhammad Sajid A contibution towads the numeical study of bubble dynamics in nucleate boiling at local scale using a consevative level set method Defended on 27 August 2010, with the juy: D. Benad PATEYON, Univ. of Limoges epote P. Hamou SADAT, Univ. of Poities epote P. Khelil SEFIANE, Univ. of Edinbugh Examine D. Adien BOUVET, Univ. of Cegy-Pontoise Examine P. achid BENNACE, Univ. of Cegy-Pontoise Supeviso Laboatoie de Mécanique et Matéiaux de Genie Civil. L2MGC-EA 4114 Univesity of Cegy Pontoise Sajid 2010

2 Abstact 1. English A contibution towads numeical study of bubble dynamics in nucleate boiling at local scale using a consevative level set method. Nucleate boiling is an efficient means of heat tansfe that has been the subject of many studies which have lead to moe empiical esults than knowledge on the physical mechanisms that goven the phenomena. In this wok, a consevative level set method (LSM) was applied to the study of bubble dynamics duing nucleate pool boiling which educes the computational cost of einitialization techniques taditionally used with LSM to limit phase loss. Also a foce-balance appoach to modelling dynamic appaent contact angle (CA) was poposed in this study based on the physics of the moving contact line (CL). It was tested against the taditional CL velocity appoach and validated in compaison to available expeimental data. In compaison to the CL velocity model ou appoach educes the non-physical stick/slip behaviou of the CL and allows a smoothe tansition fom the minimum eceding to the maximum advancing CA, which is moe akin to the physical phenomena. It was also demonstated that the heat tansfe duing bubble gowth is popotional to the appaent CA. 2. Fench Une contibution à l'étude numéique de la dynamique des bulles en ébullition nucléée à l'échelle locale, en utilisant la méthode 'level set' consevative. L ébullition nucléée est un moyen efficace de tansfet de chaleu qui a fait l objet de nombeuses études qui ont conduit ves des ésultats davantage empiiques que de connaissances su les mécanismes physiques qui égissent les phénomènes. Dans ce tavail, une méthode level-set (LSM) a été appliquée à l'étude de la dynamique des bulles los de l ébullition nucléée. Cette méthode éduit le coût de calcul en compaaison aux pécédentes techniques de éinitialisation utilisées avec LSM pou limite les petes de masse. Une nouvelle appoche, basée su la physique de la ligne de contact (CL) mobile et le bilan de foces, pemet la modélisation de l angle de contact (CA) appaent dynamique. Elle est compaée avec l'appoche classique "vitesse de CL" et validée pa appot aux données expéimentales disponibles. Cette nouvelle appoche éduit le compotement non-physique accoche-glissement de CL et pemet une tansition plus douce ente le minimum et le maximum de CA, ce qui est plus poche des phénomènes physiques obsevés. Il a également été démonté que le tansfet de chaleu au cous de la coissance des bulles est popotionnel à CA appaent. ii

3 Acknowledgements I would Iike to thank my adviso Pofesso achid Bennace fo his guidance and suppot duing the couse of this long study. His fiequent encouagement and eudite counsel was invaluable in keeping me on tack to complete this wok. I dedicate this wok to the puuit of knowledge, excellence and the ealization of deams and all those that abetted me in my own pusuit, my paents who encouaged me, my mentos who guided me and last but not least the special fiends and family membes who have always suppoted me. I am also gateful fo the financial suppot extended by the Highe Education Comission of the govenment of Pakistan. iii

4 Table of Contents Abstact... ii Acknowledgements... iii Table of Contents...iv List of Figues...vii List of Tables...x Chapte 1 Geneal intoduction Outline of the thesis...13 Chapte 2 Liteatue eview Boiling and boiling cuve Natual convection egime Nucleate boiling egime Film boiling egime (FB) Tansition boiling egime (TB) Length scales Two phase flow scale Mean bubble gowth scale Local scale Bubble dynamics Contact angle Nucleation Bubble gowth Bubble depatue Bubble elease fequency Heat tansfe mechanisms Bubble agitation Vapo-liquid exchange Evapoation Heat tansfe oveview Numeical simulation...42 iv

5 2.5.1 Poblems hindeing modelling of boiling Explicit tacking of intefaces Implicit tacking of intefaces Conclusion Chapte 3 Numeical modelling Consevative level set method Inteface epesentation Equation of the inteface motion Phase consevation Physical popeties Govening equations Contact angle Static contact angle appoach Contact line velocity appoach Foce balance appoach Implementation of dynamic contact angle Stability and nondimensionalization Chaacteistic scales Nondimensionalization Numeical implementation Defining the computational domain Implementation of the govening equations Adding the bounday conditions Validation Static bubble in zeo gavity Oscillating bubble in zeo gavity Coalescence of vapo bubbles Conclusion Chapte 4 Decoupled bubble dynamics Gowing and depating bubble Effect of contact angle on bubble gowth v

6 4.1.2 Bubble diamete compaison Local conditions and bubble gowth Effects of pofile of the inteface velocity Effects of the ate of evapoation Conclusions Chapte 5 Coupled bubble dynamics Bubble gowing on heated suface Evolution of the tempeatue field Pogession of the flow field Heat tansfe Phase consevation Dynamic contact angle Case-I: static contact angle Case-II: Contact line velocity appoach Case-III: Foce balance appoach Compaison with expeimental data Heat tansfe compaison Conclusion Chapte 6 Concluding emaks Advances made in this wok Publications duing PhD studies Outlook Bibliogaphy Appendix A Comsol scipting code vi

7 List of Figues Figue 2.1: Tempeatue distibution in pool boiling Figue 2.2: The Nukiyama boiling cuve fo satuated wate. Adapted fom Incopea and DeWitt (2006) using the data of Nukiyama (1934) Figue 2.3: Chaacteistics of the boiling cuve fo a heated hoizontal suface Figue 2.4: Natual Convection Figue 2.5: Onset of nucleate boiling Figue 2.6: Low heat flux nucleate boiling Figue 2.7: High heat flux nucleate boiling Figue 2.8: Citical heat flux (CHF) Figue 2.9: Film boiling egime (FB) Figue 2.10: Tansition Boiling egime Figue 2.11: The length scales of desciption of nucleate boiling, (a) Two phase flow scale (b) Mean bubble gowth scale and (c) Local scale Figue 2.12: Bubble gowth model of van Stalen (1966) Figue 2.13: Bubble agitation Figue 2.14: Vapo-liquid heat exchange Figue 2.15: Evapoation Figue 2.16: Explicit and implicit inteface epesentation Figue 2.17: ALE Gid Figue 2.18: FTM Gid Figue 3.1: Heaviside Function, smooth Heaviside function and delta function Figue 3.2: Suface plot of the unit cicle cented at oigin at t=0 with the level set function as (a) a distance function, (b) a Heaviside function, and as (c) a hypebolic tangent function Figue 3.3: Advancing and eceding contact angles of n-alkanes and 1-Alcohols vesus the mean contact angle, θ 0. Data fom Lam et al.(2002) vii

8 Figue 3.4: Chaacteistic expeimental esults fo contact angle measuements against contact line velocity Figue 3.5: Navie-Slip bounday condition...69 Figue 3.6: Computatoinal domain...74 Figue 3.7: elaxation of an elliptical vapo bubble of aspect atio of 2:1 at vaious times with a vecto plot of suface tension foces...80 Figue 3.8: elaxation of an elliptical vapo bubble of aspect atio of 7:4 at vaious times with a vecto plot of suface tension foces...81 Figue 3.9: elaxation of an elliptical vapo bubble of aspect atio of 3:2 at vaious times with a vecto plot of suface tension foces...82 Figue 3.10: elative eo in aea of vapo phase Figue 3.11: elative eo in aea of vapo phase fo the thee aspect atios...84 Figue 3.12: Aea of the vapo phase fo the thee aspect atios...85 Figue 3.13: elative eo in aea of vapo phase fo vaious mesh densities Figue 3.14: Initial state of the domain...87 Figue 3.15: Phase loss duing initialization of the level set function...88 Figue 3.16: Coalesence of two bubbles...88 Figue 3.17: Phase loss duing coalescence Figue 4.1: Pofile of the injected velocity Figue 4.2: Bubble base diamete fo contact angle, θ CA =50, θ CA = 55 and θ CA = Figue 4.3: Equivalent bubble diamete fo contact angle, θ CA =50, θ CA = 55 and θ CA = Figue 4.4: Equivalent bubble depatue diamete calculated and computed fom empiical elations Figue 4.5: Height of velocity pofiles, hu Figue 4.6: Velocity pofiles, u evp fo α=3,2,1,0.5 and 0.33 at time = Figue 4.7: ate of vapo injection, V & Figue 4.8: Non dimensional volume of vapo bubble viii

9 Figue 4.9: Equivalent bubble diamete fo vaious α s at θca = 52, Figue 4.10: Bubble base diamete fo vaious α s at θca = 52, Figue 4.11: Velocity pofiles, u evp fo ά = 1/2 and 2/3 at time = Figue 4.12: Bubble base diamete fo ά = 1/2 and ά = 2/3 at θca = Figue 4.13: Volume of vapou bubble fo ά = 1/2 and ά = 2/3 at θ CA = Figue 5.1: Tempeatue field (in K) at t = 0s Figue 5.2: Tempeatue field (in K) at t = 0.01s Figue 5.3: Tempeatue field (in K) at t = 0.11s Figue 5.4: Tempeatue field (in K) at t = 0.14s Figue 5.5: Tempeatue field (in K) at t = 0.16s Figue 5.6: Nomalized flow field at t = 0.0s, 0.01s and 0.11s Figue 5.7: Nomalized flow field at t = 0.0s, 0.01s and 0.11s Figue 5.8: Latent heat flux fom a gowing bubble Figue 5.9: Total heat absobed by vapo bubble Figue 5.10: Phase consevation duing simulation Figue 5.11: Appoximate inteface velocity (a), contact angle (b), foces acting on the bubble (c) and bubble dimensions (d) fo a static mean contact angle of Figue 5.12: Appoximate inteface velocity (a), dynamic contact angle (b), foces acting on the bubble (c) and bubble dimensions (d) fo an appaent contact angle as a function of the contact line velocity with umax = 0.015m/s Figue 5.13: Appoximate inteface velocity (a), dynamic contact angle (b), foces acting on the bubble (c) and bubble dimensions (d) fo an appaent contact angle as a function of the foce balance Figue 5.14: Contact angle vs time fo T = 7K, fom amanujapu and Dhi (1999) Figue 5.15: Bubble diamete vs time T = 7K, fom amanujapu and Dhi (1999) Figue 5.16: Latent heat tansfe fo diffeent contact angle models Figue 5.17: Conductive heat tansfe fo diffeent contact angle models ix

10 List of Tables Table 2.1: Single bubble cycle...28 Table 2.2: Static contact angles θ CA fo distilled wate at polished sufaces...29 Table 2.3: Static contact angle θ CA fo distilled wate at themally o chemically teated polished sufaces...29 Table 3.1: Young-Laplace pessue dop acoss liquid vapo inteface Table 4.1: Pofiles of the applied inteface velocity...99 Table 5.1: Bubble chaacteistic at depatue fo vaious test cases x

11 Chapte 1 Geneal intoduction Boiling; a liquid vapo phase change pocess has been the subject of an enomous amount of studies which have lead to moe empiical esults than knowledge on the physical mechanisms that goven the phenomena. The field is still govened by empiical coelations that ae in pactice and ae equied fo design puposes. Howeve it is now accepted that studies focused on the physics of this vey complex pocess will be equied fo futhe pogess. This tend is on one hand suppoted by new expeiments that take advantage of the gowing capacities in data acquisition, stoage and pocessing by computes in connection with miniatuized sensos and high speed camea and othe optoelectonic facilities which ae doubtless unavoidable in this field, and on the othe by bette and new mathematical tools which have been developed in ecent yeas. Thus, the basic mechanisms of boiling can now be moe pofoundly exploed leading to esults that ae being inceasingly used to develop heat tansfe models that ae based on physics. As compaed to the empiical coelations these models ae moe geneally moe valid. In this way we ae able to ceate a bette a chance to develop enhanced heat tansfe sufaces systematically and not by tial and eo methods by acquiing a deepe undestanding of the govening physical mechanisms. Most enegy convesion systems and heat exchanges use boiling heat tansfe, which is also employed in the themal contol of compact devices that have high heat dissipation ates. Applications of this type include the use of boiling to cool electonic components in mainfame computes and the use of compact evapoatos and condenses fo themal contol of aicaft avionics and spacecaft envionments. The vitually isothemal heat tansfe associated with boiling and condensation pocess makes thei inclusion in powe and efigeation cycles highly advantageous fom a themodynamic efficiency viewpoint. The y ae of citical impotance to nuclea powe plant design, both because hey ae impotant in nomal opeating cicumstances and because they dominate many of the accident scenaios that ae studied in detail as pat of design evaluation. Liquid vapo phase change pocesses ae also encounteed in petoleum and chemical pocessing, liquefaction of nitogen and othe gases at cyogenic tempeatues and duing evapoations of pecipitation of wate in the eath s atmosphee. The use of boiling heat tansfe is motivated by the high efficiency of heat exchange duing the nucleate boiling pocess. Nucleate pool boiling efes to boiling unde natual convection conditions in contast to flow boiling whee liquid flow ove the 11

12 12 suface heate is imposed by extenal means. Flow boiling can include extenal and intenal flow boiling. Nucleate pool boiling is chaacteised by the onset of cyclic gowth and depatue of vapou bubbles fom a discete site on a heated suface and is theefoe geatly influenced by the bubble gowth mechanism. Consequently fo mechanistic modelling, it is necessay to be able to pedict data fo a single bubble nucleation which involves the mechanics of multiphase fluids as well as the dynamics of phase tansition. The study of bubble dynamics is a coupled poblem as the heat tansfe detemines the ate of evapoation which in tun contols the dynamics of the inteface. The natue of the inteface between two fluids has been the subject many investigations. Initially the inteface between two fluids was epesented as a suface of zeo thickness and endowed with physical popeties such as suface tension. Bounday conditions on the inteface wee used to epesent physical pocesses occuing at the inteface, like the Young-Laplace equation elating the pessue jump acoss an inteface to the poduct of suface tension and cuvatue of the inteface. In the case whee the inteface is a plane of discontinuity that sepaates egions of diffeent phases, tacking it as a function of time becomes a challenging task. This is especially so, if the inteface meges, disappeas and/o developes holes in othe wods, if it develops topological singulaities. The main poblematic of solving the boiling flow is thus to take into account the inteface as a moving bounday. In addition the boiling pocess has all the complexity of single phase convective tanspot, i.e. nonlineaities, tansition to tubulence, thee-dimensional and time vaying behaviou. To ovecome such poblems one typically esots to level set methods (LSM). The level set method consides an inteface moving unde a given velocity field (which depends on position, time, geomety of the inteface, and so on). A smooth field Φ is defined and the contou whee Φ = 0 is defined as the inteface. The evolution equation fo the inteface is then given by evolving Φ with the given velocity field and tacking egions of Φ = 0. A complete numeical simulation of a gowing and depating bubble duing patial nucleate boiling was pefomed by Son et al. (1999) by employing the level set method (LSM), which can handle the bubble depatue pocess. Since then the level set method has been used by vaious eseaches fo studying boiling elated phenomena. A pesistent poblem with the level set method is that it leaks phase, i.e. some potion of one fluid seeps into the othe fluid, typically fom the less dense to the dense. Paticulaly, when thee is a steep contast between the density and viscosity of the two inteacting fluids. In the pesent 12

13 13 wok a consevative level set method is fomulate and employed to simulate the dynamics of a single vapou bubble at the onset of nucleate pool boiling. 1.1 Outline of the thesis Chap. 2 Liteatue eview. We pesent the backgound, context and a vey bief histoy of the nucleate pool boiling pocess as well the numeical methods employed to study it. At the end we motivate the choice of the level set method fo simulating single bubble dynamics. Chap. 3 Numeical modelling We descibe the consevative level set method, the dynamic contact angle models and the govening equations of ou model. The implementation of the model is discussed and the implemented model is validated using esults fo known poblems. The phase consevation chaacteistics of the model ae scutinized. Chap. 4 Decoupled bubble dynamics The validated consevative level set model is applied to the study of a gowing and depating vapo bubble decoupled fom enegy consevation equation, with phase change being simulated by diect vapo injection on the inteface nea the contact line. The appoach is used to study the effect of local phenomena nea the contact line on global dynamics of bubble. Chap. 5 Coupled bubble dynamics The numeical model is used to study the gowth of single bubble on a heated suface. Dynamic angle models ae consideed and the foce-balance based dynamic contact angle is substantiated. Chap 6. Concluding emaks We finalize the thesis by summaizing the wok ealized duing the couse this study and discussing possible futhe development of the code as well as futhe application aeas. 13

14 14 Chapte 2 Liteatue eview In this chapte a eview of the fundamentals of boiling is pesented togethe with the numeical methods used to pedict the phenomena. Boiling is pobably the most investigated phenomenon in the themal sciences ove 80 yeas. Expeimental data have been collected, e.g. Jakob and Fitz (1931), Nukiyama (1934), Yamagata et al. (1955), Kuihaa and Myes (1960), Gaetne and Westwate (1960), Siegel and Keshock (1964), Gaetne (1965), Coope and Lloyd (1969), Nishikawa et al. (1974), Conwell and Bown (1978), Stephan & Abdelsalam (1980), Sultan and Judd (1983), Goenflow et al. (1986), Lienhad (1988), Wang and Dhi (1993a,b), Chen and Hsu (1995), Auache and Maquadt (2002) and Buchholz et al. (2006) etc. We begin ou eview with a pesentation of the diffeent boiling heat tansfe egimes with the help of the classical epesentation of the Nukiyama cuve. Then we descibe moe pecisely the main chaacteistics of the diffeent egimes by highlighting the diffeent physical pocesses occuing in a nea wall egion, followed by a naative of bubble dynamics and an intoduction of the numeical methods used to study it. Diven by the necessity to undestand this complex phenomenon many studies have been caied on boiling and quite a few eviews of boiling pocesses have appeaed. Fo moe exhaustive teatments of boiling heat tansfe, the following books ae ecommended fo consultation: ohsenow (1973, 1985, 1998), van Stalen and Cole (1979), Caey (1992), Collie and Thome (1994), Kandlika, Shoji and Dhi (1999) and Incopea and DeWitt (2006). 2.1 Boiling and boiling cuve When heat is applied to a suface in contact with a liquid, if the wall tempeatue is sufficiently above the satuation tempeatue, boiling occus on the wall. Liquids having tempeatue lage than the satuation tempeatue coesponding to the local pessue ae called supeheated liquids. Supeheated liquids ae unstable and stat to disintegate. This pocess is in geneally called flashing. The pocess of uptuing a continuous liquid by decease in pessue at oughly constant liquid tempeatue is often called cavitation - a wod poposed by Foude. The pocess of uptuing a continuous liquid by incease the tempeatue at oughly constant pessue is often called boiling. The fluctuation of molecules having enegy lage than that chaacteistic fo a stable state causes the fomation of clustes of molecules, which afte eaching some citical size ae called nuclei. Boiling may occu 14

15 15 unde quiescent fluid conditions, which is efeed unde gavity as nucleate pool boiling, o unde foced-flow conditions, which is efeed to as foced convective boiling. Let us conside the heat tansfe pocess fom a heated solid to a boiling fluid. The pool boiling expeimental set-up consists of a boiling fluid contained in a pool and it theefoe does not include any extenal mean flow. The heating solid is eithe a plane plate, a wie, a ibbon o one of the bounding walls of the pool. It is equied that its dimensions exceed the chaacteistic dimension of the vapo inclusion (typical bubble adius) in ode to ignoe size effects. In the emainde of this study, we conside by default the pool boiling of pue fluids with a heated lowe hoizontal wall with a mean tempeatue T at the bottom of the pool that and q being the amount of heat tansmitted though the solid-fluid contact aea (Figue 2.1). Note that the tempeatue of the boiling liquid is nealy constant except at the suface of the heated wall. At the heated suface the liquid tempeatue inceases shaply. Vapo y Vapo bubbles Liquid T(y) Solid T sat T s T Figue 2.1: Tempeatue distibution in pool boiling Since we descibe boiling systems as a heat exchange pocess, it is natual to chaacteize the wall tempeatue by T= T s - T sat ; Whee T s, T sat and T ae espectively called the suface tempeatue, satuation tempeatue and wall supeheat. When heating a suface in a lage pool of liquid, the heat flux is usually plotted vesus the wall supeheat. The boiling cuve was fist constucted by Nukiyama (1934), he used a nichome wie immesed in a bath of wate. As he inceased the powe to the wie, he noticed bubbles foming at an excess tempeatue, T of about 5 C. As he inceased powe futhe, he noticed that the powe supplied to the wie could be inceased geatly without a lage 15

16 16 incease in tempeatue. As he inceased the powe futhe still, the tempeatue suddenly inceased damatically, and the wie eached its melting point. Afte expeimenting with a platinum wie with a highe melting point, he noted that the heat flux and tempeatue wee elated, as shown in Fig 2.2. Figue 2.2: The Nukiyama boiling cuve fo satuated wate. Adapted fom Incopea and DeWitt (2006) using the data of Nukiyama (1934) Nukiyama believed that a method of tempeatue contol of T, instead of a powe-contolled method, would yield a bette cuve. Dew and Muelle (1937) pefomed the expeiment using a steam pipe and obtained a cuve simila to that shown in Figue 2.3. This figue shows the elationship of heat flux and tempeatue fo most liquids. A typical Nukiyama cuve has fou distinct heat tansfe egimes between which ae thee tansition points. The fou distinct heat tansfe egimes ae natual convection and nucleate, film and tansition boiling egimes. The thee tansition points ae the onset of nucleate boiling (ONB), depatue fom nucleate boiling (DNB) and the beginning of the film boiling egime at the point of minimal heat flux (MHF). These ae shown in the following figue. 16

17 17 DNB BO CHF q ONB NB TB FB NC MHF Figue 2.3: Chaacteistics of the boiling cuve fo a heated hoizontal suface T Natual convection egime Fluid suounding the heated solid eceives heat, becomes less dense and ises due to buoyancy. The suounding, coole fluid then moves in to eplace it, which is then heated and the pocess continues, foming a convection cuent. This pocess tansfes heat enegy fom the bottom of the pool to top. The pocess is chaacteized by single-phase natual convection fom the hot suface to the satuation liquid without fomation of bubbles on the suface. Bubbles do not fom in this egion because thee is not enough vapo in contact with the liquid. Natual convection teminates at the fist tansition point called incipience of boiling o onset of nucleate boiling (ONB), at which the fist individual bubbles appea on the heated suface leading to the nucleate boiling egime. 17

18 18 Heated Suface Figue 2.4: Natual Convection Nucleate boiling egime The nucleate boiling egime coesponds to low wall supeheats T and to a limited ange of wall heat flux q. The lowe limit in tems of both q and T is the onset of nucleate boiling (ONB) and thus coesponds to the uppe limit of the convective egime 1. ONB coesponds to a tansition between the non-boiling and the boiling egimes. Heated Suface Figue 2.5: Onset of nucleate boiling The NB egime is a vey efficient heat exchange mechanism since lage amounts of heat can be extacted though a wall while keeping its tempeatue at low levels (i.e. of the ode of magnitude of the satuation tempeatue of the boiling fluid). This explains the wide use of boiling fluids in heat exchanges. 1 We undeline that the heat exchange coesponds to the phase change and convective enegy contibution. 18

19 19 The fomation of the vapo bubbles in the NB egime is the esult of successive nucleation events and consecutive heteogeneous bubble gowth dynamics until the depatue of the bubble fom the wall. The majo pat of the wall is in contact with the liquid phase even if high void factions can be eached ight above the wall. As a consequence, the wall supeheat is low (since liquid tempeatue cannot each lage supeheats values). Due to the agitation associated with bubble gowth and motion, the efficiency of the convective heat exchange is geatly enhanced with egad to a single-phase case. Heated Suface Figue 2.6: Low heat flux nucleate boiling Duing natual convection, the wall tempeatue ises as the heat flux is inceased until the fist bubbles fom (o nucleate) at small cavities in the heated suface, called nucleation sites, leading to nucleate boiling egime. By inceasing the heat flux, moe nucleation sites become activated. Thus at low heat fluxes the suface is coveed with bubbles that gow and depat in quick succession (figue 2.4). By inceasing the heat flux even futhe, the ate of depating bubbles becomes moe apid until the bubbles appea to coalesce into vapo jets, (figue 2.5) that is, bubbles fom, sepaate, and efom so quickly that continuous steams o vapo columns ae seen. The heat flux inceases damatically fo elatively modest inceases in T sat, changing the slope of the nucleate boiling cuve. 19

20 20 Heated Suface Figue 2.7: High heat flux nucleate boiling A futhe incease in q eventually pohibits the liquid fom eaching the heated suface, which such that blanketing of the heated suface by vapo occus (figue 1.6), accompanied by a apid ise in the suface tempeatue to dissipate the applied heat flux. This limit of the NB egime associated with high values of q is called depatue fom nucleate boiling (DNB) and coesponds to a tansition between the diffeent boiling egimes. The associated value fo the heat flux is called the citical heat flux (CHF). Heated Suface Figue 2.8: Citical heat flux (CHF) Following DNB, the pocess follows a path that depends on the manne in which the heat flux is applied to the suface. 20

21 21 Fo heates that impose a heat flux at the suface, such as electical-esistance elements o nuclea fuel ods, the pocess pogesses on a hoizontal line of constant heat flux so that the wall supeheat jumps to the coesponding point on the film boiling pat of the cuve. When the wall tempeatue is the extenally contolled vaiable, such as by vaying the satuation tempeatue of steam condensing inside a tube with boiling on the outside, the pocess path moves fom the DNB to the minimal heat flux (MHF) point, and vice vesa, following the tansition boiling path Film boiling egime (FB) If at depatue fom nucleate boiling, the heat flux is inceased above the CHF value, the system shifts to the film boiling egime (FB). The FB egime coesponds to vey high values of the wall supeheat and to heat tansfe coefficients that ae lowe than in the NB egime. The tansition fom NB to FB is chaacteized by a vey sudden and vey lage incease of the wall tempeatue. Typically the ode of magnitude of this incease can each seveal hundeds of Kelvin. An ulteio incease in heat flux q may bing the suface to the bunout point (BO), whee the suface tempeatue eaches the melting point of the heate. This phenomenon is also temed as the boiling cisis. In the FB egime, the suface becomes dy and the wall is coveed by a vapo film. The vapo bubbles no longe gow and depat fom the heate suface but fom the vapo liquid inteface of the vapo laye. At the liquid-vapo inteface, a dynamic pocess of vapo geneation and elease occus. Due to the ayleigh Taylo instability (TI), the suface of the vapo film is wavy and is the location of a cyclic pocess of bubble fomation as epesented on figue 1.7. The vapo film is stable in the sense that liquid does not nomally wet the heate suface and elatively lage bubbles ae fomed by evapoation at the fee vapo liquid inteface, which then depat and ise up though the liquid pool. The heat flux is tansmitted fom the wall to the liquidvapo inteface (whose tempeatue can be appoximated by the satuation tempeatue) though a combination of adiative, conductive and convective tansfes acoss the vapo laye. Conduction takes place acoss the vapo film while adiation occus fom the wall to the liquid o to the walls of the vessel. 21

22 22 Heated Suface Figue 2.9: Film boiling egime (FB) The low value of the vapo themal diffusivity (with egad to the liquid one), explains the highe wall tempeatue in the FB egime than in the NB egime. The ode of magnitude of the vey lage and vey sudden incease of the wall tempeatue, that chaacteizes the tansition fom NB to FB, can typically each seveal hundeds of Kelvin. An ulteio incease in heat flux q may bing the suface to the bunout point (BO), whee the suface tempeatue eaches the melting point of the heate. This phenomenon is also temed as the boiling cisis. If we educe the heat flux afte aiving at FB fom NB at CHF, then the film boiling cuve passes below the CHF line until eaching the MHF point. Hee again, the pocess path depends on the mode of heating. Fo an imposed heat flux, the pocess path jumps hoizontally at constant q to the nucleate boiling cuve NB. Consequently, a hysteesis loop is fomed when heating a suface up past the DNB and then binging it below MHF when q is the bounday condition imposed Tansition boiling egime (TB) The lowe limit of FB is chaacteized by the minimal heat flux (MHF) that coesponds to the tansition to the thid boiling egime, the tansition boiling (TB). The domain of existence of TB joins the CHF and the MHF points and concens intemediate values of wall tempeatue. Fo this egime, the local wall heat flux and/o tempeatue fluctuate violently aound thei mean values. Fo these easons, TB is had to chaacteize in itself and is often modeled as an unstable mix between the NB and FB egimes. Thus, in tansition boiling the pocess is consideed to vacillate between nucleate boiling and film boiling, whee each mode may coexist next to one anothe on the heated suface o may altenate at the same location on the suface. It is woth noting that steady-state TB egime can 22

23 23 be expeimentally eached by imposing the wall tempeatue and not the heat flux. If the heat flux q is expeimentally imposed, this egime is unstable and theefoe inaccessible. At the MHF, while deceasing T and coming fom the TB egime, the film configuation becomes unstable and locally beaks (i.e. locally some liquid comes into contact with the wall). As a consequence, wetted (coveed by the liquid phase) aea appeas on the solid suface. In the tansition boiling egime, the fluid/solid contact suface is the place of lage and intemittent wetting and dying dynamics, i.e. it is coveed altenatively by the liquid (wet) o the vapo (dy) phases. These hydodynamic events ae elated to the lage fluctuations of T and q. Heated Suface Figue 2.10: Tansition Boiling egime 2.2 Length scales Inode to study the physical mechanisms associated with nucleate pool boiling, we fist intoduce thee diffeent length scales as detemining thee levels of desciption of the nucleate boiling pocess. Each of this levels being elated to a diffeent typical length scale fo the bubble desciption. Fo each scale diffeent physical mechanisms can be identified. In the pevious section, we have distinguished the diffeent egimes of the pool boiling by consideing the pocess in a nea wall egion. The typical size of this egion is of the ode of magnitude of a few bubble diametes. This sets the bubble as the natual basic element of the nucleate boiling pocess. The thee diffeent length scales can be defined fom thee diffeent levels of modelling bubbles. At the fist level of desciption, denoted as two phase flow scale, the bubbles ae consideed as fixed in geomety and size. This is mainly elevant fa fom the wall, i.e. when the main pat of the bubble gowth is achieved. This level of desciption theefoe ignoes the bubble gowth dynamics. At the 23

24 24 second level of desciption, denoted mean bubble gowth scale, the bubbles ae consideed as fixed in geomety but not in size. Fo instance the bubble is assimilated to a sphee whose size depends on time, the dynamics of bubble fomation is theefoe taken into account. At the thid level of desciption, denoted local scale, both the geomety and the size of the bubble ae consideed as time dependent. The dynamics of bubble fomation aea theefoe descibed moe pecisely (with moe degees of feedom) than at the mean bubble gowth scale. An illustation of these thee levels of desciption of the nucleate boiling pocess is povided in figue 2.9. (a) (b) (c) Figue 2.11: The length scales of desciption of nucleate boiling, (a) Two phase flow scale (b) Mean bubble gowth scale and (c) Local scale Two phase flow scale In ou classification this scale is the lagest one and is associated with the most idealized level of the modeling of the nucleate boiling pocess. The basic pictue consides a population of bubbles coming fom the wall and having constant size and geomety. Mean space and time fequencies of bubble emission ae modeled. The ate of vapo incoming fom the wall is elated to the value of the wall heat flux q. It is woth noting that this scale does not actually 'see' the wall. This kind of lage scale analysis of the boiling fows has been used fo example by Zube (1963) to deive a coelation fo the low heat tansfe nucleate boiling egime (egion of isolated bubbles, whee bubbles do not inteact with each othe). At lage heat flux, bubbles become so numeous that thei inteactions ae no longe negligible. The lage numbe of individual bubbles geneated in the nea wall egion coalesce each othe and somewhat fom big masses of vapo flowing away fom the wall. These big masses can be idealized by a somewhat continuous vapo channel, the vapo columns. The model of the two- 24

25 25 phase flow fa above the wall is theefoe idealized by a counte-cuent flow of vapo inside these channels and of liquid aound those columns. The eade inteested by this appoach can efe to Caey (1992) Mean bubble gowth scale At this scale, the nea wall nucleate boiling pocess is descibed in a way that facilitates the evaluation of bubble fomation though modelling. The chaacteistic scale of this level of desciption is that of the bubble, which is egaded as having a fixed geomety but vaying in size allowing us to model it gowing on the wall. Fom the study of the bubble gowth pocess, seveal heat-exchange mechanisms can be identified including the tanspot of latent heat thus pemitting us to model moe accuately the vaious phenomena involved in the fluid wall heat exchange. The influence of physical phenomena occuing in a nea wall egion (that wee ignoed at the two-phase flow scale) ae theefoe enteing the model. At this level of desciption the nucleate boiling pocess is idealized as a set of sub-phenomena. The main one is the cyclic pocess of bubble fomation nea the heated wall. It is idealized by a sequence of events (nucleation, gowth, depatue). Each of these sub-phenomena has been the object of specific studies, eithe based on analytical models o on coelation issued fom expeimental obsevations. The ceation of bubbles is modelled as a cyclic pocess localized on sites and consisting of the following events: nucleation, gowth, depatue, and waiting. Some models also conside the inteaction between sites. This scale of analysis coves a wide ange of phenomena listed below: 1. patition of the wall heat flux between diffeent heat tansfe pocesses (a) latent heat tanspot (evapoation) (eventually two pats aound bubble and mico-laye contibution) (b) (c) tansient conduction natual convection 2. spatial fequency of the bubble fomation pocess, the nucleation site density 3. bubble gowth ate 4. bubble depatue size 5. waiting time 25

26 26 6. bubble inteactions: themal, hydodynamic, and coalescence Local scale The mean bubble gowth scale descibes the mean bubble fomation pocess in an idealized way which is elevant at low heat flux NB egime. Howeve at high heat flux bubbles ae moe spead ove the wall befoe thei depatue. This bubble behavio is vey diffeent fom the egula behavio. The model of nucleate boiling pocess at the mean bubble scale is too igid to descibe such a behavio (accoding to the fact that the bubble shape is imposed to be eithe spheical o hemispheical). We must theefoe conside a smalle level of modeling of the NB pocess. The pesent level of desciption takes into account the full time and space dependency of the bubble shape. It is woth noting that the amount of modeling is theefoe quasi-vanishing since we now conside the full set of non-isothemal Navie-Stokes equations and inteface jump conditions. The main physical mechanisms that ae taken into account at the local scale in addition to the physical mechanisms consideed at the mean bubble gowth scale ae listed below. 1. Local cuvatue of the bubble and capillay foces 2. Pessue ecoil at the inteface: the jump in pessue p = m& ρ, whee m& is the local mass tansfe ate. 3. Tiple line dynamics (static-dynamic contact angle) and associated quasi-singula heat tansfe (cf. Andeson and Davis (1994) o Mathieu et al. (2004) 4. Gavity: effect of the hydostatic pessue gadient on the bubble shape and consequently on its depatue dynamics (cf. Shikhmuzaev (1999) whose model consides the time dependent shape of the bubble and also includes a model fo the contact line dynamics) 5. Local heat conduction poblem inside the aea of contact between wall and vapo at the dy base of a bubble (cf. Blum et al. (1999)) To take into account the whole set of mechanisms and the complex time dependent geomety of the bubble at this level of desciption, it is equied to use numeical methods. Seveal numeical simulations of bubble gowth dynamics using diffeent numeical methods can be found in the liteatue, e.g. Son et al. (1999, 2002) fo the level-set method, Welch and Wilson (2000) fo the VoF method, Juic and Tyggvason (1998) fo the font tacking method. While Yang et al. (2001) used a 26

27 27 Lattice-Boltzmann model based numeical method fo simulating film boiling egime whee as Fouillet and Jamet (2005) sued a diffuse inteface method appoach fo the same egime. As a patial conclusion, in ode to descibe cetain featues of the nucleate boiling egime at high heat fluxes, it is necessay to conside the full poblem of the bubble gowth as having a time dependent geomety. 2.3 Bubble dynamics A thoough undestanding of the pocess of boiling heat tansfe in pool boiling equies an investigation of the themodynamics of bubbles and the hydodynamics of the flow esulting fom the depatue of the bubbles fom the heate suface. The gowth of vapo bubbles can be a paticulaly complex physical phenomenon. Fo example, bubble gowth and depatue ae influenced by the oientation of the suface, the contact angle, the thickness and tempeatue pofile in the themal bounday laye, the poximity of neighboing bubbles, tansient themal diffusion within the wall to the adjacent liquid, wake effects of the pevious bubble, bubble shape duing gowth, and so on. A vapo bubble foms and gows in a liquid as long as the bubble vapo pessue is highe than the ambient liquid pessue, that is, p v > p l. Stability is eached when p v - p l = 2σ/ B which foms a bubble in equilibium and is known as the Youngs-Laplace law. Boiling is a cyclic pocess. The life of a single bubble is summaized in Table 2.1. The ebullition cycle descibed in the table is followed by a waiting peiod, which occus at the nucleation site just afte the depatue of a bubble and befoe a new bubble is fomed. This was showed by a shadowgaph and Schlieen technique (Hsu and Gaham, 1961). They poduced an explanation fo the waiting time of bubbles fomed on a heated suface by boiling. The bubble depatue uptued the themal bounday laye, and a new bubble was fomed only afte the effects of the distubance had disappeaed. Hsu (1962) developed a model based aound the so-called waiting time' equied fo the liquid to attain the necessay supeheat. The nucleation poceeds when the themal bounday laye thickness eaches a citical faction of the nucleus adius. This waiting peiod between two consecutive bubbles can be descibed meaningfully only in the lowe heat flux ange whee the bubbles ae discete. A compehensive teatment of bubble gowth theoy can be found in the book by van Stalen and Cole (1979). 27

28 28 Nucleation Nucleation is a molecula scale pocess in which a small bubble (nucleus) of a size just in excess of the themodynamic equilibium is fomed. Bubble Gowth Inetially contolled Themally contolled Initial gowth fom the nucleation size is contolled by inetia and suface tension effects. The gowth ate is small at fist but inceases with bubble size as the suface tension becomes less significant. Heat tansfe becomes inceasingly impotant while inetia effects lose significance until bubble eaches an asymptotic gowth stage whee it is contolled by the ate of heat tansfeed fom the suounding liquid to facilitate the evapoation at the bubble inteface. Depatue/ Collapse If the bubble duing its gowth encountes subcooled liquid, it may collapse. The contolling phenomena fo the collapse ae much the same as fo the gowth but ae encounteed in the evese ode. Othewise the bubble depats fom the heate suface. Table 2.1: Single bubble cycle Bubble dynamics play a key ole in the development of any analytical model pupoting to pedict nucleate pool boiling heat tansfe coefficients. The simplest case to analyze is that of a single spheical bubble gowing within an infinite, unifomly supeheated liquid emote fom a wall. Befoe we descibe the cycle of single bubble it is impotant to highlight the ole of the contact angle Contact angle The contact angle chaacteizes the molecula inteaction between the liquid and the solid suface. It is the angle between the tangent to the inteface and the wall. Hydophobic sufaces, θ CA > 0, cause heteogeneous nucleation at much educed pessue diffeence. In the study of bubble dynamics two types of contact angles ae consideed, static and dynamic. Static contact angle: The shape of a bubble in a static state o a state of equilibium on a heated suface depends on the mateial popeties of the liquid suounding the bubble, the vapo inside it and the suface on which it is esting. This dependency is usually descibed as a function of the 28

29 29 intefacial tensions using the equation fist deived by Thomas Young (1805). Young s equation govens the equilibium of the thee intefacial tensions and the Young contact angle, θ CA,Y ; 1 σ sv σ sl θ = CA, Y cos σ lv (2.1) Whee σ is the suface tension between solid-vapo (sv), solid-liquid (sl) and liquid-vapo (lv) sufaces. The following tables give some infomation on how the contact angle can change fo diffeent wall mateials and diffeent suface pepaation pocedues. Steel π / 3.7 Siegel and Keshock (1964) Nickel π / 4.74 to π / 3.83 Siegel and Keshock (1964) Chome-Nickel Steel π / 3.7 Aefeva and Aladev (1958) Zinc π / 3.4 Aefeva and Aladev (1958) Bonze π / 3.2 Aefeva and Aladev (1958) Coppe π / 4 Aefeva and Aladev (1958) Coppe π / 3 Gaetne and Westwate (1960) Coppe π / 2 Wang and Dhi (1993b) Stainless steel 304 (25 C) π / 2.25 Hiose et al. (2006) Zicaloy (25 C) π / 2.46 Hiose et al. (2006) Aluminum (25 C) π / 2 Hiose et al. (2006) Table 2.2: Static contact angles θ CA fo distilled wate at polished sufaces. Coppe heated to 525 K and exposed to ai π / 5.14 Wang and Dhi (1993b) one hou: Coppe heated to 525 K and exposed to ai π /10 Wang and Dhi (1993b) two hou: Chome-Nickel Steel chemically teated: π / 2.9 Aefeva and Aladev (1958) Table 2.3: Static contact angle θ CA fo distilled wate at themally o chemically teated polished sufaces 29

30 30 Dynamic contact angle: In nucleate pool boiling bubbles nucleate at the heated suface and stay attached to it though out thei gowth peiod. Smoothness; homogeneity and igidity of the solid suface ae undelying assumptions of the Young s equation which also implies that that the solid should be chemically and physically inet with espect to the liquids to be employed. Thus a unique contact angle is expected fo a given system. Howeve, it has been expeimentally obseved that the contact angle made by the liquid vapo inteface at the bubble base with the wall vaies duing bubble gowth and depatue stages. Thus in a eal system a spectum of contact angles anging fom the advancing contact angle, θ ACA, to the eceding contact angle, θ CA, ae detected. The uppe limit of the spectum is the advancing contact angle, θ ACA, which is the contact angle found at the advancing edge of the liquid. The lowe limit is the eceding contact angle, θ CA, which is the contact angle found at the eceding edge. The diffeence between the advancing and eceding contact angles is known as the contact angle hysteesis, θ hyst : θ hyst = θ ACA θ CA (2.2) The vaiation of the contact angle may influence the chaacteistics of bubble dynamics and is theefoe an impotant paamete in suface science. In consequence, fo the numeical simulation of bubble dynamics, an appopiate contact angle model is essential fo a coect pediction of the bubble fomation Nucleation The pocess of uptuing a continuous liquid by incease the tempeatue at oughly constant pessue is often called boiling. The fluctuation of molecules having enegy lage than that chaacteistic fo a stable state causes the fomation of clustes of molecules, which afte eaching some citical size ae called nuclei. The theoy of nucleation povides us with infomation about the geneation of nuclei pe unit time and unit volume of the liquid as a function of the local paamete. It has been well established that vapoization usually begins with an embyo in the supeheated liquid, Caey (1992). Once the embyo is lage than a citical size, a vapo bubble will fom. The citical adius, B,c can be estimated by computing the incipient nucleation tempeatue in pool boiling fom the classical kinetics of nucleation, Cole (1974) as follows; Homogeneous nucleation: If the vapo embyo completely foms within a supeheated liquid, it is called homogeneous nucleation. It is usually associated with high degees of supeheat and 30

31 31 extemely apid ates of vapo geneation. Fo homogeneous nucleation in supeheated liquids, the bubble nucleation density J is given by; J kt h 16πσ exp 3kTn n = N0 2 ( p ) v pl whee Tn, σ, k and h ae the nucleation tempeatue, the suface tension, the Boltzmann constant, and the Planck constant, espectively. Heteogeneous nucleation: On the othe hand, if the vapo embyo foms at the inteface between liquid and solid o gas, it is called heteogeneous nucleation. It can occu at lowe degees of supeheat. Fo heteogeneous nucleation on a smooth suface with no cavity, the bubble nucleation density is; (2.3) J 3 3 kt ( ) n 16πσ ω ψ exp (2.4) h 3kTn pv pl 2 = N0 2 with, ( + cosθ ) 1 CA ψ = and 2 2 ( 1+ cos ) ( 2 cos ) θ CA θ ω = CA (2.5) 4 Whee θca is the contact angle. With the aid of Clausisus-Clapeyon equation, Eqs 2.1 and 2.2 can be expessed in the following compact fom, Li and Cheng (2004); T T n s Ts = h fgρv 3 16πσ ω γ N kt nψ (2.6) 0 3kTn ln Jh which is applicable fo both homogeneous nucleation (with γ = ψ = ω =1) and heteogeneous nucleation (with γ = 2/3 and ψ and ω given by Eq 2.3). The nucleation tempeatues fo homogeneous and heteogeneous nucleation can be obtained fom Eq. (2.4) iteatively if the value of J is known. Note that the nucleation tempeatue detemined fom Eq. (2.4) is not sensitive with the chosen value of J. Afte the nucleation tempeatue is obtained, the citical bubble adius fo homogeneous and heteogeneous nucleation is given by; 31

32 32 2σ B, c = (2.7) p sat ( Tn ) pl Fo wate at the atmospheic pessue, the homogeneous nucleation tempeatue and the citical bubble adius detemined fom Eqs. (2.4) and (2.5) ae: T n = C = 576.8K and B,c = 3.15 nm Bubble gowth A spheical bubble gowing in a unifomly supeheated liquid is the simplest geomety to analyze. The pessue and tempeatue of the vapou egion inside the bubble ae p V and T V ; the bubble adius is B and is a function of time t fom the initiation of gowth (gowth ate is d B /dt). The pessue and tempeatue in the liquid ae p L and T L. In this analytical model othe effects influencing the bubble ae ignoed, such as the static head of the liquid, and the cente point of the bubble is assumed to be immobile. At inception, the supeheat is sufficient fo nucleation. Then, as the bubble gows, the pessue inside the bubble deceases and with it, T sat at the bubble inteface. Enthalpy stoed in the supeheated liquid adjacent to the inteface is conveted into latent heat at the bubble inteface and hence the intefacial tempeatue falls, ceating a themal diffusion shell aound the bubble. Momentum is impated to the suounding liquid as the bubble gows and heat diffuses fom the supeheated bulk to the inteface at a ate equal to the ate at which latent heat is libeated at the inteface. In addition, the equilibium vapo pessue cuve is assumed to descibe this dynamic pocess, and it is assumed that the vapo pessue in the bubble coesponds to the satuation pessue at the vapo tempeatue [i.e., that p V = p sat (T V )]. Bubble gowth unde these conditions is contolled by two factos: 1. Inetia. The initial gowth of a bubble is vey fast, limited only by the momentum available to displace the suounding liquid fom its path. That is, inetia must be impated to the liquid to acceleate it away in font of the gowing bubble. 2. Heat diffusion. As the bubble gows in size, the effect of inetia becomes negligible, and gowth continues by vitue of diffusion of heat fom the supeheated liquid to the inteface, although at a much slowe gowth ate than duing the inetia-contolled stage of gowth. 32

33 33 Fo inetia-contolled bubble gowth duing the initial stage of bubble gowth, ayleigh (1917) modeled incompessible adially symmetic flow of the liquid suounding a bubble fo a spheical bubble with a diffeential element of adius and thickness d fo a bubble of adius B. The ayleigh equation is; B d dt 2 B 2 3 db + 2 dt 2 1 = pv ρl p L 2σ B Whee the vapo pessue in the bubble is p V and that at the inteface is p L. Since 2σ/ B << p V p L, the tem 2σ/ B can be ignoed. Using a lineaized vesion of the Clapeyon equation fo the small pessue diffeences involved, the ayleigh equation educes to; (2.8) B d dt 2 B db + 2 dt ρ V TL T = ρl Tsat sat ( p ) L hlv ( p ) L Integating fom the initial condition of B = 0 at t = 0, the ayleigh bubble gowth equation fo inetia-contolled gowth is obtained; (2.9) B ( t) 2 TL T = 3 Tsat sat ( pl ) ( p ) L h LV ρ V ρl 1 2 t (2.10) Fo heat diffusion contolled gowth, Plesset and Zwick (1954) deived the following bubble gowth equation fo elatively lage supeheats: aLt ( t) = Ja (2.11) B π Whee a L is the themal diffusivity of the liquid; a λ L L = (2.12) ρlcpl and the Jakob numbe is; 33

34 34 ρlcpl( TL Tsat ) Ja = (2.13) ρ h V LV Thus, fo heat diffusion contolled bubble gowth, the adius inceases with time as t 1/2, while it gows linealy with time duing the initial inetia-contolled stage of gowth. Mikic et al. (1970) combined the ayleigh and Plesset Zwick equations to aive at an asymptotic bubble gowth equation valid fo the entie bubble gowth peiod; [( t + 1) ( t ) 1] 2 2 B = (2.14) 3 Whee, + B = B A 2 B (2.15) + t = ta B 2 2 (2.16) 2 TL T A = 3 Tsat sat ( pl ) ( p ) L h LV ρ V ρl 1 2 (2.17) 12a B = π L 1 2 Ja The expession (2.14) educes to equations (2.10) and (2.11) at the two extemes of t +. (2.18) Bubble gowth at heated walls diffes significantly fom these ideal conditions since gowth occus in a themal bounday laye that may be thicke o thinne than the bubble itself. 34

35 35 Bulk Satuated Liquid Supeheated Laye Height θ CA Vapo Bubble B Heat Flow Bubbble Height Themal Bounday Laye Heated Suface Micolaye Figue 2.12: Bubble gowth model of van Stalen (1966). The velocity field in the liquid ceated by the gowing bubble is affected by the wall and with it the inetia foce imposed on the liquid, which may change the bubble shape fom spheical to hemispheical o to some othe moe complex shape. The hydodynamic wake of a depating bubble may distub the velocity field of the next bubble o that of adjacent bubbles. Futhemoe, apidly gowing bubbles tap a thin evapoating liquid micolaye on the heated suface. Fo example, Fig.2.12 illustates micolaye evapoation undeneath a gowing bubble and macolaye evapoation fom the themal bounday laye to the bubble as poposed in a model by van Stalen (1966) Bubble depatue Bubble depatue is anothe fundamental pocess of impotance in nucleate boiling. The tangential shea due to the volume- and time-aveaged pulsation velocity caused by the themally contolled bubble gowth is found to be esponsible fo the natual switch fom isolated bubble gowth into depatue as the tempeatue diffeence between the bubble inteface and the suounding liquid inceases. The diamete at which a bubble depats fom the suface duing its gowth is contolled by buoyancy and inetia foces (each attempting to detach the bubble fom the suface) and suface tension and hydodynamic dag foces (both esisting its depatue). Futhemoe, the shape of the bubble may deviate significantly fom the idealized spheical shape. While slowly gowing bubbles tend to emain spheical, apidly gowing bubbles tend to be hemispheical. Numeous othe shapes ae obseved using high-speed movie cameas o videos. 35

36 36 The simplest case to analyze is that of a lage, slowly gowing bubble on a flat suface facing upwad, fo which the hydodynamic and inetia foces ae negligible. Depatue occus when the buoyancy foce tying to lift the bubble off ovecomes the suface tension foce tying to hold it on. The suface tension foce also depends on the contact angle α (i.e., a contact angle appoaching 90 inceases the suface tension foce and hence the bubble depatue diamete). Fitz (1935) poposed the fist bubble depatue equation that equated these two foces. The Fitz equation, using the contact angle θ CA (i.e., θ CA = π/2 = 90 fo a ight angle) and suface tension σ, gives the bubble depatue diamete as; 1 2 σ DB, d = θ (2.19) CA g( ρl ρv ) By consideing the vaious foces influencing bubble gowth expessions fo bubble depatue diamete have been developed both empiically and analytically and can be found in liteatue. Though, these expessions ae not always consistent with each othe. The ole of suface tension in the detemination of bubble depatue diametes has been the souce of some disageement between vaious investigatos. Geneally, suface tension inhibits bubble depatue by tending to push the bubble against the wall. Howeve, Coope et al. (1978) agued that by making the bubble spheical suface tension assisted bubble depatue in some cases. Futhemoe Cole & ohsenow (1969) coelated the bubble depatue diamete with fluid popeties and found it to be independent of the wall supeheat while the expession fo bubble depatue diamete poposed by Goenflo et al. (1986) indicates weak dependency of the bubble diamete on wall supeheat. Issues egading the foces that act on a gowing bubble can be put to est only by popely accounting fo the adhesion foces and intefacial tension duing complete numeical simulation of both bubble gowth and depatue, Dhi (1998) Bubble elease fequency Using the bubble waiting time tw and the gowth time tg expessions a theoetical evaluation of the bubble elease fequency f can be appoximated, i.e. f ( + ) = 1 t w t g. As peviously explained in the beginning of section 2.3 the waiting time coesponds to the time it takes fo the themal laye to edevelop so that it can allow nucleation of a bubble. Howeve because many simplifications ae made in obtaining tw and tg the pedictions of bubble elease fequency based on waiting and gowth times do not match well with the available data, Dhi (1998). Thus both the bubble depatue diamete 36

37 37 and the bubble elease fequency ae included in coelations that epoted in the liteatue. One of the most compehensive coelations of this type is given by Malenkov (1971). Moe complex bubble depatue models include the inetia, buoyancy, dag, and suface tension foces. The suface tension may also act to pinch off the bubble as it begins to depat fom the suface. Keshock and Siegel (1964) assumed the following foce balance of the static and dynamic foces acting on a depating spheical bubble; F b + F p = F i + F σ + F D (2.20) whee F b and F p ae the buoyancy and the excess pessue foces, espectively, acting to lift the bubble off the suface and F i, F σ and F D ae the inetia, suface tension, and liquid dag foces esisting bubble depatue. The buoyancy foce is; 3 B d πd, Fb = g( ρl ρv ) (2.21) 6 The excess pessue on the dy aea whee the bubble is attached to the wall is; 2σ sinθ p = D B, b 2σ sinθ p D B, b CA CA σ + B, b (2.22) The fist tem comes fom the Young-Laplace equation law applied to the bubble base diamete, including the contact angle α, while the second tem accounts fo the effect of cuvatue at the base of the bubble. Keshock and Siegel assumed that the second tem is negligible compaed to the fist, such that the excess pessue foce acting on the base aea of diamete D B,b is; F p πd σ θ B, b = sin CA (2.23) 2 The inetia foce impated on the suounding liquid by the gowing bubble is; 37

38 Heat tansfe mechanisms In this section the heat tansfe mechanisms playing a ole in nucleate pool boiling ae descibed. In patial nucleate boiling, o in the isolated bubble egime, tansient conduction into liquid adjacent to the wall is an impotant mechanism fo heat tansfe fom an upwad-facing hoizontal suface (Foste & Geif 1959). These mechanisms ae descibed below; Bubble agitation Afte bubble inception, the supeheated liquid laye is pushed outwad and mixes with the bulk liquid. The bubble acts like a pump in emoving hot liquid fom the suface and eplacing it with cold liquid. The systematic pumping motion of the gowing and depating bubbles agitates the liquid, pushing it back and foth acoss the heate suface, which in effect tansfoms the othewise natual convection pocess into a localized foced convection pocess. Sensible heat is tanspoted away in the fom of supeheated liquid and depends on the intensity of the boiling pocess. Heated Suface Figue 2.13: Bubble agitation Vapo-liquid exchange The wakes of depating bubbles emove the themal bounday laye fom the heated suface, and this ceates a cyclic themal bounday laye stipping pocess. Sensible heat is tanspoted away in the fom of supeheated liquid, whose ate of emoval is popotional to the thickness of the laye, its mean tempeatue, the aea of the bounday laye emoved by a depating bubble, the bubble depatue fequency, and the density of active boiling sites. 38

39 39 Heated Suface Figue 2.14: Vapo-liquid heat exchange Buchholz (2005) found faste liquid tempeatue fluctuations with smalle amplitudes at a lowe liquid tempeatue level fo highe heat fluxes to the heate. In thei study the liquid above the heate was significantly supeheated with ponounced but slow tempeatue fluctuations at low heat flux. Thus, at low heat fluxes with a small numbe of active nucleation sites a significant potion of the total heat flux is tansfeed by convection Evapoation Heat is conducted into the themal bounday laye and then to the bubble inteface, whee it is conveted to latent heat. Maco-evapoation occus ove the top of the bubble while micoevapoation occus undeneath the bubble acoss the thin liquid laye tapped between the bubble and the suface, the latte often efeed to as micolaye evapoation. A hypothesis fo the mechanism of nucleate boiling consideing the vapoization of a micolaye of wate beneath the bubble was fist suggested by Mooe and Mesle (1961). They measued the suface tempeatue duing nucleate boiling of wate at atmospheic pessue and found that the wall tempeatue occasionally dops C in about 2msec. Fom these osciallations in the tempeatue measued at the bubble elease site they deduced the existence of a micolaye unde the bubble. This hypothesis was futhe veified by Shap (1964). 39

40 40 Heated Suface Figue 2.15: Evapoation The existence of a micolaye undeneath isolated bubbles fomed on glass o ceamic sufaces was not only confimed by Coope and Lloyd (1969) but its thickness was also deduced by the authos fom the obseved esponse of the heate suface themocouple. They expessed the local thickness of the micolaye in the following fom: δm ν l t (2.24) Whee v l is the kinematic viscosity of the liquid, and t is the bubble gowth time. It was futhe demonstated that bubble gowth was mostly due to evapoation fom the micolaye. Howeve, even afte moe than thee decades of eseach, we still do not have an effective, consistent model fo bubble gowth on a heated suface that appopiately includes the micolaye contibution and time-vaying tempeatue and flow field aound the bubble, Dhi (1998). Plesset and Pospeetti (1976) concluded that in subcooled boiling, evapoation at the micolaye accounts fo only 20% of the total heat flux. Lee & Nydahl (1989) calculated the gowth of spheical bubbles with a micolaye using Coope and Lloyd s fomulation and came to the same conclusion as Coope and Lloyd, that micolaye evapoation is a significant contibuto to the heat tansfe duing bubble gowth. While Son et al. (1999) epoted micolaye contibution in evapoation to be about 20% fo a cetain set of calculations. A theoetical pediction of the micolaye thickness was caied out by Zhao (2002) which focuses on individual bubbles and can be used to explain the pool boiling mechanism and to pedict the heat flux in the nucleate boiling egion at high heat flux and to the minimum heat flux point. Despite these effots, studies on the effect of the micolaye ae inconclusive and accuate expeimental data on the thickness of the micolaye unde the bubble not widely available. 40

41 Heat tansfe oveview The mechanisms descibed above compete fo the same heat in the liquid and hence ovelap with one anothe, themally speaking. At low heat fluxes chaacteistic of the isolated bubble egion, natual convection also occus on inactive aeas of the suface, whee no bubbles ae gowing. The ate of latent heat tanspot depends on the volumetic flow of vapo away fom the suface pe unit aea. ohsenow (1962) poposed the fist widely quoted coelation. He assumed the boiling pocess is dominated by the bubble agitation mechanism, whose bubble-induced foced convective heat tansfe pocess could be coelated with the standad single-phase foced-convection coelation elation: Mostinski (1963) applied the pinciple of coesponding states to the coelation of nucleate pool boiling data, aiving at a educed pessue fomulation without a suface fluid paamete o fluid physical popeties. This coelation gives easonable esults fo a wide ange of fluids and educed pessues. Stephan and Abdelsalam (1980) developed individual coelations fo fou classes of fluids (wate, oganics, efigeants, and cyogens), utilizing a statistical multiple egession technique. These coelations used the physical popeties of the fluid (evaluated at the satuation tempeatue) and ae hence said to be physical popety based coelations. Coope (1984) poposed a educed pessue expession in dimensionless fom fo the nucleate pool boiling heat tansfe coefficient. His coelation uses educed pessue, molecula weight, and suface oughness as the coelating paametes. Inceasing the suface oughness has the effect of inceasing the nucleate boiling heat tansfe coefficient. The coelation coves a database of educed pessues fom to 0.9 and molecula weights fom 2 to 200 and is highly ecommended fo geneal use. Goenflo (1993) poposed a educed pessue type of coelation that utilizes a fluid-specific heat tansfe coefficient, defined fo each fluid at the fixed efeence conditions. This method is accuate ove a vey wide ange of heat flux and pessue and is pobably the most eliable of those pesented. Howeve, this appoach is not extendable to boiling of mixtues, which is of inteest in numeous industial pocesses. Auache et al. (2002) and Bucholz et al. (2005), taking advantage of the gowing capacities in data pocessing by computes in connection with miniatuized sensos and high speed cinematogaphy and othe opto-electonic facilities, conducted a seies of expeiments to povide a deepe insight into some of the aspects of the extemely complex mechanisms of boiling. 41

42 42 Chu and Yu (2009) poposed a factal model to pedict the heat tansfe in nucleate boiling of a pue liquid at low to high heat fluxes including the CHF, based on the factal distibution of nucleation sites on heating sufaces. The model pesents an impoved undestanding of the mechanisms of nucleate pool boiling by incopoating contibutions of latent heat flux, tansient conduction and natual convection. 2.5 Numeical simulation In this section, we pesent the diffeent numeical methods that allow us to study the dynamics of bubble gowth. Fist we intoduce the poblematic of solving of the bubble gowth dynamics. We then conside the diffeent numeical techniques allowing us to solve this dynamic poblem. Thee ae vaious computational methods available to solve incompessible two-phase poblems. These ae mainly of two types accoding to the mathematical epesentation of the liquid-vapo inteface which can be eithe explicit o implicit. Such as the font tacking method, the bounday integal method, the volume of fluid method, the Lattice Boltzmann method, diffuse inteface modelling and the level set method. Each of them has advantages and limitations. Most ae limited to elatively small aveage o slip velocity between the two phases. We ediect ou attention to the level set method and motivate this choice. In the next chapte the level set method is elaboated in detail Poblems hindeing modelling of boiling Multiphase flows ae often difficult to model computationally, especially because thee is a steep change in physical popeties such as density, viscosity etc. between one phase and the othe which makes govening equations hade and the computations stiffe. Futhemoe thee is the difficulty of tacking the fluid-fluid inteface. These ae detailed below: Govening equations of the nucleate boiling flow: The boiling fluid is locally eithe liquid o vapou. Fom the mathematical point of view, the two phases can be consideed as a field that has two single phase egions, with moving boundaies that sepaate the phases. The diffeential equations, say the Navie-Stokes equations, hold fo each of the fields sepaately, but cannot be applied to the whole field without violating the condition of continuity at the boundaies of each of the fields, the inteface location. These boundaies, i.e. the geomety of the intefaces ae unknown a pioi as a function of time and space. At the inteface a set of jump conditions ae actually satisfied, the ankine-hugoniot 42

43 43 jump conditions. The main poblematic of solving the boiling flow is thus to take into account the inteface as a moving bounday. Let us now conside the mathematical epesentation of the inteface. epesentation of the inteface location: The location of an inteface x i in a thee space dimensions system can always be elated to the location of a suface. The mathematical desciption of a suface can be eithe explicit, i.e. x i = f (t), o implicit, i.e. F(x i ; t) = 0, whee x i denotes the position of the suface. This distinction is at the basis of the two families of numeical method fo the solving of the boiling flows by tacking intefaces. The fist categoy consists of Lagangian methods whee the intefaces ae explicitly tacked though out the simulation, meaning that the cell edges ae always aligned with the intefaces (Fig 2.16a). The coesponding methods ae theefoe denoted explicit tacking methods. a-explicit tacking of inteface b-implicit tacking of inteface Figue 2.16: Explicit and implicit inteface epesentation. The second family consists of Euleian inteface tacking methods whee the intefaces ae allowed to intesect the cells abitaily (Fig 2.16b). This allows fo fixed gids to be used which is moe desiable fom both computational and implementational points of view. In the following we will pesent vaious Euleian and Langangian methods that have been applied to the study of two-phase flow poblems. 43

44 Explicit tacking of intefaces When the inteface is epesented explicitly, its location is tacked with the help of a moving mesh. This appoach peseves a shap inteface epesentation but does so at the cost of intoducing additional algoithmic complexities to handle beak up and meging of inteface segments. The govening equations also have to be modified to account fo the gid movement, and peiodic emeshing may be necessay if the intefaces defom too much, which may intoduce additional eos. Thee exists two main ways of tacking the inteface dynamically. Eithe a pat of the discetized elements used fo descibing the physical domain epesents the inteface, the method is eithe puely Lagangian o mixed method called ALE (Abitay Lagangian Euleian). O a moving additional Lagangian gid is supeposed to the fixed Euleian gid. The esolution on the Euleian gid must be coupled with the time dependent position of an additional Lagangian gid which epesents the inteface. These ae the so-called font-tacking methods. Lagangian and ALE methods: The most inconvenient featue of the puely Lagangian methods is that the mesh becomes apidly highly distoted. As a consequence they ae to ou knowledge, not applicable to the study of the liquid-vapo phase change simulations. Fo the second categoy of method, namely the ALE methods, a paticula sub-element of the mesh is associated to the inteface as epesented on the figue below. The inteface being of time dependent geomety, the mesh is distoted with time. As a consequence the govening equations need to be solved on a cuvilinea moving mesh. If this method allows actually a vey accuate desciption of the inteface its numeical handling is costly and complex. Moeove the ability to take into account topological tansitions as well as seveal bubbles is eally limited. 44

45 45 Figue 2.17: ALE Gid Font-tacking method: In the font tacking method, make paticles ae explicitly intoduced to keep tack of the font that educes the esolution needed to maintain accuacy. Howeve e-gidding algoithms should be employed with font tacking method to pevent make paticles fom coming togethe, especially at the points of lage cuvatue. Figue 2.18: FTM Gid In these methods the geomety of the inteface is descibed with the help of a Lagangian mesh (dots on the HS figue) supeposed on a fixed Euleian gid, the squaes in the above figue, Tyggvason et al. (2001). In the algoithm, the motion of the inteface is an independent step of the calculation. This engendes majo difficulties to apply constaints to this motion that ae consistent with the main physical balances (mass, momentum, enegy). The distotion of the Lagangian mesh due to the motion of the intefaces equies to egulaly econstuct this mesh. Moeove, even though the inteface is epesented as a shap suface, a necessay smeaing of some souce tems, like the suface tension foce, must be intoduced in ode to insue that the solving of the govening equations on the Euleian gid takes into account the physics of the inteface. This smeaing is numeically contolled and its consequences on the main balances cannot be analyzed. The topological tansitions ae not natually taken into account. 45

46 Implicit tacking of intefaces These add dynamics to implicit sufaces. The main idea behind inteface captuing methods is to use an implicit definition of a suface to descibe the inteface, athe than an explicit definition. In these methods we take advantage of the fact that the phase field F is not unique and only its zeo isocontou detemines the location of the inteface. Fo example, the zeo isocontou of Φ (x)=x²-1 is the is the set of all points whee Φ (x)=0; i.e., it is exactly Ω={-1,1}. The location of the inteface is only a pat of the field F. The knowledge of the field F is sufficient to captue the inteface location. The coesponding methods ae denoted font captuing methods. Since the definition of F can be extended to the whole two-phase system as an Euleian field, F can be numeically teated as any othe physical main vaiable of the two-phase fluid desciption and thee is no need to use a specific numeical discetization fo the inteface. This makes the numeical detemination of the time dependent location of the intefaces easie. In addition to this, coalescence and beak up is, fo most Euleian methods, implicitly contolled by the gid esolution and is thus teated natually. The dawback of using Euleian methods is that the intefaces potentially ae smeaed acoss one o moe cell widths equiing highe esolution and coection methods to achieve the same accuacy as fo the Lagangian methods. In this section, we povide a shot pesentation of the basic ideas allowing such an inteface desciption. The volume of fluid method: The volume of fluid method (VOF) is based on the discetization of the volume faction of one of the fluids. The motion of the inteface is captued by solving a consevative law fo volume faction navie-stokes equations simultaneously. Since the inteface is epesented in tems of volume faction, mass should be conseved. But cetain stages of the algoithm ae appoximative and the consevation is inexact in most of the existing algoithms, Aulsia (2003). The VOF method needs to have accuate econstuction algoithms to solve fo the advection of volume faction. A disadvantage of the VOF method is that it is difficult to compute accuate local cuvatue fom volume faction. This is due to the shap tansition in volume faction nea the inteface. Lattice Boltzman method: The Lattice Boltzman method is a mesoscopic appoach to the numeical simulation of fluid motions based on the assumption that a fluid consists of many paticles whose epeated collision, tanslation, and distibution convege to a state of local equilibium yet al.ways 46

47 47 emaining in flux. LBM has advantages such as implementation on a complex geomety, vey efficient paallel pocessing, and ease of epoduction of the inteface between the phases. Howeve, LBM is not yet a widely used computational method to tack the fluid motion in multiphase systems due to its computational intensity. Level set method (LSM): The level set appoach is anothe potential numeical method to solve incompessible two-phase flow incopoating suface tension tem. In the level set method, the inteface is epesented as the zeo level set of a smooth function. This has the effect of eplacing the advection of physical popeties with step gadients at the inteface with the advection of level set function that is smooth in natue. The level set method does not have the same consevation popeties of VOF method o font tacking method. Phase leakage is an unelenting dawback of the level set method which causes the less dense phase to leak into the dense phase, mainly when thee is a steep contast between the density and viscosity of the two inteacting fluids. Phase leakage is a numeical atefact intoduced by the smoothing opeatos at finite gid esolution. At infinitesimal gid esolution this poblem would cease. The standad emedy fo phase leakage is einitialization. In einitialization, the thickness of the inteface is maintained by iteating between time steps, an auxiliay patial diffeential equation is used to condition Φ to be close to a distance function fom the inteface, intoduced by Sussman et al. (1994). einitialization is a computationally expensive wokaound to e-distibute Φ to get close to phase consevation. The poblem of phase consevation is fathe complicated when dealing with inte species phase change, as in the case of boiling. The majo stength of level set method lies in its ability to compute cuvatue of the inteface easily. Futhemoe level set does not equie complicated font tacking egidding algoithms o VOF econstuction algoithms. Level set method is based on continuum appoach in ode to epesent suface tension and local cuvatue at the inteface as a body foce. This facilitates the computations in captuing any topological change due to change in suface tension. Diffuse inteface method: The diffuse inteface method is a kinded notion to the level set method and VOF in that it computes the tanspot of anothe function that vaies between the phases, the chemical o themodynamic potential. In diffuse inteface models, the inteface is consideed as a continuous tansition zone. The main issue is then to descibe the behaviou of fluid paticles that ae 47

48 48 located within the tansition egion. To do that, the two-phase system is fist studied themodynamically and the coesponding equations of motion ae then deived. The suface tension between two fluids is also the excess patial mola Gibbs fee enegy pe unit suface aea; so that the change of chemical potential acoss an inteface between immiscible fluids is teated by the notion of suface tension as infinitely steep. The diffuse inteface method pemits this condition to be to be meely elaxed to be steep, and then a field equation fo chemical potential is tacked, athe then the imposition of topology and stess balance equations implied by the notion of suface tension. The late method still equies gid adaptation, which in state of the at computational methods employ auxiliay equations fo elliptic mesh geneation, but ae fagile in the face of topological change, e. g. coalescence o beakage phenomenon, Kolev (1993). It is aguable whethe at the same level of computational intensity; geate accuacy is assued by monitoing the fee suface position using topological methods o by solving an auxiliay tanspot equation fo a field vaiable. The latte method is easie to code. Of these the Level Set method is by fa the most popula of the inteface captuing methods, e.g. Sethian (2001). In this method the necessay smooth extension of the main vaiables aound the inteface is made using a numeical distance function as the field F. To contol the smeaing of this field, it is egulaly e-initialized. One of the difficulties of the level-set method is to ensue that the mass is actually conseved when the smeaing of the distance function is such e-initialized. 2.6 Conclusion Theoetically both explicit and implicit methods of inteface epesentation can appoximate the inteface with the same accuacy. Howeve, in pactice that is not always achievable. In case of the discontinuous implicit appoach the indicato step-function has to be smoothed to avoid numeical oscillations, which esults in an inteface of finite thickness, of the ode of mesh size. If we use an explicit epesentation this dawback is absent, and can be eliminated with the finite element vesion of continuous implicit appoach. In the case of changes in inteface topology like mege o beakup, the explicit epesentation appoach fails completely, and all the known emedies consist in vey sophisticated ad hoc algoithms. While the implicit appoaches (both continuous and discontinuous) ae capable of captuing the inteface independently of its topology. 48

49 49 These obsevations lead us to conclude that continuous implicit epesentation method combined with the finite element discetization with suitable phase coection is the best choice fo numeical simulation of bubble dynamics in nucleate pool boiling. 49

50 50 Chapte 3 Numeical modelling In ode to model dynamic physical phenomena such as fluid flow the involved poblems is posed in the fom of patial diffeential equations based on the fundamental physical pinciples of continuity, momentum and enegy upon which all of fluid mechanics is based. In this chapte the fundamental equations govening the dynamics of single bubbles and the associated heat tansfe will be descibed. The liquid-vapo phase change poblem involves both fluid flow and heat tansfe and equies the solution of the govening equations coupled with appopiate inteface bounday conditions. The numeical esults pesented in this thesis equied a significant amount of wok to be employed in development of a moving inteface pogam that could manage the computationally complex phenomena of bubble gowth. Despite the fact that the basic concepts behind moving intefaces ae athe simple, the implementation poses many pactical challenges. The main pupose of this chapte is thus twofold: fistly, to descibe how the code woks and, secondly, to detail the schemes and numeical ticks developed to make the code eliable and accuate. This chapte pesents the computational model used in this study. The model is based on a consevative level set fomulation which is used to tack the inteface. Isothemal flow without phase change is a special case of the fomulation pesented hee. We wite one set of tanspot equations valid fo both phases. This local, single field fomulation incopoates the effect of the inteface in the equations as delta function souce tems, which act only at the inteface. Let us conside the mathematical epesentation of the inteface. 3.1 Consevative level set method This section pesents the computational model used in this wok to study single bubble dynamics duing nucleate pool boiling. The model is based on the level set fomulation of Oshe and Sethian (1988). In the level set method fo a two phase flow an auxiliay field the level set function Φ is intoduced to keep tack of which pats of the computational domain the two phases ae occupying. Fom a computational viewpoint, the most convenient option fo choosing the implicit function Φ, is the signed nomal distance to the inteface. Such a vaiant of defining the function Φ. was advocated by A. Devieux and F. Thomasset (1981) but fully exploed and justified by Oshe and Sethian (1988) 50

51 51 who called this method the level-set appoach and the signed distance function Φ was called the level set function. The sign of the level-set function was chosen to be positive on one side of the inteface and negative on the othe, and the modulus of the function's gadient is equal to 1 eveywhee (as it is a nomal distance function). The level set function Φ, defined by a signed distance function d(x) between a position x and the inteface, Γ at time t is descibed in the following tems; ( x x ( ) ) d( x, t) = sign min t x i Γ i, (3.1) < 0 if x vapo φ d = = 0 if x Γ (3.2) > 0 if x liquid { x ( x, t) = 0} Γd = φ (3.3) Whee x i ae the coodinates of the inteface and sign is positive in liquid phase and negative in vapo, using which the phase can be epesented by a Heaviside function (Fig 3.1). 1 fo d < 0 φ 0 = HS = 0 fo d = 0 (3.4) 1 fo d > 0 The delta function δ(φ) is defined as the deivative of the one-dimensional Heaviside function i.e. δ(φ)=h (Φ). The delta function δ(φ) is identically zeo eveywhee except at δ(φ) = Inteface epesentation Use of the Heaviside function descibed by Eq 3.4 has an abupt jump at the inteface and leads to poo esults due to the assumed zeo thickness (Fig 3.1) of the inteface. Which plots of the initial level set function, the smooth level set function and the smooth delta functions fo ε = 0.01 and with the inteface cented at d = 0. 51

52 52 Figue 3.1: Heaviside Function, smooth Heaviside function and delta function. Also, since δ(φ 0 ) = 0 almost eveywhee, i.e., except on the inteface which has measue zeo, it seems unlikely that any standad numeical appoximation will give a good appoximation to its integal. An altenative is to use a fist-ode accuate smeaed-out appoximation of δ(φ). Fist, the smeaed-out o smooth Heaviside function Φ s (d) needs to be defined in place of the shap Heaviside function Φ 0 (d). The smooth Heaviside function is defined as; 0 fo d < ε 1 d 1 dπ φ = 1+ + sin fo ε < d < ε (3.5) 2 ε π ε 1 fo ε > d Whee ε is a paamete that contols the inteface thickness and is of the ode of the aveage mesh element size. Then the delta function is defined as the deivative of the Heaviside function as; δ ( φ ) s dπ = + cos 2 ε ε ε 1 fo d < ε fo ε < d < ε (3.6) fo ε > d This definition of the smooth delta function and the smeaed-out Heaviside function allows the suface integal to be evaluated using standad techniques such as the midpoint ule. The smooth Heaviside function and the smooth delta function ae shown in Fig 3.1 along with the shap Heaviside function. It should be noted that the smeaed-out Heaviside and delta functions appoach to the calculus of implicit functions leads to fist ode accuate methods. With the smeaed-out Heaviside the eos in the calculation of integals ae O( x) egadless of the accuacy of the integation 52

53 53 method used. Since highe-ode accuate methods can be complex, we pefe the smeaed-out Heaviside and delta function methods wheneve appopiate. The smooth continuous natue of the level set function (Fig 3.2-c) which is defined at all points of the computational domain Ω, makes it easy to calculate the inteface nomal vecto and inteface cuvatue which ae obtained fom the level set function gadient and the divegence of the gadient. n Γ = φ φ (3.7) κ Γ = n Γ φ = (3.8) φ To avoid the steep jump while epesenting physical popeties acoss the inteface esulting fom Eq 3.2 Olsson et al. (2007) smea the inteface ove a egion of width 2ε using a hypebolic tangent function. Thus the level set function takes the following fom d φ = tanh 2 ε 1 ( x) + 1 fo fo fo d < ε ε < d < ε ε < d We have chosen to epesent Φ by the hypebolic tangent function. It is zeo in vapo phase, unity in {, ~ = 0.5 } liquid phase and has intemediate values fo the inteface, Γ = xφ ( x t) (3.9). Fo example, if the inteface is the cicle with the cente at the oigin and unit adius, the distance function can be computed as; 2 2 ( x + ) 1 d t 0( x, y) = (3.10) = y The above equation is elaboated in Figue 3.2 which plots the unit cicle cented at oigin using a distance function, a Heaviside function and the hypebolic tangent function. 53

54 54 (a) (b) (c) Figue 3.2: Suface plot of the unit cicle cented at oigin at t=0 with the level set function as (a) a distance function, (b) a Heaviside function, and as (c) a hypebolic tangent function. 54

55 Equation of the inteface motion We now focus on the poblem of advection of the inteface. The inteface velocity nomal to the inteface diection can be found by computing the full fist diffeential of the level set function, giving us; u Γ φ = t φ (3.11) If we combine the expession fo the nomal intefacial velocity with the equation of continuity fo fluid velocity (which is a diect consequence of mass balance) we get; uγ = u n on Γ, (3.12) Whee u is the fluid velocity, we immediately obtain the following evolution equation fo the levelset function; φ + t ( u n) φ = 0 on Γ, (3.13) This equation is defined ove the whole domain Ω, since all vaiables ae defined thee. Substituting the nomal vecto fom (Eq 3.7) into the above expession, we deive the following pue convection equation fo the level-set function Φ; that descibes the evolution of Φ coesponding to the motion of the liquid vapo inteface (the Φ = 0.5 iso-contou). φ + u t Γ φ = 0, in Ω (3.14) To descibe the inteface motion we have taken into consideation only the nomal component of the inteface velocity, as it should be evident in (Eq 3.12). This is a consequence of using the implicit appoach fo inteface epesentation. If we elied on the explicit (paametic) epesentation, the inteface points would be convected with fluid velocity, i.e. moving along the tajectoies of fluid paticles (along the chaacteistics); the motion of inteface points would not be esticted to the nomal diection. Howeve, it is evident that only the motion of inteface in its nomal diection may change the inteface shape; the tangential motion has no influence on the fom of inteface. In addition, Hou et al. (1994) and Stockie and Wetton (1999) showed: it is the tangential mode of the 55

56 56 inteface motion that causes main stability poblems. These aguments should futhe justify ou choice of the level-set appoach. When choosing a suitable numeical method to solve Eq 3.14 one has to conside: The method should be consevative. No spuious oscillations should be intoduced. The thickness of the inteface and the pofile of Φ should emain constant Phase consevation While using the advection equation (Eq 3.15) descibed in the pevious section to tanspot the inteface of a distance level set function we note that the pofile and thickness of the inteface do not emain constant. This is because when the signed distance function d(x,t) is advected by a nonunifom flow, it does not necessaily coespond to a distance function any longe. To elaboate this we conside one dimensional advection with some constant velocity. φ t ( x, t) φ( x, t) + u x = 0, (3.15) Assume that initially d(x,t) is a egulaized chaacteistic function denoted by d 0 (x) that undegoes an initial petubation δ(x), theefoe at t=0 we have; ( x 0) φ ( x) δ ( x) φ, +, (3.16) = 0 The solution to (Eq 3.14) is then;,, (3.17) ( x t) = φ ( x ut) + δ ( x ut) φ 0 Thus the petubations will not be dampenedd but simply advected with the constant velocityu. Mulde et al. (1992) pointed out that new petubations ae constantly intoduced in numeical calculations in the fom of lage of o small gadients of the distance level set function and theefoe the shape of the pofile will become moe and moe distoted as time evolves. It is theefoe mandatoy to ensue that the level set emains a distance function without changing the zeo-level set, and as a consequence, eseaches have focused on methods that fobid the level set function fom developing lage o small gadients. When this e-distancing of the level set function is pefomed at 56

57 57 the beginning of the calculation it is called initialization, and when it is pefomed duing the computation it is called einitialization, which is a popula way to wok aound the phase loss poblem in the level set method. Thee ae many diffeent appoaches to einitialize the level set function as a distance function: Diect appoach: Chopp (1993) was the fist to ecognize and exploit the potential of einitialization using a diect appoach late used by Meiman (1994). It is a staightfowad appoach that consists in simply standing at each computational mesh point and finding the signed distance to the inteface. The dawback of this appoach is its computational complexity which is of the ode O(N 3 ) fo an N x N gid. Pseudo time appoach: This involves solving the following fist ode patial diffeential equation in pseudo-time to steady state. φ + sign ( φ 1 ) = 0, (3.18) τ This was intoduced by Sussman et al. (1994). Its vitue is that the level set function is einitialized with out explicitly finding the zeo-level set. The pincipal dawback when solving this equation is that the zeo level set, i.e.,the position of the inteface, might be shifted due to numeical diffusion of the undelying numeical scheme. So, eventhough the method does not equie the localization of the inteface, the computational cost of solving the nonlinea hypebolic equation (3.19) is athe high. Moeove, any numeical scheme fo the solution of (3.19) will intoduce some numeical diffusion (some pat of the diffusion comes also fom the appoximation of sign see, e.g., Sussman et al. (1994), which esults in small inaccuacy of the inteface location and some loss of mass, e.g., Tonbeg (2000). Fast maching method: In ode to cicumvent these limitations, Sethian (1999) poposed a fast maching appoach fo the distance e-initialization based on locally solving the Eikonal equation φ = 1 while employing only points close to the inteface in the numeical stencil. Fast maching methods solve the Eikonal equation 57

58 58 ove the whole computational domain in O(N log N) opeations, whee N is the numbe of gid points in the domain. In this method a data stuctue is ceated that allows one to mach though cells in an ode that pemits only one pass pe cell. Using a heap sot algoithm, this pocedue can be made highly efficient. Howeve, the accuacy of this appoach is limited, and the e-distancing of the points closest to the inteface induces a displacement of the font. In the context of multiphase flows the pominent limitations of the distance function level set appoach is that neithe the level set tanspot no the e-initialization inheently conseve the volume of the egion enclosed by the zeo level set. Fo liquid-vapo flows in boiling, this can lead to gains o losses in the mass of the liquid, which can lead to substantial eos in many applications. In this wok, we appoach the poblem of phase consevation by choosing a stable and accuate numeical scheme based on finite element appoach, fo advecting the level set function. Instead of a signed distance function, a hypebolic tangent level set function Φ (Eq 3.9) has been employed which allows us to wite the advection equation (Eq 3.14) in consevative fom as follows; φ + t Γφ ( u ) = 0, (3.19) With the level set tanspot equation witten in consevative fom, and the given definition of Φ, it is clea that the scala Φ should be a conseved quantity. As in the case of the distance level set function, nothing insues that solving Eq. (3.18) will peseve the fom of the hypebolic tangent pofile Φ. As a esult, an additional e-initialization equation needs to be intoduced to e-establish the shape of the pofile. As oiginally poposed by Haten (1977) it is possible to add atificial compession in ode to maintain the esolution of contact discontinuities. This can be viewed as an intemediate step in pseudo time whee one is solving the consevation law: φ + f ( φ ) = 0, (3.20) τ whee f coesponds to the compessive flux. In ou case, we want the atificial compession flux to act in egions whee 0 < Φ < 1 and in the nomal diection of the inteface. To achieve this, we f = φ 1 φ n. We denote the time vaiable by τ to stess that this is an atificial time, not choose ( ) Γ 58

59 59 equivalent to the actual time t. We note that (3.20) is a hypebolic diffeential equation. As τ inceases, stationay shocks will develop at the intefaces. To avoid discontinuities at the inteface, we add a small amount of viscosity o diffusive flux thus modifying the consevation law to; φ + f ( φ ) = ε ( φ ), (3.21) τ The diffusive flux on the ight hand size of the above equation balances the nomal compessive flux that aims at shapening the pofile and ensues that the pofile of the liquid vapo inteface emains of chaacteistic thickness ε. Eq can be ewitten in consevative fom, as follows; φ + f ( φ) = 0, (3.22) τ Whee, ( φ ) = ( φ ) ε ( φ) f f, (3.23) By solving (3.22) to steady-state the inteface thickness will emain constant and popotional to ε. Taking advantage of the divegence fee velocity field Olsson and Keiss (2005) set the atificial time as equal to the eal time in ode to combine Eqs. (3.22) and (3.19) esulting in the following stabilized level set advection equation, witten in consevative fom: φ + t [ u φ + φ( 1 φ ) n ε ( φ) ] = 0 Γ Γ (3.24) Since both the advective as well as the einitilization steps ae fomulated as consevation laws, φdx is constant in time in the continuous case. As a esult, solving successively fo Eq. (3.24) Ω accomplishes the tanspot of the Φ = 0.5 iso-contou, while peseving the shape of the pofile and ensuing the consevation of Φ without einitializing the level set function at evey time step Physical popeties The density and viscosity of the liquid and vapo phases is defined by scala fields that use the level set function to distinguish between phases. 59

60 60 ρ φ = ρ + ( ρ ρ )φ (3.25) v l v = µ + ( µ µ )φ (3.26) µ φ v l v These ae defined though out the domain and not just on the inteface, because of which a delta function is used with the cuvatue duing numeical implementation. The delta function used (Fig 3.1) is the absolute value of the gadient of the level set function. 3.2 Govening equations The level set equation is coupled to the Navie Stokes equation though the level set function and the velocity field. The Navie Stokes equation eads. u T ρ + u u = p + ρ g + F s β T ( T T sat ) g + µ ( u + u ) (3.27) t Whee the density and viscosity ae given by Eq (3.25,3.26) and F s is the local suface tension foce at the inteface. The suface tension foces ae evaluated using the continuum suface foce method of Backbill et al. (1992). F s = σκδ n Γ (3.28) Whee, δ is the Diac delta function localizing the suface tension foce to the inteface. In bubble dynamics the accuate epesentation of the suface tension foces is citical fo the numeical model. An inaccuate calculation of these foces geneates subtle spuious cuents that act as paasites and contaminate the solution leading to oscillations that can be stong enough to destoy the inteface Shin et al. (2005). A pope epesentation of F s should satisfy the Young-Laplace law fo an inteface in equilibium. To achieve this, an impotant ingedient is the accuate estimation of the inteface cuvatue. In the level set method the inteface cuvatue can be accuately calculated since the chosen smooth Heaviside function has continuous fist and second deivates. The continuity equation is deived by including the effects of volume expansion due to liquid-vapo phase change at the inteface. Applying the chain ule we can wite the continuity equation as 60

61 61 ρ φ + ( ρu) = 0 φ t Taking φt fom Eq (3.19) whee the inteface velocity is given byu = u + u Γ evp. (3.29) φ = t (( u + ) φ ) u evp And inseting into Eq (3.29) we obtain the following elation. ρ φ ( u + u ) φ + ρ u + u ρ = 0 evp (3.30) (3.31) ( ) + ρ u = 0 u evp ρ (3.32) = uevp u ρ ρ (3.33) Whee the vapo velocity at the inteface due to evapoation is defined in tems of the vapo mass flux and is detemined by the expession: u evp m& = whee ρ v λ T m = δ h l & (3.34) fg The enegy equation is T ρ C p + u T = λ T + Q (3.35) t The computations ae pefomed using a finite element method-based softwae COMSOL. The advection equation of the level set function; the incompessible Navie Stokes equations and the enegy equation ae coupled with each othe and ae solved using the multiphysics utility. The advection equation uses the velocity field calculated using the incompessible Navie Stokes equations plus the vapo velocity at the inteface fom the enegy equation (Eq. 21). The dependence of the level set function in the incompessible Navie Stokes equations comes though the physical popeties as epesented by Eq. (3.25, 3.26). 61

62 Contact angle In suface science the contact angle is an impotant paamete as its vaiation may influence bubble depatue diametes and gowth ates duing bubble gowth. Sefiane et al. (1998) demonstated that the contact angle plays a cucial ole in the magnitude of the citical heat flux. Theefoe, fo the numeical simulation of bubble dynamics, an appopiate contact angle model is essential fo a coect pediction of the bubble fomation. Vaious appoaches have been employed in liteatue Static contact angle appoach Because of the lage uncetainties associated with detemination of advancing and eceding contact angles, eseaches have often used a static contact angle, a constant value of the contact angle while simulating bubble dynamics duing nucleate pool boiling. Son et al. (1999) assumed a static contact angle at the bubble duing thei numeical simulation of a single bubble on a hoizontal suface duing nucleate pool boiling. Thei esults showed that the depating bubble became lage with incease in contact angle. Abaajith and Dhi (2002) fixed the contact angle thoughout the bubble gowth and depatue pocess while elating it to the Hamaka constant which was found to change with suface wettability. A caeful study of contact angle hysteesis (Eq-2.2), using dops of hydocabon chains was conducted by Lam et al. (2002) in ode to investigate the elation between the contact angle of a liquid and the size of its molecule. The authos pefomed contact angle measuements of 21 liquids fom two homologous seies (n-alkanes and 1-alcohols) on FC-732-coated sufaces. They obseved that the contact angle hysteesis deceases with the chain length of the liquid while the eceding contact angle equals the advancing angle when the liquid molecules ae infinitely lage. This implies that as the advancing and eceding contact angles incease the hysteesis between them deceases, such that at 90 the liquid will have a cosntant contact angle. To elaboate this we plot the advancing and eceding contact angles (ACA and CA) measued by Lam et al. (2002) against the mean static contact angle, θ 0 calculated fom diffeent pais of expeimentaly obtained ACA and CA given in the pape. The mean static contact angle is simply defined as the the aithmetic mean between ACA and CA, measued as; θ ACA + θ θ CA 0 = (3.36) 2 62

63 63 Figue 3.3 thus demonstates that advancing and eceding contact angles become vey close when extapolated to 90. Thus a static contact angle appoach to simulating bubble dynamics is moe accuate at high values of contact angle, i.e. nea 90. Figue 3.3: Advancing and eceding contact angles of n-alkanes and 1-Alcohols vesus the mean contact angle, θ 0. Data fom Lam et al.(2002) Contact line velocity appoach The expeimental data typically obtained fo contact angle vaiation (e.g. Dussan 1979) in the case of a doplet is shown in Figue

64 64 θ θ ACA θ CA u CL >0 0 u CL <0 u CL,max -u CL,max Figue 3.4: Chaacteistic expeimental esults fo contact angle measuements against contact line velocity. The contact angle is plotted as a function of contact line velocity. The fitted cuve shows a quasilinea vaiation of contact angle between limiting inteface velocities. Hee u CL,max denotes the limiting value of the contact line velocity at which a measuement of the contact angle was made. Based on these expeimental obsevations the dynamic contact angle has been typically modelled as a function of the inteface speed. In the case of dynamic contact angle at the base of a vapou bubble, amanujapu and Dhi (1999) deived a contact angle model based on thei expeiments using a silicon wafe as the test suface with micomachined cavities fo nucleation, duing nucleate pool boiling. They calculated the inteface velocity and measued the bubble base diamete as a function of time. They obseved the dynamic contact angle as function of the inteface velocity and obtained a figue simila to Figue-2. amanujapu and Dhi (1999) found that the contact angle vaied duing diffeent stages of bubble gowth and was weakly dependent on the inteface velocity thus the contact angle could be detemined based on the sign of the inteface velocity. Moe ecently, Mukhejee and Kandlika (2007) studied the effects of dynamic contact angles on the fomation of single bubbles duing nucleate boiling using two kinds of contact line models without consideing the evapoation in the micolaye. In the fist appoach they use a constant advancing and 64

65 65 constant eceding contact angle depending on the sign of the contact line velocity, u CL which is appoximated as the ate of change of bubble base diamete. θ CA θ = θ ACA CA if if u u CL CL 0 > 0 (3.37) In the second appoach the dynamic contact angle is a function of the ate of change of bubble base diamete and thus contact line velocity based on Fig-3.4 with a smooth tansition fom eceding contact angle to advancing contact angle. θ CA = u CL θ ACA θ CA θ ACA + θ 2umax θ CA ACA + θ 2 CA if if if u u max CL u < u u CL CL > u max u max max (3.38) Dynamic contact angle models based on contact line velocity have been used by othe eseaches as well. Fancois and Shyy (2003) used a method simila to one descibed above to simulate doplet speading. Huang et al. (2004) computed the motion of a two-dimensional dop/bubble attached to a plane suface. Fo the motion of contact line and the detemination of the dynamic contact angle, they used ad hoc models based on a elationship between moving contact angle and contact line speed. Smith et al. (2005) employed a contact line velocity based model to study the contact line dynamics of an inteface between immiscible fluids. One pominent aspect of the contact line velocity appoach is stick-slip behaviou of the liquid vapou inteface at the contact point. The change in the contact angle affects the balance of foces by alteing the suface tension which acts though the contact angle. The sudden distubance in the competing foces, the buoyancy foce that pulls the bubble up and the contact line foce that keeps the bubble down, causes petubations in the liquid vapou inteface, which in tun affects the velocity of contact line. This cyclic pocess is at the oot of the stick-slip behaviou of the liquid vapou inteface Foce balance appoach In this study we implement a dynamic contact angle based on the balance between suface tension foces and buoyancy foces. It is a hypothesis that has not yet been justified in liteatue. In this model as well, the pescibed values of the appaent eceding and the appaent advancing contact angles 65

66 66 define the limits of the instantaneous appaent contact angle. The tansition between eceding contact angle towads the advancing contact angle takes place based on the balance of buoyancy foces and suface tension foces. The undelying idea in this model is that; The advancing and eceding contact angles ae detemined by the chaacteistics of the inteaction between liquid/vapou inteface and the suface, and can be detemined expeimentally. The sign of the contact line velocity is a weak indicato of the advancing/eceding contact angle, amanujapu et al. (1999). θ CA The switch fom CA towads ACA can be modelled based on the atio of the buoyancy foce to the contact line foce. θ = θ ACA CA if if F F g g F F σ σ (3.39) The instantaneous shift fom eceding contact angle to advancing contact angle causes numeical petubations to avoid which one can model it as a smooth tansition. Thus in the foce balance appoach the liquid-vapo inteface has the eceding contact angle fo a buoyancy foce less than o equal to the minimum contact foce, i.e. contact foce in effect at eceding contact angle. While the inteface foms advancing contact angle with the suface when the buoyancy foce is geate than o equal to the maximum contact foce, i.e. contact foce in effect at advancing contact angle, with intemediate values of θ CA fo buoyancy foce in between the minimum and maximum values. θ CA θ = θ θ ACA tansition CA if if if F F σ,max F g F > F σ,min σ,max g > F F g σ,min (3.40) Instead of defining a smooth contact angle based on conditions on the foce balance we can model the contact angle as a hypebolic tangent function which will be valid though out the simulation. This is shown as follows; 66

67 67 θ CA θ = + θ F + tanh Fσ F θ θ 2 ACA CA g σ, min ACA CA 2,max Fσ, min (3.41) Whee, F g is a function of the equivalent bubble diamete and the gavitational constant, g. While F σ,min and F σ,max ae functions of the bubble base diamete and the known values of the minimum eceding contact angle and maximum advancing contact angle espectively Implementation of dynamic contact angle A complication in simulating bubble gowth is that the Navie Stokes equations in combination with the no-slip bounday condition, pedicts that an infinite foce is equied to move a contact line. Lauga (2005) pesent a thoough eview of expeimental, numeical and theoetical investigations on the subject of the fallibility of the no-slip bounday condition and existence of the slip. Slip is said to occu when the value of the tangential component of the velocity appeas to be diffeent fom that of the solid suface immediately in contact with it. Physically, thee is a diffeence between thee diffeent types of slip: 1. Micoscopic slip at the scale of individual molecules, using hydodynamics to foce liquid molecules to slip against solid molecules by involving lage foces. 2. Actual continuum slip at a liquid-solid bounday (i.e., beyond a few molecula layes) and 3. Appaent (and effective) slip due to the motion ove complex and heteogeneous boundaies. The phenomenon of slip has been encounteed in thee diffeent contexts. Gass flow in devices with dimensions of the mean fee path of gas molecules, flow of non-newtonian fluids such as polymes and contact line motion. In the context of Newtonian liquids, solving the equations of motion with a no-slip bounday condition in the neighbohood of a moving contact line leads to the conclusion that the viscous stesses and the ate of enegy dissipation have non-integable singulaities. When the singulaity was explicitly fomalized (Dussan and Davis 1974), it became clea that such models ae physically unacceptable. To undestand this it can be theoized that at the micoscopic scale, suface oughness of the solid bounday poduces the macoscopic slip condition which shows that at the contact line a Newtonian fluid must exhibit local slip in ode to move, Davis (2003). The idea of using local slip bounday condition to emove the singulaity was fist suggested by Huh and Sciven 67

68 68 (1971). Moe ecently the slip bounday condition has been used by Noman and Miksis (2005) to model a gas bubble with a moving contact line ising in an inclined channel using the level set method and by Chen et al. (2009) to simulate bubble fomation on oifice plates with moving contact lines using the level set method. While Afkhami et al. (2009) poposed a mesh dependent model of dynamic contact angle using local slip conditions fo volume of fluid simulations. Consequently the flow nea the contact line should be modelled to allow slip locally on the fluid-solid inteface. This should not be consideed as a molecula slip but as an effective slip that pemits the desciption of mathematical models which give easonable pedictions fo macoscopic flow quantities. It is a device that makes the macoscopic dynamics well defined though it is also an admission that the macoscopic physics nea the contact line is not well undestood no well descibed. A well posed poblem at the moving contact line thus equies two essential ingedients: 1. A slip law that elieves the foce singulaity. 2. A condition on the contact angle, θ CA. The simplest slip bounday condition is the so-called fee slip condition, an ad-hoc elationship valid fo the entie wall; n wall u = 0 (3.42) This does not set the tangential velocity component to zeo. To estict the tangential velocity we tun to the Navie slip bounday condition (Stokes 1845) which says that thee is a degee of slip at a solid bounday that depends on the magnitude of the tangential stess. Thus at a solid wall the component u t of the fluid velocity in the diection of the unit tangent 't' is popotional to the ate of stain in that diection, the constant of popotionality (λ) being the slip length, shown in Fig 3.5. u t λ = T t µ and (3.43) 68

69 69 Whee T is the stess vecto. Theefoe a zeo slip length coesponds to zeo tangential velocity (no-slip condition), while fee slip is obtained as λ tends towads infinity. In ou simulations, the slip length has been set equal to the mesh element size. Thus the extapolated tangential velocity component is 0 at the distance λ (equal to one mesh element) fom the wall, as indicated in the figue below. u θ CA λ Figue 3.5: Navie-Slip bounday condition Now, a contact angle of θ CA has to be applied. This bounday condition can be expessed by the dot poduct between the level set nomal vecto at the inteface and the wall nomal vecto; n wall nγ = cos ( θ ) CA The contact angle and slip length constaints ae implemented in weak fom as follows. (3.44) σ ( n cos( θ ) n ) δ u u~ CA Γ µ λ (3.45) Whee u ~ is the test function. 3.4 Stability and nondimensionalization Each intefacial flow poblem is chaacteized by some set of non-dimensional paametes (citeia) as well as by some specific time scale. That time scale is extemely impotant, as it eflects the dynamics of the consideed physical pocess; it also seves as a guideline fo selecting a time-step 69

70 70 size in numeical simulations. Although the physical time scale is always dependent on the poblem at hand, thee ae some common scales connected with typical physical phenomena like gavity, suface tension, viscosity. Let l be the length scale of the consideed poblem. The gavity phenomenon is well epesented by the dimensionless Foude numbe F = u gl, whee u is the chaacteistic velocity scale and g is the magnitude of the gavitational acceleation. If we denote by t the typical time scale of a gavity-diven physical pocess, then u = l /t. A chaacteistic time-scale should be such that all physical effects foming coesponding non-dimensional citeion countebalance each othe; thus, fom l F = 1, t gl (3.46) We can deduce l t g, (3.47) The suface tension phenomenon in viscous flow is chaacteized by the capillay numbe, Ca = ( u µ ) σ whee µ is the dynamic viscosity and σ is the coefficient of suface tension. If we denote by ' t the typical time scale fo a capillaity-diven physical pocess, then fom stability citeion we have; u = l t. And ' / lµ Ca = 1, ' t σ (3.48) Fom whee we deduce t ' l µ, σ (3.49) To captue the dynamics of the modelled physical pocess, the numeical method should use a timestep not exceeding the time-scales of all involved physical phenomena. In bubble dynamics duing nucleate pool boiling gavity diven flow is moe significant than capillaity diven flow theefoe we will conside the chaacteistic time fom Eq

71 Chaacteistic scales In nucleate boiling the chaacteistic length is the size of the bubble when it sepaates fom the suface. To find this dimension, we balance the bubble suface tension to the bubble buoyancy foce, as shown in Fig-xx; theefoe [Notes Heat tansfe with phase change (2006) pg 5/36]The length scale is of the ode of bubble diamete. Fom which we can obtain the chaacteistic velocity using the chaacteistic time descibed peviously. Length scale l σ = = m g ( ρ ρ ) l v Chaacteistic time l t = g = s Chaacteistic velocity u = l g = m/s Chaacteistic tempeatue diffeence T = T w T sat Nondimensionalization The govening equations ae made dimensionless by using the chaacteistic scales and subsequent escaling of the involved vaiables. Dimensionless numbes help to identify which physical effects ae dominating, and also assist when classifying diffeent model poblems. The non dimensional quantities used to make the equations dimensionless ae the eynolds, Foude, Webe and Peclet numbes. The govening equations, (Eqs ) ae nondimensionalized using the following definitions fo vaious dimensionless quantities. t u t = = t t l x x =, y = l y l u u =, v = u v v ρ ρ =, µ = ρ µ µ p p = ρu T T T = T sat λ p λ =, c p = κ κ λ c = l p c Q = l Q u ρ 3 Whee the chaacteistic physical popeties like density, viscosity and themal conductivity etc ae taken to be those of the vapo phase. Putting these definitions in Eq 3.27 we get: 71

72 72 72 ( ) ( ) ( ) = + T sat T I l u u l u u l g T T l n l u p l u u u u u l u t u µ µ β ρ ρ δ σκ ρ ρ ρ (3.50) Multiplying both sides by l /(u ² ρ ) we get: ( ) ( ) ( ) T I sat T u u l u n l u T T u g l p u u t u = + ρ µ µ δ κ ρ σ β ρ ρ (3.51) O, ( ) ( ) ( ) T I sat T u u n We T T F p u u t u 2 e = + µ δ κ β ρ ρ (3.52) Similaly the enegy equation is ewitten as: p p l u Q l T T l l T T u u t T u l T C C 3 1 ρ λ λ ρ ρ + = + (3.53) p p l u Q T l T T u t T C u l T C 3 2 ρ λ λ ρ ρ + = + (3.54) Multiplying both sides by l /(u Cp ρ T) we get:. 2 Q T C u T l u C T u t T C p p p + = + λ ρ λ ρ (3.55) 1 Q Ec T Pe T u t T C p + = + λ ρ (3.56) The non dimensional continuity equation is:

73 73 u = u evp ρ ρ (3.57) Whee, u evp u = u evp and λl T m& = δ h fg T l (3.58) The level-set equation is aleady nondimensional. φ + t [ ε ( φ n ) n φ ( 1 φ ) n ] ui φ = Γ Γ Γ (3.59) 3.5 Numeical implementation This section descibes the numeical implementation of the physical equations using COMSOL Multiphysics (fomely FEMLAB), a finite element analysis and solve utility fo vaious physics and engineeing applications, especially coupled phenomena (multiphysics). Fist the model is set up with the necessay appoximations. Then the equations ae ewitten in ode to implement them in the coect fom and then initial and bounday conditions ae descibed. In the finite element method it is possible to fomulate the equations and bounday conditions in stong o weak fom. The stong fom is what one would think of as the nomal way to wite the equations. The weak fom is a special integal fomulation of the equations that in some cases ae moe suited fo numeical simulations. It povides the use with moe possibilities to contol and enfoce given constaints. Nevetheless, the weak fom is not as accuate as the stong fom. Both foms ae used to solve the pesent poblem Defining the computational domain A vapou bubble gowing on a heated suface duing nucleate boiling in the absence of flow can be assumed to exhibit axially symmetic dynamics with efeence to the vetical axis. As a consequence we can educe the poblem to a plane intesecting the bubble at its cente. Futhemoe, one has to calculate the solution in half of the plane cutting though the cente of the bubble and the full solution fo the plane can be obtained by mioing the symmety plane while the full spatial solution can be obtained by evolving the solution obtained fo half of the intesecting plane ove 360 aound the 73

74 74 axis of symmety. In this way the poblem is educed to two independent vaiables (adial) and z (vetical) coodinates. The computational domain is shown in figue 3.6. Figue 3.6: Computatoinal domain The above geomety can be dawn using the gaphical use inteface Implementation of the govening equations In two dimensions the poblem is solved fo five fields the level set function, the adial and vetical components of velocity, the pessue and the tempeatue govened by Navie stokes; level set and enegy consevation equations descibed in the pevious section. In Comsol the govening equations ae epesented in the following fom; U FU, in Ω i t i d U + Γ i U = i (3.60) Whee the coefficient d Ui and the vaiable vecto U i ae as follows, 74

75 75 75 = p U C d i ρ ρ ρ 1 0 and = T p v u U φ, (3.61) The othe paametes i.e. the flux tenso Γ Ui is; ( ) ( ) + + = Γ z T Pe T Pe z n n z v p z u v v z u u p z U i φ φ φ φ φ φ λ λ φ ε φ φ φ ε φ φ µ µ µ µ e 2 e e e 2 (3.62) The genealized souce tem F Ui is; = Q T v C T u C z n u n u z u z v u z u v v u F T u z v v u u p F l p l p z evp evp evp T U i ρ ρ φ φ ρ ρ ρ ρ β ρ µ ρ φ 2 e 2 (3.63) In the Comsol scipting language the above fomulation is given as: equ.dim = {'u','v','p','phi','t'}; fem.shape = {'shlag(2,''u'')','shlag(2,''v'')','shlag(1,''p'')','shlag(2,''phi'')', 'shlag(2,''t'')'};

76 76 equ.da = {{'ho_ns';'ho_ns';0;'dts_phi_ls';'dts_ccho_ccc_cc'}}; equ.ga = {{{'(-2eta_nsu+p)';'-eta_ns(uz+v)'};{'- eta_ns(v+uz)'; '(-2eta_nsvz+p)'};{0;0};{'gamma(phi(1-phi)nojac(nom)-eephi)'; 'gamma(phi(1-phi)nojac(nomz)-eephiz)'};{'-k_t_cct';'- k_t_cctz'}}}; equ.f = {{'-ho_ns(uu+vuz)-2eta_nsu/+p';'(f_z_nsho_ns(uv+vvz))'; '-(divu_ns-usouce)';'(-u_phi_lsphi-v_phi_lsphiz)';'(- ho_ccc_ccu_cct-ho_ccc_ccv_cctz+q_cc)'}}; The matix Γ is enteed in equ.ga and F in the stuctue equ.f. The field equ.shape defines the shape and ode of the basis functions used fo u, v, p, Φ and T espectively. Finally, the dependent vaiables ae definned in equ.dim. The foce density fom the suface tension tem (Eq 3.64) in the Navie Stokes equation is included by taking advantage of the weak fomulation 1 (3.64) κδn Γ We Equation 3.63 is multiplied with a test/basis function and then integated ove the domain. This integal can be split into an integal ove the domain and one ove the bounday. It is the integal ove the domain that defines the weak contibution. In Comsol scipt language this weak contibution is added as; equ.weak = {{'-sigma(-test(u)(-1+nom^2)- test(uz)nomznom+test(u)/)delta'; '-sigma(-test(v)nomnomz-test(vz)(-1+nomz^2))delta';'- test(divu_ns)tauc_nsho_nsdivu_ns'}}; Hee sigma is equal to the invese of the Webe numbe and DELTA has a positive non zeo value only on the inteface such that the integal of DELTA acoss the inteface is always equal to one. 76

77 Adding the bounday conditions The bounday conditions coesponding to the geneal fom of Eq. (3.60) ae; U i n b ΓU = G i U µ i U i T Ui, on Ω (3.65) u i = 0, on Ω (3.66) Hee n b is the nomal vecto on the domain bounday. The µ Ui s ae Lagange multiplies. These Lagange multiplies ae only elevant when the Ui s ae diffeent fom zeo. Moe geneally speaking one can ceate a Lagange multiplie fo each constaint the poblem is subjected to. The Ui could fo example be equal to one of the velocity components Ui = v x on a specific bounday segment. This means that the x-component of the velocity must be zeo on that domain bounday pat. The Lagange multiplie coefficients µ Ui s ae calculated by Comsol such that the constaint in Eq. (3.67) is fulfilled. Equation (3.66) is the Neumann condition and Eq. (3.67) is the Diichlet condition o constaint. The bounday constaints ae given by specifying the Ui in Eq. (3.64) fo each bounday segment. In the cuent set-up thee ae fou boundaies denoted Ω k fo k = 1-4, see Fig One may wite; =, fo i = 1 5 and k = 1 4 Ωk Ui (3.67) Which gives a total of 20 vaiables. One Lagange multiplie is intoduced fo Ω 2 in ode to enfoce the contact angle condition on this bounday. This esults in a weak equation on this bounday. The fist bounday, Ω 1, is the symmety bounday. The condition fo symmety is that the component of the velocity nomal to the bounday, u is zeo and the gadient of tempeatue nomal to the wall is zeo. Fo the second bounday, Ω 2, the component of the velocity nomal to the wall, u has to be zeo. The no-slip condition is elaxed such that the component of the velocity paallel to the wall is fee. A n n = cos θ. The weak fom of contact angle of θ CA has to be applied accoding to Eq ( ) which is given in Eq No conditions ae applied to the pessue and the level set function while the tempeatue on the second bounday is fixed at T wall. Bounday thee, Ω 3, has open bounday and a constant tempeatue of T sat. The last bounday, Ω 4, has a slip bounday condition and tempeatue gadient nomal to the wall is zeo. 77 wall Γ CA

78 78 The tempeatue constaints ae added using stong fom while emaining constaints ae implemented using the weak fomulation. In Comsol scipting language the bounday conditions ae simply given as: bnd.ind = [1,2,3,4]; bnd. = {0,{0;0;0;0;'-T+T0_cc'},{0;0;0;0;'-T+T0_cc'},0}; bnd.const = {'-u','-un_ns-vnz_ns',0,'-un_ns-vnz_ns'}; bnd.constf = {'test(-u)','test(-un_ns-vnz_ns)',0,'test(-un_nsvnz_ns)'}; bnd.weak = {0,{'test(u)(sigmaDELTA(n_ns-cos(theta)nom)- ueta_ns/h)'; 'test(v)(sigmadelta(nz_ns-cos(theta)nomz)- veta_ns/h)'},0,0}; Hee bnd. defines the Ui. T0_cc is equal to T wall fo the second bounday and T0_cc is equal to T sat fo the thid bounday. These ae the only two boundaies that have stong bounday constaints. The weak constaints ae defined in bnd.const and the constaint foce is defined in bnd.constf. The contact angle is implemented though a weak foce in bnd.weak, sigma is equal to the invese of the Webe numbe and DELTA has a positive non zeo value only on the inteface such that the integal of DELTA acoss the inteface is always equal to one. 3.6 Validation To check if the implementation descibed above is coect we will use it on some simple configuations in ode to benchmak the code. Fo simulation of bubble dynamics the impotant paametes ae the suface foces on the liquid vapou inteface and the computation of the flow aound the bubble. We will use the deived esult fo the pessue in a vapou bubble to check the eliability of the code using analytical esults to validate the computation of the bubble diamete Static bubble in zeo gavity Ou fist test case is a cicula bubble in static equilibium. The net suface foce should be zeo, since at each point on the bubble suface the tension foce is counteacted by an equal and opposite foce at 78

79 79 a diametically opposed point. A zeo velocity field was obseved and a pessue field that ises fom a constant value of p out outside the bubble to a constant value of P in = P out + σ/ B inside the bubble was noted which is in accodance to the Laplace-Young law Oscillating bubble in zeo gavity A mass of fluid in a zeo gavity field and a suface tension geate than zeo, whose shape deviates fom cicula/spheical should ecove the stable cicula/spheical shape. The final stage of the fluid mass should have no velocity field and the pessue dop at the inteface should satisfy the Young- Laplace Law. p = σ in 2D, and p = σ 2 in 3D (3.68) Whee is the adius of cicula o spheical bubble in equilibium condition. We consideed a two dimensional case using wate as fluid in zeo gavity conditions. This test case thus deals with the exteme conditions of 1000:1 density atio, 100:1 viscosity atio and a lage suface tension coefficient. The initial shape of the bubble is elliptical with axis atios of 3:2, 7:4 and 2:1 in a squae computational domain of 5x5mm with tiangula mesh elements. Initially the suface tension which depends on the inteface cuvatue is geate at the naow end of the bubble; this is shown by the vecto plot in Figue 3.7. These foces compel the bubble inteface to oscillate like an elastic band. Then afte some time the inteface comes to est in a cicula shape with a adially unifom distibution of suface tension foces. The pessue dop acoss the inteface at the end of the simulation fo the thee cases is given in the following table. Aspect atio Elliptical Aea adius of equivalent cicula bubble 79 Young-Laplace pessue dop Actual pessue dop % Eo 2: x : x : x Table 3.1: Young-Laplace pessue dop acoss liquid vapo inteface. The pecentage eo in the last column is the diffeence between the Young-Laplace pessue dop and the calculated pessue dop as faction of the Young-Laplace pessue dop. The oveshoot in calculation can be explained due to the mino loss in vapo phase which is chaacteistic of the level set method caused by paasitic cuents.

80 80 t = 0.00 t = 0.01 t = 0.02 t = 0.03 t = 0.04 t = 0.05 Figue 3.7: elaxation of an elliptical vapo bubble of aspect atio of 2:1 at vaious times with a vecto plot of suface tension foces 80

81 81 t = 0.00 t = 0.01 t = 0.02 t = 0.03 t = 0.04 t = 0.05 Figue 3.8: elaxation of an elliptical vapo bubble of aspect atio of 7:4 at vaious times with a vecto plot of suface tension foces 81

82 82 t = 0.00 t = 0.01 t = 0.02 t = 0.03 t = 0.04 t = 0.05 Figue 3.9: elaxation of an elliptical vapo bubble of aspect atio of 3:2 at vaious times with a vecto plot of suface tension foces 82

83 83 Next we will examine the phase loss duing the simulations and evaluate the phase consevative popeties of the level set method duing the dynamic pocess of bubble oscillation. Figue 3.10 shows the consevative popeties of ou model fo the case of elliptical bubble with aspect atio of 3:2. It is the diffeence of the initial vapo aea and the integal of vapo phase ove the computational domain as a faction of the oiginal aea of vapo phase, plotted against time fo the consevative and non consevative level set method. Figue 3.10: elative eo in aea of vapo phase. The non consevative level set method shows significant phase loss which is continuously inceasing due to the paasitic natue in which numeical discepancies popagate Shin et al. (2005). The consevative level set model conseves 99% of the initial phase. 83

84 84 Figue 3.11: elative eo in aea of vapo phase fo the thee aspect atios Note that most of the phase loss occus at the initial stages of the computation aftewads the phase is almost entiely conseved though most of the dynamic pocess with only slight vaiations. This is moe evident when we compae the aeas of vapo egion fo the thee cases, as follows: 84

85 85 Figue 3.12: Aea of the vapo phase fo the thee aspect atios Fo the aspect ation of 2:1 the inteface is vey dynamic though out most of the simulation (Fig 3.7) whee as the inteface is less dynamic fo the case of an aspect atio of 3:2 as shown in Fig 3.9. Thus it may be said that the stong topological changes duing the dynamics of vapou bubbles do not pose a theat to the phase consevative popeties of the model. 85

86 86 Figue 3.13: elative eo in aea of vapo phase fo vaious mesh densities. Figue 3.12 shows the mesh dependency of phase consevation. We obseve that fo about a 100% incease in mesh density the elative eo deceases by about 1% Coalescence of vapo bubbles In this section the consevative level set method has been applied to captue the coalescence of two vapo bubbles in a liquid column. This test pemits the study of the evolution of the inteface afte the meging of the two bubbles allowing the assessment of the model in captuing the liquid vapou inteface duing exteme topological changes demonstated by the uptue of the inteface duing coalescence. Initialization: At the beginning of the simulation a 2-D axisymmetic domain is consideed, with two equally sized bubbles placed at a distance of 1.5 times thei diamete (Fig 3.14). The bubbles ae suounded by a liquid with the density atio between them equal to 1:1000 the viscosity atio is 1:10. 86

87 87 Figue 3.14: Initial state of the domain. The level set function is initialized ove a couse of time, independent of the Navie Stokes, until the phase is unifomly conseved fom one time step to the next and so on. In section we noted that most of the phase loss occus at the initial stages of the computation and aftewads the phase is almost entiely conseved. In this section we analyze the phase loss sepaately between the initialization of the model and the actual simulation of the dynamic poblem. Figue 3.14 plots the non dimensional volume of vapo phase against the non dimensional time. We obseve that some phase loss occus duing the initialization phase. Howeve once the compessive and diffusive tems in the level set fomulation ae in equilibium the phase loss stops and the volume of the vapo phase emains constant ove a elatively longe peiod of time. Thus if the solution obtained at the end of the initialization phase, when the compessive and diffusive tems ae balance, is used as the initial solution to the dynamic poblem one can avoid the initial phase loss. 87

88 88 Figue 3.15: Phase loss duing initialization of the level set function. Coalescence: The ise of the two bubbles in a liquid column is captued in figue The velocity vecto field is also shown and the Φ = 0.5 isocounte plots the motion of liquid-vapo inteface that ultimately esults in the coalescence of the two bubbles. (a) (b) (c) (d) Figue 3.16: Coalesence of two bubbles. 88

89 89 The gavity diven motion of the two bubbles is captued at diffeent time steps. Although the two bubbles ae of same density and unifom size the lowe bubble ises faste than the uppe one because the lowe bubble becomes entapped in the wake egion of the uppe one and expeiences its velocity field theeby educing the effective velocity of the uppe bubble. In this manne the two feely bubbles suspended in a liquid column eventually coalesce and the subsequent coalesced bubble ises again. In this way the level set method eadily captues the vibant topological changes associated with coalescence. The consevative popeties of the dynamic simulation of this pocess ae shown in figue Figue 3.17: Phase loss duing coalescence. We note that the thee ae mino fluctuations in the vapo phase at the time when the bubbles coalesce this can be explained by the stetching and shinking of the inteface duing the coalescence pocess. Once the dynamic pocess is complete the volume of the vapo phase stabilizes to a unifom value and we see that the vapo phase is conseved. 3.7 Conclusion A consevative appoach to the level set method that ules out the necessity of einitialization, was numeically implemented to study moving intefaces. The Level Set and Navie Stokes model was tested fo accuacy in the epesentation of liquid vapou inteface, suface tension foces and phase consevation. The model was found to be obust and accuate in calculating suface tension foces acoss a vapo-liquid inteface. The inteface model was also tested to see if it could suppot 89

90 90 flexibility of the inteface and it was found that the model is capable of handling topological changes involving the stetching, shinking, uptue and ejoining duing coalescence. The density and viscosity atios fo the tests wee compaable to that of wate and wate vapou and the tests wee caied out in compaison with analytical solutions. We can theefoe conclude that the pesent model is suitable fo simulation of single bubble dynamics duing nucleate pool boiling. 90

91 91 Chapte 4 Decoupled bubble dynamics The ate of bubble gowth and the subsequent bubble motion has a temendous influence on heat tansfe. The study of bubble dynamics is a coupled poblem. The ate of evapoation contols the inteface speed. One appoach to study bubble dynamics is to decouple the poblem fom enegy consevation equation and use an input value of ate of evapoation. The objective is to obseve how iegula evapoation ate contols bubble dynamics and the shape of bubble. As an added advantage the computational cost of simulating boiling flow is educed by decoupling the themal tanspot fom the momentum and mass tanspot. 4.1 Gowing and depating bubble The consevative level set model descibed in the pevious section was applied to study the dynamics of a vapo bubble gowing on a hoizontal suface with vapo phase injection taking place at the inteface at the contact point. The bubble gowth and depatue pocess is essentially a balance of competing buoyancy foces and suface tension foces. It is theefoe possible to leave out the enegy tansfe, fo a late stage while concentating on dynamics of a gowing bubble. The vapo velocity at the inteface in Eq 3.30 is chosen based on the tempeatue gadient equied to initiate boiling, it has the following expession; u 25 evp ( t> 1) ( 0. z) Γ = δ The expession insues a linea pofile fo the inteface velocity (Fig 4.1), while the total evapoative vapo flux at the liquid vapou inteface inceases as a function of the bubble base diamete as well as local cuvatue by a facto of appoximately 2π. The influence of the local phenomena at the contact line on the global dynamics of gowing bubbles will be studied in section 4.2. (4.1) 91

92 92 u evp Figue 4.1: Pofile of the injected velocity. Accodingly the cicumfeence of the bubble which u evp pushes outwad begins to gow as the bubble volume inceases, consequently inceasing the ate of vapo phase injection and mimicking coupled bubble dynamics. Even though the ate of vapou phase injection inceases, the pofile of the inteface velocity is still govened by Eq Effect of contact angle on bubble gowth In ode to study the gowth and depatue of a single vapo bubble the model was solved fo a contact angles, θ CA = 50, θ CA = 55 and θ CA = 60 while keeping a consistent velocity of the inteface though Eq 4.1. It should be noted that while the velocity pofile of the ate of evapoation was maintained the same in all the cases, the ate of evapoation itself was not maintained the same. If thee been a condition govening the ate of evapoation fo all the cases then the bubble should gow at the same ate iespective of the contact angle. By allowing the evapoation ate to vay as a function of local dynamic conditions we wee able to study the effects of the contact angle on the bubble gowth. The evolution of the bubble base diamete fo the thee contact angles is shown in the following figue. It is evident that the bubble lift-off time inceases with the incease in contact angle. Obseving the gaph at easonable amount of time afte the initiation of bubble gowth, i.e. t=6, we note that the ate of expansion of the bubble base is geate fo lage contact angles. 92

93 93 Figue 4.2: Bubble base diamete fo contact angle, θ CA =50, θ CA = 55 and θ CA = 60. Howeve the evolution of the bubble volume fo the thee contact angles is quite simila, as shown in figue 4.3 which plots the equivalent bubble diamete. We note that the bubble diamete fo the thee cases follow a simila path fo most of the initial bubble gowth stage. It is only at the late stage of bubble gowth when the buoyancy foce is geate then the contact line foce (i.e. appoximately t = 10) that the bubble diamete fo the lowe contact angle falls below that of geate contact angle. What we can deduce fom these two figues is that fo a given volume of vapo bubble the bubble base diamete is lage fo lage contact angles because of the incease contact line foce that ties to pull the bubble outwad. As the contact line inceases with the incease in bubble volume (until the bubble eaches depatue conditions) theefoe the ate of expansion of the bubble base is geate, in popotion to contact line foce. In section 4.2 we will examine in moe detail the inteaction of local phenomena ove global bubble gowth dynamics. 93

94 94 Figue 4.3: Equivalent bubble diamete fo contact angle, θ CA =50, θ CA = 55 and θ CA = 60. Fig 4.4 shows the state of the vapou bubble afte vaious intevals of time fo a contact angle of 60 highlighting the initial and final stages of bubble gowth and depatue. In the initial stage of bubble gowth the contact point moves adially outwad. This hoizontal movement of the bubble base is slowed down by suface tension focing the bubble to comply with the geometic condition imposed by the contact angle. As the adial bubble gowth is esticted the bubble begins to push upwads. The pocess of adial expansion of the bubble base and upwad/outwad gowth of the bubble happens simultaneously and continuously. The volumetic expansion of the vapou bubble causes an incease in the buoyancy foce pulling the bubble upwad with espect to suface tension foce which ties to hold the bubble down. The buoyancy foces ae a function of the bubble volume and chaacteized by the bubble diamete whee as the suface tension foces ae chaacteized by the bubble base diamete. The shape of the bubble evolves, fom a small tuncated sphee attached to the suface at the vey beginning of ou calculations, to a tuncated ballo0n attached to the suface half way though the calculations, to a complete balloon shape just above the suface ight afte detachment and culminates into a mushoom shape a few millimetes away fom the suface at the end of the calculation. 94

95 95 Figue 4.4: Bubble gowth and depatue patten fo a contact angle, θ CA = 60 95

96 96 The non unifom bubble motion induces a clockwise votex in the liquid phase. When the liquid votex becomes stonge with the incease of bubble diamete, the bubble base begins to move inwads. The inwad motion of the contact point is also motivated by the incease in buoyancy foce that is pulling the bubble upwad. The depatue pocess occus vey apidly because of the dominance of buoyancy foce ove suface tension foce acceleates as the bubble base shinks, the bubble neck foms nea the wall (Fig 4.3) and soon afte that the bubble beaks off (Fig 4.3). The small vapo potion left afte the bubble has taken off seves as a nucleation point fo the next bubble as the cyclic pocess of bubble depatue duing ebullition commences Bubble diamete compaison The diametes of single bubbles and the bubble base wee computed continuously though out the gowth and depatue pocess fo vaying contact angles (Fig 4.5). Seveal theoetical elations have been poposed fo pedicting the size of bubbles at depatue fom a hoizontal suface, and some of these will now be consideed. Fitz (1935) was the fist to develop a citeion fo bubble depatue based on a balance of buoyancy and suface tension foces acting on a static bubble. Based on expeimental obsevations he poposed the following empiical expession fo the bubble depatue diamete: D d = 0208 σ 0. θca (4.2) g( ρl ρv ) Whee the contact angle θ CA is measued in degees. It has been obseved that pedictions of bubble diamete at depatue, made fom Equation (4.2) deviate fom the data epoted in liteatue, Son et al. (1999). Cole (1969) gives the following equation fo bubble depatue diamete as function of bubble ise velocity at depatue. D d 0.22 σ gσ = 0.04θ CA 4 (4.3) g( ρl ρv ) ub, d ( ρl ρv ) Figue 4.5 compaes the equivalent bubble depatue diamete computed by ou model with the ones pedicted by the Fitz and Cole. 96

97 97 Figue 4.4: Equivalent bubble depatue diamete calculated and computed fom empiical elations. Cole s expession was consideed because it ties to take into account the dynamic effects of bubble shape, howeve its esults deviate fom the computed values as well as values pedicted by Fitz at contact angle geate than 52. The bubble depatue obtained by the coelation of Fitz (1935) is only a faction less than the computed values which coesponds with othe esults found in liteatue. Seigel and Keshock (1964), fo example found though thei expeiments that the Fitz coelation undeestimated bubble depatue diamete by a small pecentage. We can theefoe conclude that the decoupled model is able to accuately pedict the diamete of depating vapo bubbles ove a boad ange of contact angles. Thus a decoupled model may be used to study the physics of bubble gowth dynamics. 4.2 Local conditions and bubble gowth The dynamic aspects of the dependence of bubble gowth mechanism on the local conditions nea the contact line can be studied using the decoupled bubble gowth model which computes dynamic gowth of bubbles while simulating liquid-vapo phase change at the inteface as a unifom phase injection at the contact point as a velocity tem. 97

98 Effects of pofile of the inteface velocity Hee, the idea is to vay the value of the ate of unifom injection ove the contact point egion by vaying the velocity pofile while ensuing that the oveall phase injection is the same. And in ode to etain the vapou phase injection at a constant value we modify Eq 4.1 as follows: u evp ( hu z) Γ = α δ, whee h u V& = (4.4) απ B, b Whee h u is the height of the velocity pofile, α is the atio between the base and the pependicula of the tiangula pofile of the inteface velocity. The desied volume gowth ate is denoted by V & and B,b is the bubble base adius. The volume gowth ate V & can be chosen to be a constant o a function of anothe vaiable. In ode to best mimic natual bubble gowth we have chosen V & to vay as a function of time. Thus, 8 t V & = t 3 (4.5) l The constant tem at the beginning of Eq 4.5 has units of m 3 /s and is the amount of volumetic incease that the bubble be subjected at t=1s. The chaacteistic time and cube of the chaacteistic length at the end of the equation ae thee to adimensionalize the ate of volumetic expansion of bubble. V & is designated as a function of the squae oot of time in ode to esemble the non-linea behavio of volumetic expansion of vapo bubbles obseved in liteatue. It should be noted hee that Eq 4.5 assumes continuous gowth of the vapo bubble and doesn t take into account the eventual detachment and depatue of bubble fom the suface. In ode to incopoate bubble detachment into this model, Eq 4.5 would have to have a tem that inveses the powe on time t at a given condition. Howeve this non-tivial execise is not necessay to study the effects local conditions on bubble gowth. Theefoe we etain Eq 4.5 in its cuent fom and highlight that the validity of this simulation is limited to the peiod of bubble gowth whee the bubble base is gowing outwads o until the buoyancy foce is geate than the suface tension foce. This limitation still gives us the oppotunity to examine vaiations in the bubble gowth mechanism caused by the changes in the pofile of the inteface velocity. In ode to do so, the liquid vapo 98

99 99 inteface nea the contact point was subjected to five diffeent velocity pofiles shown in table 4.1, having same ate of vapo mass incease. α = 3 u evp = 3 z δ Γ B, b α = 2 u evp = 2 z δ Γ B, b α = 1 u evp = z δ Γ B, b α = 1/2 u evp = z δ Γ 2 B, b α = 1/3 u evp = z δ Γ 3 B, b Table 4.1: Pofiles of the applied inteface velocity 99

100 100 While implementing the ate of evapoation, u evp fom table 4.1, the pependicula height of the velocity pofile h u fo vaious atios of pependicula to base α, fom Eq 4.4 ae plotted in figue 4.5. Whee as the effect of h u fom Eq 4.4 on the inteface velocity is visible in Figue 4.6 which shows the inteface velocity fo all α s at time = 3. Figue 4.5: Height of velocity pofiles, hu We can see fom figue 4.6 that the intended velocity pofile is obustly implemented. Thus the maximum inteface velocity fo the case α = 3, at time = 3 in Fig 4.6-(a) is 0.47 indicating that the height of the velocity pofile h u should be aound 1.5. Which coesponds well with the h u indicated in figue 4.6 fo α = 3, at time = 3. Similaly thee is good ageement between velocity pofiles at othe α s though out the simulation and we ae assued that the idea of vaying velocity pofiles has been coectly implemented. We also note that the maximum velocity at the base of the bubble nea the contact line fo α = 1/3 is 0.14 while fo α = 3 the maximum velocity at the contact line 0.474, showing in incease of 338%. This povides a easonable ange to study the effect of local chaacteistics on global dynamics of a gowing bubble. 100

101 101 (a) α = 3 (b) α = 2 (c) α = 1 (d) α = 1/2 (e) α = 1/3 Figue 4.6: Velocity pofiles, u evp fo α=3,2,1,0.5 and 0.33 at time = 3 101

102 102 Having scutinized the obust local application of the ate of evapoation, we examine the evolution of the non-dimensional volumic ate of vapo injection, V & vesus the non-dimensional time fo all cases of the vapo injection velocity pofile atio α. This is plotted in Figue 4.8. The ate of vapo injection is by design is the same fo all the cases and this is accuately concued by the obtained gaph. Figue 4.7: ate of vapo injection, V &. In the above figue while the vapo injection pe unit time inceases ove time, the ate of incease in vapo injection decays steadily. This is designed to imitate natual bubble gowth and should lead to a quasi-linea incease in the bubble volume ove time, shown in figue 4.9. The volume of the bubble at the end of the simulation can be estimated by calculating the aea unde the cuve fom figue 4.8 and adding it to the initial volume of bubble. Altenatively we can integate Eq 4.3 between the given time inteval and add it to the initial bubble volume. This yields; V = t t = dt (4.6) l 102

103 103 Thee is vey good ageement between this estimated value of the incease in vapo volume and the volume of vapo bubble fo the five cases plotted in figue 4.9. Figue 4.8: Non dimensional volume of vapo bubble The contact angle fo all the test cases was kept at θ CA = 52, indicating a bubble depatue diamete of m accoding to Eq 4.2. The equivalent bubble diametes fo the calculations ae shown in figue Thee is vey good ageement between the bubble diametes obtained fom calculations involving diffeent values of α. Futhe moe at bubble diamete at the end of the simulation is vey close to the bubble depatue diamete pedicted by the Fitz equation (Eq 4.2). Thus the simulation coves a lage pat of the bubble gowth pocess even though it was limited to the time when the bubble is gowing outwads o when the buoyancy foce is less than the contact line foce. So fa, it has been clea that the local velocity pofile at the contact line does not affect the global bubble dynamics, thus the bubble diamete continues to gow in a simila fashion despite ove 300% incease in maximum bubble base velocity at the contact line. 103

104 104 Figue 4.9: Equivalent bubble diamete fo vaious α s at θca = 52, We now examine a chaacteistic moe susceptible to local phenomena at bubble base, the bubble base diamete, plotted in figue The evolution of the bubble base diamete ove time fo all the cases follows a simila path, thee is slight vaiation in the exact values of the diamete that is spead out ove a vey small faction of a millimete. If we assume that the α = 1 case povides the mean value of the bubble base diamete ove the couse of time, then we see that the othe cases vay on both sides of the this mean path iespective of thei base to pependicula atio α. Thus while thee is vaiation in the bubble base diamete between the diffeent cases having diffeent values of maximum velocity at the contact point, we can not assume that the vaiation in the bubble base diamete fom the mean path is due entiely due to the vaiation in the maximum velocity at the contact point. Howeve, we also can not completely ule out the influence of maximum contact line velocity on the bubble base diamete. It would be easonable assume that the vaiation in the bubble base diamete fom the mean path is moe likely the esult of discetization eos than the vaiation of maximum inteface velocity nea the contact point. This hypothesis is stengthened by the minute natue of the vaiation in bubble base diamete that is of the ode of a small faction of the mean value compaed to the vey lage vaiation in the inteface velocity nea the contact point of the ode of above 300%. 104

105 105 Figue 4.10: Bubble base diamete fo vaious α s at θca = 52, In consequence we can conclude that the vaiation of the inteface velocity nea the contact point does not have a significant influence on the dynamics of a gowing bubble which ae moe dependent on the ate of evapoation Effects of the ate of evapoation Hee the ate of evapoation V & is vaied while the pofile of the inteface velocity is maintained at α = 1. The idea is to see if the only effect of vaying the ate of evapoation is to vay the speed of the dynamics of gowing bubble. The ate of evapoation is vaied by ewiting Eq 4.3 as; 8 t V& α = t 3 (4.7) l The change in the powe of t ceates allows us to vay the ate of vapo phase injection as a function of time. We have consideed two cases, one whee ά =0.5 as in Eq 4.3 and the second whee ά = 2/3. This will incease the ate of evapoation though the change will only be pominent at highe values of t. The effect of this change is shown in the following figue which plots the inteface velocity at t =

106 106 (a) ά = 1/2 (b) ά = 2/3 Figue 4.11: Velocity pofiles, u evp fo ά = 1/2 and 2/3 at time = 3.5 We note that the maximum inteface velocity at t = 3.5 fo the fist case is 0.280; whee as it was at t=3.5 in the pevious section (Fig 4.7-c). This is explained by the expansion of the bubble base. A lage bubble base diamete povides the liquid vapo inteface with a geate cicumfeence ove which evapoative mass flux passes. So, in ode to have the same ate of evapoation the height of the velocity pofile is diminished. Similaly in case-ii whee ά = 2/3 and the ate of evapoation is geate than the evapoative ate in case-i whee ά = ½, we see that the maximum inteface velocity is 0.26 i.e. lesse than case-ii. Again this is because the bubble base has expanded moe quickly due to the inceased ate of evapoation. And in ode to povide the same evapoative mass flux as dictated by Eq 4.5, the height of the inteface velocity is lesse. To claify this we next plot the bubble base diametes fo the two cases. Fom the following figue (4.13) it is clea that at the instant t = 3.5 the bubble base diamete of case- II is geate than that of case-i and hence the diffeence in the maximum inteface velocity. Consideing the evolution of the bubble base diamete ove time, we obseve that it follows the same patten fo the two cases. At times the ά = 2/3 cuve leads, how eve the diffeence is little. To eassue the coect implementation and esolution of the incease ate of evapoation we next plot the volume of the vapou bubble fo the two cases. 106

107 107 Figue 4.12: Bubble base diamete fo ά = 1/2 and ά = 2/3 at θca = 52 Figue 4.13: Volume of vapou bubble fo ά = 1/2 and ά = 2/3 at θ CA =

108 108 Scutinizing the evolution of bubble volume ove the couse of time fo the two cases we note that the incease evapoation ate becomes evident at t > 3. Pio to that the two cases follow an almost simila path of bubble gowth. Once t is lage enough the inceased ate of evapoation manifests itself and the bubble begins to gow faste fo case-ii. Oveall, the only visible effect of the inceased evapoation ate at the bubble base nea the contact point is the quickening of the bubble gowth pocess. 4.3 Conclusions In this chapte we pesented the idea of studying bubble dynamics independent of heat tansfe. The concept was validated by compaing numeical solutions to existing widely accepted empiical solutions fo the equivalent bubble depatue diamete ove a wide ange of appaent contact angles. We then focused on the effect of local phenomena on the global dynamics of bubble gowth assuming a constant angle. The maximum inteface velocity at the bubble base, u evp nea the contact point was vaied by a atio of ove 300% and its effects wee obseved while keeping the oveall ate of evapoation constant pe unit of time fo vaying u evp. It was noted that the local conditions have vey little effect on the gowth dynamics of a vapo bubble. We also elaboated the effect of the ate of evapoation in quickening the bubble gowth pocess and its lack of influence ove othe aspects of the dynamics of gowing bubbles. 108

109 109 Chapte 5 Coupled bubble dynamics While the decoupled bubble gowth model povided useful insight into the dynamics of evolving bubbles the complete numeical modelling of the single bubble gowth duing nucleate pool boiling equies the coupling of the themal tanspot with the momentum and mass tanspot. In this chapte the intefacial mass tansfe elated with evapoation of liquid at the liquid vapo inteface is implemented as a function of the tempeatue gadient acoss the inteface. Fom which the inteface velocity is calculated and used in the level set equation to displace the inteface. In the late potion of this chapte the gowing bubble is endowed with a dynamic contact angle, an impotant chaacteistic of bubble dynamics, which depends on the suface and liquid popeties and influences the oveall heat tansfe. Numeical modelling of the dynamic contact angle allows the evaluation of these effects. Anothe contibuto to the dynamics of gowing and depating bubbles is the evapoation in the micolaye. Expeimental obsevations of the contibution of the micolaye to the oveall liquid vapou evapoation have placed its influence as low as 20% to as Plesset and Pospeetti (1976), Son et al. (1999). Howeve as shown in the pevious chapte the ate of evapoation nea the contact point at the bubble base has vey little influence on the global dynamics of an evapoating bubble while affecting only the ate at which these dynamics take place. Theefoe keeping in mind the lack of eliable expeimental data egading the natue and contibution of the micolaye coupled with the not-too significant influence of local phenomena on global dynamics as elaboated in the pevious chapte and fo the sake of educing the complexity and computational cost of the model we have not taken into account the evapoation in the micolaye. Instead the focus of this wok is a obust coupling of the themal tanspot with the momentum and mass tanspot, taking into account the effects of dynamic contact angle on bubble dynamics with the undestanding that the tansient conduction due to liquid motion is the majo contibuto to wall heat tansfe duing nucleate pool boiling. 5.1 Bubble gowing on heated suface The model of dynamic bubble gowth is extended to include heat tansfe. The heat exchange mechanisms that must be consideed include heat tansfe fom liquid to vapo in the fom of latent 109

110 110 heat duing evapoation as well as the ole of the liquid vapo inteface motion in agitating the fluid and causing convection. All these phenomena that goven the heat tansfe duing nucleate boiling will be studied in the context of single bubble gowth on a heated suface fo which simulations will be caied out. In compaison with the decoupled model exploited in the pevious chapte, in this case the vapo phase tansfomation is a function of the tempeatue gadient acoss the inteface as descibed in Section 3.2. The heated suface is maintained at a tempeatue of T w = T sat + T. The liquid and vapo ae initially at a tempeatue T sat. Because of heat loss duing evapoation by the liquid phase in ode to tansfom into vapo, the vapo phase is assumed to emain at T sat though out the calculation; this is done by activating Ec Q (Eq3.57) as a sink in the vapo phase that bings its tempeatue down to T sat. Also a themal bounday laye is imposed at the beginning of the simulation, the height of which is established though an initialization pocedue Evolution of the tempeatue field In this section we study the evolution of the tempeatue field and its influence of bubble dynamics, duing the bubble gowth and depatue cycle fo a contact angle of 54.5 and T = 7K. Figue 5.1 shows the initial state of the computational domain. Figue 5.1: Tempeatue field (in K) at t = 0s. Once the simulation stats the tempeatue gadient acoss the liquid vapo inteface causes phase change. As a esult the inteface advances outwads. This pushes the liquid back and ceates 110

111 111 convective foces which futhe help the heat tansfe pocess. Figue 5.2 shows the state of the system a few instants into the simulation at eal time = 0.01 seconds. Figue 5.2: Tempeatue field (in K) at t = 0.01s. We can see that the convective foces caused by the agitation of the vapo bubble ae m ixing the supeheated liquid laye with the bulk liquid suounding the bubble. This causes inceased evapoation acoss as a lage aea of the liquid vapo inteface is now exposed to a highe tempeatue gadient. Thus the bubble begins to gow in a dynamic manne much the same way as decoupled bubble gowth was descibed in section 4.2. Figue 5.3: Tempeatue field (in K) at t = 0.11s. Figue 5.3 shows the bubble at an intemediate gowth stage when the bubble base diamete is nea its maximum value. Soon afte this point, the liquid vapo inteface begins to advance to wads the vapo 111

112 112 phase and away fom the liquid. This is due to the ise of buoyancy foces that ae pulling the bubble upwads in compaison with the contact line foces that ae keeping the bubble down. Figue 5.4: Tempeatue field (in K) at t = 0.14s. Once the lift foces ae sufficiently lage the bubble depats fom the heated suface. The cowding of the isothems nea the contact point at the bubble base shows that the heat flux in that egion is vey high (Figue 5.4). As the bubble begins to detach, these isothems fom a naow themal laye of supe heated liquid unde the bubble. Figue 5.5: Tempeatue field (in K) at t = 0.16s. The mixing effect is moe ponounced when the bubble depats and the subsequent wake effects beak the themal bounday laye futhe distibuting heat (Figue 5.4), the wake effects caused by 112

113 113 viscous foces also compel the bubble to adopt a mushoom like shape athe than a spheical one as visible in Fig 5.5. Afte the bubble ises away and colde fluid fom the suounding aea fills the place vacated by the bubble, the themal laye thickens and the isothems elax ove a geate height Pogession of the flow field The nomalized flow field at the initial bubble gowth stage, in and aound a vapo bubble gowing on a heated suface is shown in Figue 5.6. Duing the ealy peiod of bubble gowth, the liquid aound the bubble is pushed away (Fig 5.6-a). A ciculatoy flow patten inside the bubble as well as in the suounding liquid is obseved (Fig 5.6-b) fo the feely gowing bubble. (a) (b) Figue 5.6: Nomalized flow field at t = 0.0s, 0.01s and 0.11s. The vapo velocity vectos in the bubble ae eflective of the bulk movement of the bubble in upwad diection and the changes in the bubble shape as the bubble ises in the pool. Thee is no inflow o outflow of vapo at the vapo-liquid inteface. Figue 5.7 plots the nomalized flow fields fom the intemediate bubble gowth stage to depatue. At this stage the flow cuents have been established in and aound the bubble (Fig 5.7-a). The flow field just befoe bubble detachment (Fig 5.7-b) shows that the fluid is being pushed outwads on the uppe potion of the bubble, wheeas the liquid flow is adially inwad in the lowe potion of the bubble. 113

114 114 (a) (b) (c) Figue 5.7: Nomalized flow field at t = 0.0s, 0.01s and 0.11s. As the base of the bubble shinks, a clock-wise liquid ciculation is geneated that daws in coole, satuated liquid fom the suoundings towads the heate suface. This ciculation is popotional to the contact angle. Geate the contact angle, lage will be the velocity vectos and they will involve lage amounts of liquid. Thus the incease in bubble depatue diamete with lage contact angles is also due to the inceased heat flux which causes geate vapo volume gowth ate. In the pevious chapte it was also obseved that fo unifom evapoation ate bubble depatue is essentially a function of the buoyancy and suface tension foces, hee we note that in the coupled model the vapo volume gowth ate depends on the wall heat flux that supplies the latent heat fo phase change. One is theefoe able to find simila bubble dynamics by adjusting the wall heat flux to obtain an aveage evapoation ate that gives a vapo mass flux compaable to mass flux in the decoupled model of the pevious section Heat tansfe The total heat flux duing the gowth and depatue of a single bubble duing nucleate pool boiling is assumed to be contibuted by the following mechanisms: Latent heat by bubbles (q LH ) because of the evapoation of liquid. This was fist suggested by Mooe and Mesle (1961) based on thei obsevation of apid suface-tempeatue fluctuations in nucleate boiling. Conductive heat tansfe (q CON ) fom the heated suface to the supeheated liquid laye in contact with suface. 114

115 115 The conductive heat tansfe fom this simulation will be discussed at the end of chapte in compaison with conductive heat tansfe fom othe cases. Fo now we focus on the latent heat tansfe poduced duing the stage of bubble expansion due to the vapoization of the themal bounday laye nea the liquid vapo inteface. The heat flux emoved by a single bubble can be obtained analytically by the following expession: 1 qlh = hfgρ vvb = hfgρvπd 3 B = J 6 (5.1) Fom the numeical simulation the latent heat flux absobed by the evapoating bubble can be obtained by integating the heat souce tem Qs (afte edimensioning it) ove the computational domain and evolving it aound the axis of symmety. The total latent heat tansfe can then be calculated by integating the obtained cuve ove eal time. This can be expessed as; q LH = Qs 3 t Ω (5.2) The following figue shows the heat tansfe fom the liquid to the bubble as a function of time. The negative sign indicates that the heat is being absobed by the bubble. Figue 5.8: Latent heat flux fom a gowing bubble. 115

116 116 In figue 5.9 the total heat absobed by the bubble is plotted against time. Figue 5.9: Total heat absobed by vapo bubble. At the end of the simulation the total heat absobed by the bubble is Joules. Accoding to Eq 5.1, fo a bubble depatue diamete, D B = 2.56mm (fom Table 5.1), the latent heat, Q LH is Joules. Thee is vey good ageement between the calculated latent heat tansfe and the analytically known value Phase consevation In this section the consevative popeties of ou implementation of the level set method ae tested in the context of bubble gowth and depatue with phase change. Assessing the phase consevation popeties duing a dynamic pocess in which the amount of phase is continuously inceasing is a challenging task. One way to appoach this is to obseve the amount phase at evey time step and compae to an estimated value of the amount of that phase based on the amount of phase injection. We accomplish this be taking obsevations of the volume of the vapo bubble at evey instant. And by calculating the incease in vapo volume at evey instant caused by the ate at which the liquid vapo inteface is pulled outwads, i.e. u Γ. An estimate of the volume of the vapo bubble is then obtained 116

117 117 by integating the ate of expansion pe unit time ove time thus pedicting the volume of the bubble at evey time step. This can be expessed as; V B = u Γ t 3 Ω (5.3) Thus, by compaing the obseved and estimated volume of vapo bubble we can appoximate the phase loss. In figue 5.9 the obseved and estimated volume of the vapo bubble ae pesented. Figue 5.10: Phase consevation duing simulation. The compaison shows good ageement up to the point whee the buoyancy foces ovetake the contact line foces. The oveshoot in estimation of the volume of vapo bubble is caused by incements in inteface velocity that ae not due to evapoation but ae due to the ise in buoyancy foces which influence the ate of displacement of the liquid vapo inteface. Thus thee is little numeical phase loss and the model is consevative duing the dynamic gowth of a vapo bubble with phase change. 117

118 Dynamic contact angle In suface science the contact angle is an impotant paamete as its vaiation manipulates bubble depatue diametes and gowth ates duing bubble gowth by influencing the contact line foce as well as the heat flux. Theefoe, fo the numeical simulation of bubble dynamics, an appopiate contact angle model is essential fo a coect pediction of the bubble fomation. In chapte 2, thee appoaches to contact angle modeling wee highlighted. Static contact angle appoach. Hee the appaent contact angle θ CA is taken as the equivalent to the mean value of the advancing and eceding contact angles. θ CA θ = ACA + θ 2 CA (5.4) Contact line velocity appoach. In this case θ CA is a dynamic contact angle that vaies as a function of contact line velocity u CL as defined by: θ CA = u CL θ ACA θ CA θ ACA + θ 2umax θ CA ACA + θ 2 CA if if if u u max CL u < u u CL CL > u max u max max (5.5) Whee u max is an input vaiable that decides the ate of change of the appaent contact angle fom its minimum eceding value to its maximum advancing value. Foce balance appoach. The appaent contact angle in this case is modelled as a smooth hypebolic tangent function of the foce balance between the contact line foce and the buoyancy foce as defined by the following expession: θ CA θ = + θ F + tanh Fσ F θ θ 2 ACA CA g σ, min ACA CA 2,max Fσ, min (5.6) 118

119 119 In this section we apply the thee appoaches to a single bubble gowing on a heated suface duing nucleate pool boiling with the minimum eceding contact angle, θ CA = 48, a maximum advancing contact angle, θ ACA = 61 and T = 7K Case-I: static contact angle The esults obtained fom the numeical model of a single bubble gowing on a heated suface with a static contact angle of 54.5 ae shown in figue (a) (b) (c) (d) Figue 5.11: Appoximate inteface velocity (a), contact angle (b), foces acting on the bubble (c) and bubble dimensions (d) fo a static mean contact angle of

120 120 Because the inteface is endowed with a thickness in the level set method exact values of the contact line velocity can not be obtained. The appoximate value of the inteface velocity is obtained by integating the velocity tem ove the inteface nea the contact point and dividing by the length of the inteface coveed. The fluctuations in the contact line velocity ae caused by discete appoach which can not accuately captue continuous phenomena. Thee foe it would be difficult to model the dynamic contact angle on the basis of the contact line velocity. By compaing Fig 5.11(a) with Fig 5.11(c) we obseve that the absolute value of the inteface velocity is popotional to the absolute diffeence between the buoyancy foce and contact foce. We also note that as the buoyancy foce ovetakes the contact foce, at non-dimensional time appoximately equal to 7, the inteface velocity changes signs. Thus, thee is a stong elation between the balance of foces and the contact line velocity. To be moe pecise, the contact line velocity is invesely popotional to F g F σ. Theefoe it is possible to model the dynamic contact angle as function of the Fg and Fσ, as in Case-III, instead of contact line velocity Case-II: Contact line velocity appoach A sudden change in the contact angle fom the minimum eceding value to the maximum advancing value causes unsteady behaviou in the bubble gowth mechanism. Theefoe in this case we conside a dynamic contact angle model in which the affects of the tansfomation of the appaent contact angle ae dampened by defining the instantaneous appaent contact angle, θ CA as a function of the inteface velocity that changes linealy fom the eceding contact angle value, θ CA to the advancing contact angle value, θ ACA as the inteface velocity goes fom a chosen positive limiting value, u max to a negative one, -u max. Out side the ange of the pescibed minimum/maximum contact line velocity the appaent contact angle is assigned a value of θ CA o θ ACA depending on the sign of the contact line velocity, as descibed in Eq 5.4. The limiting value of the inteface velocity, u max is chosen to be 0.015m/s afte consideing the evolution of the inteface velocity fo a static contact angle of 54.5 shown in Fig 5.11(a). The esults of the numeical simulation ae shown in Fig

121 121 (a) (b) (c) (d) Figue 5.12: Appoximate inteface velocity (a), dynamic contact angle (b), foces acting on the bubble (c) and bubble dimensions (d) fo an appaent contact angle as a function of the contact line velocity with umax = 0.015m/s. In this case, since the contact angle is a function of the contact line velocity alone, it is sensitive to even the minute changes in the inteface velocity. The inteface velocity depends on the ate of expansion o contaction of bubble. Which in tun depend on many factos, including the balance of foces i.e. suface tension, bubble momentum etc. Thus, the inteface behaviou is caught in a cyclic pocess whee the inteface velocity affects the contact angle and the contact angle affects the inteface velocity by alteing the suface tension and bubble momentum. The esult of this cyclic pocess is visible in the fluctuation of the dynamic contact angle between θ CA and θ ACA plotted in 121

122 122 figue 5.12(b) as well as in the slight levelling of the bubble base diamete at a steady value, visible in figue 5.12(d). Compaing Fig 5.11(a) and Fig 5.12(a) we note that the inteface velocity dops below zeo at an ealie stage, due to the decease in the effective contact angle Fig 5.11(b) and Fig 5.12(b) which esults in lesse contact foce. Also at this point the contact angle changes to advancing contact angle Fig 5.12(b) and a lot of things happen simultaneously: 1. The negative inteface velocity inceases the contact angle to its maximal value. 2. The contact foce inceases due to inceased contact angle. 3. The inceased contact foce ties to pull the bubble down thus expanding its base. 4. The expansion in bubble base inceases the inteface velocity to a positive value. 5. The positive value of the inteface velocity educes the contact angle to its minimal value. 6. The eduction in contact angle educes the contact foce. 7. In the absence of the inceased contact foce the bubble pulls upwads due to buoyancy. 8. The inteface begins to move inwad with a negative value of inteface velocity, and the cycle estats at step 1. The esults of this cyclic pocess ae visible in the fluctuations in the inteface velocity, Fig 5.12(a). In the pulsating contact angle Fig 5.12(b). In the vaiation of the contact foce Fig 5.12(c) and in the levelling of the bubble base diamete in Fig 5.12(d). This behaviou is moe a esult of the butal/sudden natue of the numeical implementation of the dynamic contact angle and is computationally exhaustive. The foce balance nea the contact point influences the ability of the bubble base to expand o contact, inducing the slip/stick behaviou descibed in the pevious case. Dussan (1979) explains slip/stick behaviou as when the motion of a contact line appeas to be unsteady and spasmatic when the contact angle is changing below a cetain smallest epoted speed. Beyond this ange the contact line movement ought to be smooth. Howeve we note that the contact angle is oscillating continuously leading to a solution that is computationally expensive and shows ponounced slip/stick behaviou of the contact line. 122

123 123 We also note that despite the fluctuation in the contact line velocity and the contact angle the buoyancy and contact line foce cuves ae quite simila fo both Case-I and Case-II. In both cases the maximum and minimum values of buoyancy foce and the contact line foce ae oughly the same and the buoyancy ove takes the contact line foce at about the same time, ( t = 6). This shows that the foce balance is take a lot of influence fom the fluctuations in the contact angle o contact line velocity due to slip/stick behaviou and motivates the modelling of the dynamic contact angle based on a foce balance appoach. (a) (b) (c) (d) Figue 5.13: Appoximate inteface velocity (a), dynamic contact angle (b), foces acting on the bubble (c) and bubble dimensions (d) fo an appaent contact angle as a function of the foce balance. 123

124 Case-III: Foce balance appoach Numeical calculations ae caied out with the appaent instantaneous contact angle at the bubble base obtained as a function of the foce balance between buoyancy foce and the contact line foce, caused by evapoative mass flux and suface tension espectively, accoding to Eq The esults fo obtained fo the appaent contact line velocity, the contact angle and the foce balance and bubble dimensions ae plotted in Fig Compaing F g and F σ between the thee cases, we note that the buoyancy foce, F g ovetakes the contact foce F σ at a slightly late time fo Case-III, Fig 5.13(c), this is because as the buoyancy foce ties to coss the contact line foce the contact angle begins to change fom minimum eceding value towads maximum advancing value 5.13(b). This in tun ties to bubble base outwads causing an incease in the bubble base diamete, Fig 5.13(d). This causes an incease in the contact line foce which is esponsible fo the delay in the buoyancy foce cossing ove the contact line foce. We not that as the contact angle changes fom θ CA to θ ACA the inteface velocity changes signs, Fig 5.13(a). The slip/stick behaviou at the beginning and intemediate stages of bubble gowth stops as soon as the liquid-vapo inteface eaches θ ACA and keeps it. At aound the same time the bubble base diamete shows a maked incease in ate of expansion Compaison with expeimental data amanujapu and Dhi (1999) expeimentally studied the dynamic contact angle at the base of a vapo bubble gowing on a hoizontal suface duing nucleate pool boiling of deionized wate, using a silicon wafe as the test suface with mico-machined cavities fo nucleation. They measued the bubble base diamete and appaent contact angle as a function of time and agued that the dynamic contact angle could be modelled as a function of the inteface velocity even though the vaiation of the contact angle with inteface velocity seemed to be weak because they obseved a tend in the contact angle dependency on the sign of the inteface velocity. Duing thei expeiments the contact angle was found to vay between fo T = 7K, which coesponds to ou test case. Figue 5.14 shows thei esults fo the measuement of appaent contact angle calculated fom expeimentally obtained still images and plotted as a function of time. 124

125 125 Figue 5.14: Contact angle vs time fo T = 7K, fom amanujapu and Dhi (1999) The advancing and eceding fonts clealy vay between 48 and 61, while the mean angle between is them is about 54 which coesponds to the static contact angle of the liquid on the suface at the base of a doplet, measued by the authos just pio to the commencement of the boiling expeiment. The expeimentally obseved evolution of the appaent contact angle ove time esembles that of Case-III with a foced balance appoach to modelling the dynamic contact angle. In both cases the tansition fom eceding contact angle to advancing contact angle is smooth. Next, the pogession of the bubble diamete obtained fom the same expeiments by amanujapu and Dhi (1999) is shown in Figue Each bubble diamete cuve begins at the point of bubble nucleation and ends when the bubble depats fom the heate suface, the thee cuves coespond to thee successive bubble gowth cycles. Vaiation in the bubble depatue diamete between the thee cycles is mino. 125

126 126 Figue 5.15: Bubble diamete vs time T = 7K, fom amanujapu and Dhi (1999) The following table summaizes the esults of the fou studies (thee numeical and one expeimental) of the dynamic contact angle and its effects on the dynamics of a single bubble gowing on a heated suface duing nucleate pool boiling. Thee is excellent ageement between the expeimentally obseved bubble depatue diamete and the bubble depatue diamete obtained by numeical simulation using a foce balance model fo the dynamic contact angle, Case-III. Diamete [mm] Time [s] Case I Case II Case III Expeimental esults Table 5.1: Bubble chaacteistic at depatue fo vaious test cases. 126

127 127 Howeve thee is consideable diffeence in bubble depatue time between the expeimental and numeical studies. This is due to the absence of a micolaye evapoation model as explained at the beginning of the chapte. The close ageement between the bubble depatue diametes veifies ou assetion that the ate of evapoation nea the contact point at the bubble base has little influence on the global dynamics of an evapoating bubble while affecting the ate at which these dynamics take place. Now focusing on the esults obtained fom the thee diffeent cases of numeical simulations we obseve that depatue diamete and time ae slightly smalle fo the case of the static contact angle. Thus the use of static contact angle will unde-pedict the bubble depatue diamete and time. In case II it is clea that duing the tansitoy peiod the contact angle will have an appoximately mean value. This is also evident fom Fig 5.12(b). Thus choosing contact line velocity appoach to model dynamic contact angle diminishes the advantages of a dynamic contact angle model and the model esembles a static mean contact angle appoach. It is also evident fom Eq 5.2 that a choice of a vey low u max would esemble a constant advancing and constant eceding contact angle appoach. This would explain the slight inaccuacy of contact line appoach of modelling dynamic contact angles in pedicting bubble depatue chaacteistics Heat tansfe compaison The heat absobed by the system duing the tansfomation fom the liquid to the vapo phase fo the thee cases of dynamic contact angle appoaches, i.e. static contact angle, contact line velocity based dynamic contact angle and foce balance based dynamic contact angle is shown in figue While the conductive heat tansfe fom the heated suface to the supeheated liquid just above it, fo the thee cases is plotted in figue Fom both these plots the effect of the contact angle on heat tansfe duing the dynamic gowth of a single in nucleate pool boiling is quite evident. 127

128 128 Figue 5.16: Latent heat tansfe fo diffeent contact angle models. Figue 5.17: Conductive heat tansfe fo diffeent contact angle models. 128

129 129 In the case of the static mean contact angle, the both the conductive and latent heats ae high at the beginning since the static mean contact angle of 54.5 is geate than the eceding contact angle 48. Howeve at the end of the simulation the heat tansfe fo the static contact angle is less as the mean contact angle is less than the advancing contact angle, 68. Thus the plots show that the heat tansfe is popotional to the contact angle. 5.3 Conclusion Heat tansfe mechanisms at the local scale of a bubble gowing on a heated suface have been highlighted. These mechanisms consist of the absoption of heat available at the heate suface by the liquid in the themal bounday laye and its tanspot though natual convection. This heat is tansfeed to vapo phase as latent heat duing evapoation. Once the vapo bubble begins to gow, its agitation causes convective foces that futhe mix the fluid and help tansfe heat fom supeheated liquid in laye to the suounding liquid. This themal mixing is inceased at bubble depatue when wake effects beak the themal bounday laye causing additional heat distibution. The influence of a dynamic appaent contact angle on the dynamics of a single vapo bubble nucleating on a heated wall wee included in the numeical simulation consideing diffeent appoaches of contact angle at the bubble base. The static mean contact angle model based on the maximum advancing and minimum eceding contact angle is the simplest to implement but unde pedicts the bubble depatue chaacteistics. Modeling the dynamic contact angle as a function of the contact line velocity equies a obust estimation of the velocity of the liquid vapo inteface that is endowed with a thickness in the level set method. This appoach is susceptible to discetization eos and ove ponounced stick/slip behavio leading to loss of accuacy in the detemination of bubble depatue chaacteistics. The foce balance appoach povides an excellent altenative to modeling dynamic contact angles with the most accuate esults fo bubble depatue chaacteistics. Howeve, inode to completely pedict bubble dynamics ove time the micolaye evapoation needs to be taken into account though the absence of micolaye evapoation povides qualitative and, in cetain cases, quantative data egading bubble dynamics. 129

130 130 Chapte 6 Concluding emaks This study was undetaken ecognizing the need to have moe physics based appoach to the undestanding of boiling heat tansfe that is based on the undelying physical mechanisms and is moe eliable then the existing, pedominantly empiical appoach. The focus of the wok was the nucleate pool boiling pocess at the macoscopic level with the objective to numeically simulate the dynamics of single bubble gowing on heated suface duing nucleate pool boiling at the macoscopic scale. The wok can oughly be divided into two categoies o stages: Development of level set model to simulate bubble dynamics Application of the model to the study of bubble dynamics. The elative accomplishments made against these two objectives ae summaized in the following section. 6.1 Advances made in this wok A consevative level set method was applied to the study of bubble dynamics. This libeates the use fom the computational cost associated with einitialization techniques peviously used to limit phase loss duing the utilization of the level set method. The numeical model thus obtained was found to be obust and efficient. Futhe moe it is flexible and can theefoe be used to study diffeent kinds of fluids o fluid combinations in system whee axial symmety can be assumed. Futhe moe the use of commecially available solve fo the implementation ensues its compatibility and facilitates its extension. The validated model was used to study the effects of local phenomena on global dynamics of a gowing bubble in the context of a decoupled appoach. This appoach pemitted the study of inteface dynamics independent of the themal tanspot that causes it. Though this model the local conditions nea the contact point wee demonstated to have vey little effect on the gowth dynamics of a vapo bubble. The effect of the ate of evapoation in quickening the bubble gowth pocess was elaboated and its lack of influence ove othe aspects of the dynamics of gowing bubbles was highlighted. 130

131 131 Next, the level set and Navie Stokes model was coupled to the enegy equation to study the dynamics of bubble gowth in coelation with the associated heat tansfe. The effects of wall heat flux on bubble dynamics has been the subject of many studies in the past. In this wok ou focus has been on exploing the implementation of a consevative level set appoach, without einitialization to the study of bubble dynamics at the macoscale level. A foce-balance appoach to the modeling of a dynamic appaent contact angle, based on the physics of the moving contact line dynamics, was poposed in this study. It was tested against the taditional contact line velocity appoach of modeling dynamic contact angle which mimics actual physics as expeimental data indicates a weak elationship between contact line velocity and contact angle. The foce-balance appoach was validated in compaison to expeimental data available in the liteatue. This appoach educes the non-physical stick/slip behavio of the contact line in compaison to the contact line velocity model, and allows a smooth tansition fom the minimum eceding contact angle to the maximum advancing contact angle which is moe akin to the physical phenomena. The affects of the dynamic contact angle on heat tansfe wee also obseved. It was demonstated that the heat tansfe duing bubble gowth in nucleate pool boiling is popotional to the appaent contact angle. 6.2 Publications duing PhD studies Pee eviewed eseach papes 1. Sajid, M., Bennace,.(2009). Numeical study of decoupled bubble dynamics in nucleate boiling. Defect and Diffusion Foum , Pee eviewed confeence poceedings 1. Sajid, M., and Bennace,. (2010). Numeical study of single bubble dynamics using a consevative level set method. Accepted fo 2010 Intenational Confeence on Multiphase Flow (ICMF). Tampa, Floida, USA. 2. Sajid, M., and Bennace,. (2009). Simulation of single bubble dynamics in nucleate pool boiling using a consevative level set method. Poceedings of the 2009 ASME Summe Heat Confeence, San Fancisco, Califonia, USA 131

132 Outlook The focus of this wok has been the study of bubble dynamics at the macoscale level by exploiting a consevative level set appoach, without einitialization. The next step will be to include the contibution of the inteacting sub-pocesses while moving towads a complete numeical simulation of bubble dynamics and the associated heat tansfe pocesses. Most impotant, is the need to account fo evapoation in the liquid micolaye. Once that is done a complete thee dimensional numeical simulation can be tageted with flow conditions in ode to espond to a wide ange of industial applications. The model that has been developed in the wok is vey flexible and geneic and allows fo the inclusion of many inteesting physical phenomena such as mico gavity, multi body coalescence etc. Thus anothe possibility is to use the cuent model to study othe multiphase phenomena at the macoscopic scale besides boiling, like doplet dynamics. 132

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139 139 Appendix A Comsol scipting code % COMSOL Multiphysics Model M-file % Geneated by COMSOL 3.5a (COMSOL , $Date: 2008/12/03 17:02:19 $) % Some geomety objects ae stoed in a sepaate file. % The name of this file is given by the vaiable 'flbinayfile'. flclea fem % COMSOL vesion clea vsn vsn.name = 'COMSOL 3.5'; vsn.ext = 'a'; vsn.majo = 0; vsn.build = 603; vsn.cs = '$Name: $'; vsn.date = '$Date: 2008/12/03 17:02:19 $'; fem.vesion = vsn; flbinayfile='1 NS LS HT DCA Fsg dt7 - t12p0 TBL.mphm'; % Constants fem.const = {'ho_l','958[kg/m^3]',... 'ho_v','0.59[kg/m^3]',... 'miu_l','0.2818e-3[pas]',... 'miu_v','0.0125e-3[pas]',... 'g','9.81[m/s^2]',... 'sig','0.0611[n/m]',... 'a','0.0003/l0',... 'b','0.0003/l0',... 'gam','1',... 'ee','0.047',... 'cc','0.5',... 'CA0','ca',... 'l0','sqt(sig/(g(ho_l-ho_v)))',... 'u0','sqt(gl0)',... 'e0','ho_l0u0/miu_',... 'F0','u0/sqt(gl0)',... 'We0','ho_u0^2l0/sig',... 'sigm','1/we0',... 'tt0','5ee/gam',... 'cp_v','1889.2',... 'cp_l','4279',... 'k_v','0',... 'k_l','0.679',... 'hl','2.260e6',... 'Tsat','373.15[K]',... 'q','1e5[w/m^2]',

140 140 'p_atm','101325[pa]',... 't0','l0/u0',... 'Tw','Tsat+dT',... 'Tsat0','(Tsat-Tsat)/(Tw-Tsat)',... 'Tw0','(Tw-Tsat)/(Tw-Tsat)',... 'Pe0','l0u0ho_cp_/k_',... '0','(a+b)0.5cos(CA0)',... 'ho_','0.5(ho_v+ho_l)',... 'miu_','0.5(miu_v+miu_l)',... 'k_','k_l',... 'cp_','0.5(cp_v+cp_l)',... 'dt','7',... 'ca','48pi/180',... 'aca','61pi/180',... 'u_clmx','0.02'}; % Geomety clea daw g4=flbinay('g4','daw',flbinayfile); daw.s.objs = {g4}; daw.s.name = {'1'}; daw.s.tags = {'g4'}; fem.daw = daw; fem.geom = geomcsg(fem); fem.mesh = flbinay('m1','mesh',flbinayfile); % (Default values ae not included) % Application mode 1 clea appl appl.mode.class = 'NavieStokes'; appl.mode.type = 'axi'; appl.name = 'ns'; appl.module = 'CHEM'; appl.gpode = {4,2}; appl.cpode = {2,1}; appl.assignsuffix = '_ns'; clea pop clea weakconst weakconst.value = 'off'; weakconst.dim = {'lm1','lm2','lm3','lm4','lm5','lm6','lm7','lm8','lmpsi'}; pop.weakconst = weakconst; appl.pop = pop; clea bnd bnd.symtype = {'ax','sym','sym'}; bnd.type = {'sym','walltype','open'}; bnd.walltype = {'noslip','slip','noslip'}; bnd.ind = [1,2,3,2]; appl.bnd = bnd; clea equ equ.eta = 'eta'; equ.gpode = {{1;1;2}}; 140

141 141 equ.ho = 'ho0'; equ.cpode = {{1;1;2}}; equ.cdon = 0; equ.sigma = 0; equ.f_z = 'Fz'; equ.epsilon = 'hmax_ns'; equ.ind = [1]; appl.equ = equ; fem.appl{1} = appl; % Application mode 2 clea appl appl.mode.class = 'ConvDiff'; appl.mode.type = 'axi'; appl.dim = {'phi'}; appl.name = 'ls'; appl.module = 'CHEM'; appl.assignsuffix = '_ls'; clea pop clea weakconst weakconst.value = 'off'; weakconst.dim = {'lm9'}; pop.weakconst = weakconst; appl.pop = pop; clea bnd bnd.type = {'ax','nc'}; bnd.ind = [1,2,2,2]; appl.bnd = bnd; clea equ equ.d = 0; equ.init = 'phi0'; equ.v = 'v+uintnomz'; equ.u = 'u+uintnom'; equ.name = 'default'; equ.ind = [1]; appl.equ = equ; fem.appl{2} = appl; % Application mode 3 clea appl appl.mode.class = 'ConvCond'; appl.mode.type = 'axi'; appl.name = 'cc'; appl.module = 'CHEM'; appl.assignsuffix = '_cc'; clea pop clea weakconst weakconst.value = 'off'; weakconst.dim = {'lm10'}; pop.weakconst = weakconst; appl.pop = pop; clea bnd bnd.type = {'qc','ax','t','t'}; 141

142 142 bnd.t0 = {0,0,'Tw0','Tsat0'}; bnd.ind = [2,3,4,1]; appl.bnd = bnd; clea equ equ.c = 'cp0'; equ.q = 'Qs'; equ.sdtype = 'supg'; equ.ho = 'ho0'; equ.sdiff = 'off'; equ.init = 'Tsat0'; equ.k = 'kappa'; equ.v = 'v(phi>cc)'; equ.u = 'u(phi>cc)'; equ.sdon = 0; equ.ind = [1]; appl.equ = equ; fem.appl{3} = appl; fem.sdim = {'','z'}; fem.fame = {'ef'}; % Shape functions fem.shape = {'shlag(2,''u'')','shlag(2,''v'')','shlag(1,''p'')','shlag(2,''phi'')','sh lag(2,''t'')'}; % Integation ode fem.gpode = {4,2}; % Constaint ode fem.cpode = {2,1}; fem.bode = 1; % Equation fom fem.fom = 'geneal'; % Subdomain settings clea equ equ.cpode = {{1;1;2;1;1}}; equ.f = {{'-ho_ns(uu+vuz)-2eta_nsu/+p';'(f_z_nsho_ns(uv+vvz))';... '-(divu_ns-usouce)';'(-u_phi_lsphi-v_phi_lsphiz)';'(- ho_ccc_ccu_cct-ho_ccc_ccv_cctz+q_cc)'}}; equ.da = {{'ho_ns';'ho_ns';0;'dts_phi_ls';'dts_ccho_ccc_cc'}}; equ.c = {{{'-d((-2eta_nsu+p),u)','-d((-2eta_nsu+p),uz)';'-d(- eta_ns(uz+v),u)',... '-d(-eta_ns(uz+v),uz)'},{'-d((-2eta_nsu+p),v)','-d((- 2eta_nsu+p),vz)';... '-d(-eta_ns(uz+v),v)','-d(-eta_ns(uz+v),vz)'},{'-d((- 2eta_nsu+p),p)',... '-d((-2eta_nsu+p),pz)';'-d(-eta_ns(uz+v),p)','-d(- eta_ns(uz+v),pz)'},

143 143 {'-d((-2eta_nsu+p),phi)','-d((-2eta_nsu+p),phiz)';'-d(- eta_ns(uz+v),phi)',... '-d(-eta_ns(uz+v),phiz)'},{'-d((-2eta_nsu+p),t)','-d((- 2eta_nsu+p),Tz)';... '-d(-eta_ns(uz+v),t)','-d(-eta_ns(uz+v),tz)'};{'-d(- eta_ns(v+uz),u)',... '-d(-eta_ns(v+uz),uz)';'-d((-2eta_nsvz+p),u)','-d((- 2eta_nsvz+p),uz)'},... {'-d(-eta_ns(v+uz),v)','-d(-eta_ns(v+uz),vz)';'-d((- 2eta_nsvz+p),v)',... '-d((-2eta_nsvz+p),vz)'},{'-d(-eta_ns(v+uz),p)','-d(- eta_ns(v+uz),pz)';... '-d((-2eta_nsvz+p),p)','-d((-2eta_nsvz+p),pz)'},{'-d(- eta_ns(v+uz),phi)',... '-d(-eta_ns(v+uz),phiz)';'-d((-2eta_nsvz+p),phi)','-d((- 2eta_nsvz+p),phiz)'},... {'-d(-eta_ns(v+uz),t)','-d(-eta_ns(v+uz),tz)';'-d((- 2eta_nsvz+p),T)',... '-d((-2eta_nsvz+p),tz)'};0,0,0,0,0;{'-d(gamma(phi(1- phi)nojac(nom)-eephi),u)',... '-d(gamma(phi(1-phi)nojac(nom)-eephi),uz)';'-d(gamma(phi(1- phi)nojac(nomz)-eephiz),u)',... '-d(gamma(phi(1-phi)nojac(nomz)-eephiz),uz)'},{'- d(gamma(phi(1-phi)nojac(nom)-eephi),v)',... '-d(gamma(phi(1-phi)nojac(nom)-eephi),vz)';'-d(gamma(phi(1- phi)nojac(nomz)-eephiz),v)',... '-d(gamma(phi(1-phi)nojac(nomz)-eephiz),vz)'},{'- d(gamma(phi(1-phi)nojac(nom)-eephi),p)',... '-d(gamma(phi(1-phi)nojac(nom)-eephi),pz)';'-d(gamma(phi(1- phi)nojac(nomz)-eephiz),p)',... '-d(gamma(phi(1-phi)nojac(nomz)-eephiz),pz)'},{'- d(gamma(phi(1-phi)nojac(nom)-eephi),phi)',... '-d(gamma(phi(1-phi)nojac(nom)-eephi),phiz)';'- d(gamma(phi(1-phi)nojac(nomz)-eephiz),phi)',... '-d(gamma(phi(1-phi)nojac(nomz)-eephiz),phiz)'},{'- d(gamma(phi(1-phi)nojac(nom)-eephi),t)',... '-d(gamma(phi(1-phi)nojac(nom)-eephi),tz)';'-d(gamma(phi(1- phi)nojac(nomz)-eephiz),t)',... '-d(gamma(phi(1-phi)nojac(nomz)-eephiz),tz)'};{'-d(- k_t_cct,u)',... '-d(-k_t_cct,uz)';'-d(-k_t_cctz,u)','-d(-k_t_cctz,uz)'},{'- d(-k_t_cct,v)',... '-d(-k_t_cct,vz)';'-d(-k_t_cctz,v)','-d(-k_t_cctz,vz)'},{'- d(-k_t_cct,p)',... '-d(-k_t_cct,pz)';'-d(-k_t_cctz,p)','-d(-k_t_cctz,pz)'},{'- d(-k_t_cct,phi)',... '-d(-k_t_cct,phiz)';'-d(-k_t_cctz,phi)','-d(- k_t_cctz,phiz)'},... {'-d(-k_t_cct,t)','-d(-k_t_cct,tz)';'-d(-k_t_cctz,t)','-d(- k_t_cctz,tz)'}}}; equ.a = {{'-d(-ho_ns(uu+vuz)-2eta_nsu/+p,u)','-d(- ho_ns(uu+vuz)-2eta_nsu/+p,v)',

144 144 '-d(-ho_ns(uu+vuz)-2eta_nsu/+p,p)','-d(-ho_ns(uu+vuz)- 2eta_nsu/+p,phi)',... '-d(-ho_ns(uu+vuz)-2eta_nsu/+p,t)';'-d((f_z_nsho_ns(uv+vvz)),u)',... '-d((f_z_ns-ho_ns(uv+vvz)),v)','-d((f_z_nsho_ns(uv+vvz)),p)',... '-d((f_z_ns-ho_ns(uv+vvz)),phi)','-d((f_z_nsho_ns(uv+vvz)),t)';... '-d(-(divu_ns+usouce),u)','-d(-(divu_ns+usouce),v)','-d(- (divu_ns+usouce),p)',... '-d(-(divu_ns+usouce),phi)','-d(-(divu_ns+usouce),t)';'-d((- u_phi_lsphi-v_phi_lsphiz),u)',... '-d((-u_phi_lsphi-v_phi_lsphiz),v)','-d((-u_phi_lsphiv_phi_lsphiz),p)',... '-d((-u_phi_lsphi-v_phi_lsphiz),phi)','-d((-u_phi_lsphiv_phi_lsphiz),t)';... '-d((-ho_ccc_ccu_cct-ho_ccc_ccv_cctz+q_cc),u)','-d((- ho_ccc_ccu_cct-ho_ccc_ccv_cctz+q_cc),v)',... '-d((-ho_ccc_ccu_cct-ho_ccc_ccv_cctz+q_cc),p)','-d((- ho_ccc_ccu_cct-ho_ccc_ccv_cctz+q_cc),phi)',... '-d((-ho_ccc_ccu_cct-ho_ccc_ccv_cctz+q_cc),t)'}}; equ.init = {{0;0;0;'phi0';'Tsat0'}}; equ.ga = {{{'(-2eta_nsu+p)';'-eta_ns(uz+v)'};{'- eta_ns(v+uz)';... '(-2eta_nsvz+p)'};{0;0};{'gamma(phi(1-phi)nojac(nom)- eephi)';... 'gamma(phi(1-phi)nojac(nomz)-eephiz)'};{'-k_t_cct';'- k_t_cctz'}}}; equ.al = {{{'-d((-2eta_nsu+p),u)';'-d(-eta_ns(uz+v),u)'},{'- d((-2eta_nsu+p),v)';... '-d(-eta_ns(uz+v),v)'},{'-d((-2eta_nsu+p),p)';'-d(- eta_ns(uz+v),p)'},... {'-d((-2eta_nsu+p),phi)';'-d(-eta_ns(uz+v),phi)'},{'-d((- 2eta_nsu+p),T)';... '-d(-eta_ns(uz+v),t)'};{'-d(-eta_ns(v+uz),u)';'-d((- 2eta_nsvz+p),u)'},... {'-d(-eta_ns(v+uz),v)';'-d((-2eta_nsvz+p),v)'},{'-d(- eta_ns(v+uz),p)';... '-d((-2eta_nsvz+p),p)'},{'-d(-eta_ns(v+uz),phi)';'-d((- 2eta_nsvz+p),phi)'},... {'-d(-eta_ns(v+uz),t)';'-d((-2eta_nsvz+p),t)'};{0;0},{0;0},... {0;0},{0;0},{0;0};{'-d(gamma(phi(1-phi)nojac(nom)-eephi),u)';... '-d(gamma(phi(1-phi)nojac(nomz)-eephiz),u)'},{'- d(gamma(phi(1-phi)nojac(nom)-eephi),v)';... '-d(gamma(phi(1-phi)nojac(nomz)-eephiz),v)'},{'- d(gamma(phi(1-phi)nojac(nom)-eephi),p)';... '-d(gamma(phi(1-phi)nojac(nomz)-eephiz),p)'},{'- d(gamma(phi(1-phi)nojac(nom)-eephi),phi)';... '-d(gamma(phi(1-phi)nojac(nomz)-eephiz),phi)'},{'- d(gamma(phi(1-phi)nojac(nom)-eephi),t)';... '-d(gamma(phi(1-phi)nojac(nomz)-eephiz),t)'};{'-d(- k_t_cct,u)';

145 145 '-d(-k_t_cctz,u)'},{'-d(-k_t_cct,v)';'-d(-k_t_cctz,v)'},{'-d(- k_t_cct,p)';... '-d(-k_t_cctz,p)'},{'-d(-k_t_cct,phi)';'-d(- k_t_cctz,phi)'},{'-d(-k_t_cct,t)';... '-d(-k_t_cctz,t)'}}}; equ.gpode = {{1;1;2;1;1}}; equ.weak = {{'-sigma(-test(u)(-1+nom^2)- test(uz)nomznom+test(u)/)delta';... '-sigma(-test(v)nomnomz-test(vz)(-1+nomz^2))delta';'- test(divu_ns)tauc_nsho_nsdivu_ns'}}; equ.be = {{{'-d(-ho_ns(uu+vuz)-2eta_nsu/+p,u)';'-d(- ho_ns(uu+vuz)-2eta_nsu/+p,uz)'},... {'-d(-ho_ns(uu+vuz)-2eta_nsu/+p,v)';'-d(- ho_ns(uu+vuz)-2eta_nsu/+p,vz)'},... {'-d(-ho_ns(uu+vuz)-2eta_nsu/+p,p)';'-d(- ho_ns(uu+vuz)-2eta_nsu/+p,pz)'},... {'-d(-ho_ns(uu+vuz)-2eta_nsu/+p,phi)';'-d(- ho_ns(uu+vuz)-2eta_nsu/+p,phiz)'},... {'-d(-ho_ns(uu+vuz)-2eta_nsu/+p,t)';'-d(- ho_ns(uu+vuz)-2eta_nsu/+p,tz)'};... {'-d((f_z_ns-ho_ns(uv+vvz)),u)';'-d((f_z_nsho_ns(uv+vvz)),uz)'},... {'-d((f_z_ns-ho_ns(uv+vvz)),v)';'-d((f_z_nsho_ns(uv+vvz)),vz)'},... {'-d((f_z_ns-ho_ns(uv+vvz)),p)';'-d((f_z_nsho_ns(uv+vvz)),pz)'},... {'-d((f_z_ns-ho_ns(uv+vvz)),phi)';'-d((f_z_nsho_ns(uv+vvz)),phiz)'},... {'-d((f_z_ns-ho_ns(uv+vvz)),t)';'-d((f_z_nsho_ns(uv+vvz)),tz)'};... {'-d(-(divu_ns+usouce),u)';'-d(-(divu_ns+usouce),uz)'},{'-d(- (divu_ns+usouce),v)';... '-d(-(divu_ns+usouce),vz)'},{'-d(-(divu_ns+usouce),p)';'-d(- (divu_ns+usouce),pz)'},... {'-d(-(divu_ns+usouce),phi)';'-d(-(divu_ns+usouce),phiz)'},{'-d(- (divu_ns+usouce),t)';... '-d(-(divu_ns+usouce),tz)'};{'-d((-u_phi_lsphiv_phi_lsphiz),u)';... '-d((-u_phi_lsphi-v_phi_lsphiz),uz)'},{'-d((-u_phi_lsphiv_phi_lsphiz),v)';... '-d((-u_phi_lsphi-v_phi_lsphiz),vz)'},{'-d((-u_phi_lsphiv_phi_lsphiz),p)';... '-d((-u_phi_lsphi-v_phi_lsphiz),pz)'},{'-d((-u_phi_lsphiv_phi_lsphiz),phi)';... '-d((-u_phi_lsphi-v_phi_lsphiz),phiz)'},{'-d((-u_phi_lsphiv_phi_lsphiz),t)';... '-d((-u_phi_lsphi-v_phi_lsphiz),tz)'};{'-d((-ho_ccc_ccu_cctho_ccc_ccv_cctz+q_cc),u)';... '-d((-ho_ccc_ccu_cct-ho_ccc_ccv_cctz+q_cc),uz)'},{'-d((- ho_ccc_ccu_cct-ho_ccc_ccv_cctz+q_cc),v)';... '-d((-ho_ccc_ccu_cct-ho_ccc_ccv_cctz+q_cc),vz)'},{'-d((- ho_ccc_ccu_cct-ho_ccc_ccv_cctz+q_cc),p)';

146 146 '-d((-ho_ccc_ccu_cct-ho_ccc_ccv_cctz+q_cc),pz)'},{'-d((- ho_ccc_ccu_cct-ho_ccc_ccv_cctz+q_cc),phi)';... '-d((-ho_ccc_ccu_cct-ho_ccc_ccv_cctz+q_cc),phiz)'},{'-d((- ho_ccc_ccu_cct-ho_ccc_ccv_cctz+q_cc),t)';... '-d((-ho_ccc_ccu_cct-ho_ccc_ccv_cctz+q_cc),tz)'}}}; equ.ind = [1]; equ.dim = {'u','v','p','phi','t'}; equ.va = {'U_ns','sqt(u^2+v^2)',... 'V_ns','uz-v',... 'divu_ns','u+vz+u/',... 'celle_ns','ho_nsu_nsh/eta_ns',... 'es_u_ns','(ho_ns(ut+uu+vuz)+p-f ns)+2eta_ns(u/-u)- eta_ns(2u+uzz+vz)',... 'es_v_ns','(ho_ns(vt+uv+vvz)+pz-f_z_ns)- eta_ns((v+uz)+2vzz+uz+v)',... 'beta ns','ho_nsu',... 'beta_z_ns','ho_nsv',... 'Dm_ns','eta_ns',... 'da_ns','ho_ns',... 'taum_ns','nojac(min(timestep/ho_ns,0.5h/max(ho_nsu_ns,6eta_ns/h)))',... 'tauc_ns','nojac(0.5u_nshmin(1,ho_nsu_nsh/eta_ns))',... 'es_p_ns','ho_nsdivu_ns',... 'gad_phi ls','phi',... 'dflux_phi ls','-d_phi_lsphi-dz_phi_lsphiz',... 'cflux_phi ls','phiu_phi_ls',... 'tflux_phi ls','dflux_phi ls+cflux_phi ls',... 'gad_phi_z_ls','phiz',... 'dflux_phi_z_ls','-dz_phi_lsphi-dzz_phi_lsphiz',... 'cflux_phi_z_ls','phiv_phi_ls',... 'tflux_phi_z_ls','dflux_phi_z_ls+cflux_phi_z_ls',... 'beta_phi ls','u_phi_ls',... 'beta_phi_z_ls','v_phi_ls',... 'gad_phi_ls','sqt(gad_phi ls^2+gad_phi_z_ls^2)',... 'dflux_phi_ls','sqt(dflux_phi ls^2+dflux_phi_z_ls^2)',... 'cflux_phi_ls','sqt(cflux_phi ls^2+cflux_phi_z_ls^2)',... 'tflux_phi_ls','sqt(tflux_phi ls^2+tflux_phi_z_ls^2)',... 'cellpe_phi_ls','hsqt(beta_phi ls^2+beta_phi_z_ls^2)/dm_phi_ls',... 'Dm_phi_ls','(D_phi_lsu_phi_ls^2+Dz_phi_lsu_phi_lsv_phi_ls+Dz_phi _lsv_phi_lsu_phi_ls+dzz_phi_lsv_phi_ls^2)/(u_phi_ls^2+v_phi_ls^2+eps)',... 'es_phi_ls','(-d_phi_lsphi-dz_phi_lsphiz+phiu_phi_ls- Dz_phi_lsphiz-Dzz_phi_lsphizz+phizv_phi_ls-_phi_ls)',... 'es_sc_phi_ls','(phiu_phi_ls+phizv_phi_ls-_phi_ls)',... 'da_phi_ls','dts_phi_ls',... 'gad_t cc','t',... 'dflux_t cc','-k_t_cct-kz_t_cctz',... 'cflux_t cc','ho_t_ccc_t_cctu_t_cc',... 'tflux_t cc','dflux_t cc+cflux_t cc+sdflux_t cc',... 'gad_t_z_cc','tz',... 'dflux_t_z_cc','-kz_t_cct-kzz_t_cctz',

147 147 'cflux_t_z_cc','ho_t_ccc_t_cctv_t_cc',... 'tflux_t_z_cc','dflux_t_z_cc+cflux_t_z_cc+sdflux_t_z_cc',... 'beta_t cc','ho_t_ccc_t_ccu_t_cc',... 'beta_t_z_cc','ho_t_ccc_t_ccv_t_cc',... 'gad_t_cc','sqt(gad_t cc^2+gad_t_z_cc^2)',... 'dflux_t_cc','sqt(dflux_t cc^2+dflux_t_z_cc^2)',... 'cflux_t_cc','sqt(cflux_t cc^2+cflux_t_z_cc^2)',... 'tflux_t_cc','sqt(tflux_t cc^2+tflux_t_z_cc^2)',... 'cellpe_t_cc','hsqt(beta_t cc^2+beta_t_z_cc^2)/dm_t_cc',... 'kmean_t_cc','k_cc',... 'taue_t_cc',0,... 'helem_t_cc','h',... 'Dm_T_cc','ho_T_cc^2C_T_cc^2(k_T_ccu_T_cc^2+kz_T_ccu_T_ccv_T_cc +kz_t_ccv_t_ccu_t_cc+kzz_t_ccv_t_cc^2)/((ho_t_ccc_t_ccu_t_cc)^2+(h o_t_ccc_t_ccv_t_cc)^2+eps)',... 'es_t_cc','(-k_t_cct-kz_t_cctz+tho_t_ccc_t_ccu_t_cckz_t_cctz-kzz_t_cctzz+tzho_t_ccc_t_ccv_t_cc-q_t_cc)',... 'da_t_cc','dts_t_ccho_t_ccc_t_cc','ho_ns','ho0',... 'eta_ns','eta',... 'F ns',0,... 'F_z_ns','Fz',... 'F_phi_ns',0,... 'eta0_ns',1,... 'm_ns',0,... 'n_ns',0,... 'lambda_ns',0,... 'eta_inf_ns',0,... 'type_visc_ns','powe',... 'kappadv_ns',0,... 'hofcnp_ns',0,... 'hofcnt_ns',0,... 'epsilonp_ns',1,... 'Q_ns',0,... 'binkmaneqns_ns',0,... 'delid_ns',0.5,... 'delsd_ns',0.25,... 'cdtype_ns','sc',... 'delcd_ns',0.35,... 'delps_ns',1,... 'ck_ns',0.1,... 'impeson_ns',0,... 'usedelement_ns','lagp2p1',... 'finestmcelem_ns','lagp2p1',... 'gls_ns','- test(nojac(ho_ns)(nojac(u)u+nojac(v)uz)+p+2nojac(eta_ns)(u/u)/-nojac(eta_ns)(2u+uzz+vz))taum_nses_u_nstest(nojac(ho_ns)(nojac(u)v+nojac(v)vz)+pznojac(eta_ns)(v+uz+2vzz+(uz+v)/))taum_nses_v_nstest(divu_ns)tauc_nses_p_ns',... 'gijgij_ns',0,... 'usedelm_ls_ns','lagp2p1',... 'ho1_ns',1,

148 148 'ho2_ns',1,... 'eta1_ns',1,... 'eta2_ns',1,... 'sigma_ns',0,... 'gamma_ns',1,... 'chi_ns',1,... 'epsilon_ns','hmax_ns',... 'dfdphi_ns',0,... 'kappab_ns',1,... 'U_ef_ns','U_ns',... 'D_phi_ls',0,... 'D_phi_ls',0,... 'Dz_phi_ls',0,... 'Dz_phi_ls',0,... 'Dzz_phi_ls',0,... 'Dts_phi_ls',1,... 'u_phi_ls','u+uintnom',... 'v_phi_ls','v+uintnomz',... '_phi_ls',0,... 'k_cc','kappa',... 'Dts_cc',1,... 'ho_cc','ho0',... 'C_cc','cp0',... 'Q_cc','Qs',... 'fluidtype_cc','usedefined',... 'ptype_cc','gauge',... 'pgaugeef_cc',101325,... 'p_cc',0,... 'QpwokOn_cc',0,... 'QviscOn_cc',0,... 'eta_cc',0,... 'etat_cc',0,... 's_cc',287,... 'Mn_cc',0,... 'u_cc','u(phi>cc)',... 'v_cc','v(phi>cc)',... 'usedelement_cc','lag2',... 'finestmcelem_cc','lag2',... 'idon_cc',0,... 'delid_cc',0.5,... 'sdon_cc',0,... 'sdtype_cc','supg',... 'delsd_cc',0.25,... 'cdon_cc',0,... 'glim_cc','0.01[k]/helem_t_cc',... 'gamma_cc',1,... 'sdiff_cc','off',... 'kt_cc',0,... 'k_t_cc','kappa',... 'k_t_cc','kappa',... 'kz_t_cc',0,... 'kz_t_cc',0,... 'kzz_t_cc','kappa',

149 149 'ho_t_cc','ho0',... 'C_T_cc','cp0',... 'Dts_T_cc',1,... 'u_t_cc','u(phi>cc)',... 'v_t_cc','v(phi>cc)',... 'Q_T_cc','Qs'}; % Inteio mesh bounday settings equ.bnd.gpode = {{1;1;2;1;1}}; equ.bnd.ind = [1]; fem.equ = equ; % Bounday settings clea bnd bnd.cpode = {{1;1;2;1;1}}; bnd.shape = [1;2;3;4;5]; bnd.h = {0,{0,0,0,0,0;0,0,0,0,0;0,0,0,0,... 0;0,0,0,0,0;'-d(-T+T0_cc,u)','-d(-T+T0_cc,v)','-d(-T+T0_cc,p)',... '-d(-t+t0_cc,phi)','-d(-t+t0_cc,t)'},{0,0,0,0,0;0,0,0,0,... 0;0,0,0,0,0;0,0,0,0,0;'-d(-T+T0_cc,u)','-d(-T+T0_cc,v)',... '-d(-t+t0_cc,p)','-d(-t+t0_cc,phi)','-d(-t+t0_cc,t)'},0}; bnd.constf = {'test(-u)','test(-un_ns-vnz_ns)',0,'test(-un_nsvnz_ns)'}; bnd.gpode = {{1;1;2;1;1}}; bnd. = {0,{0;0;0;0;'-T+T0_cc'},{0;0;0;0;'-T+T0_cc'},... 0}; bnd.const = {'-u','-un_ns-vnz_ns',0,'-un_ns-vnz_ns'}; bnd.weak = {0,{'test(u)(sigmaDELTA(n_ns-cos(theta)nom)- ueta_ns/h)';... 'test(v)(sigmadelta(nz_ns-cos(theta)nomz)-veta_ns/h)'},0,0}; bnd.ind = [1,2,3,4]; bnd.va = {'K ns','eta_ns(2n_nsu+nz_ns(uz+v))',... 'T ns','-n_nsp+2n_nseta_nsu+nz_nseta_ns(uz+v)',... 'K_z_ns','eta_ns(n_ns(v+uz)+2nz_nsvz)',... 'T_z_ns','-nz_nsp+n_nseta_ns(v+uz)+2nz_nseta_nsvz',... 'ndflux_phi_ls','n_lsdflux_phi ls+nz_lsdflux_phi_z_ls',... 'ncflux_phi_ls','n_lscflux_phi ls+nz_lscflux_phi_z_ls',... 'ntflux_phi_ls','n_lstflux_phi ls+nz_lstflux_phi_z_ls',... 'ndflux_t_cc','n_ccdflux_t cc+nz_ccdflux_t_z_cc',... 'ncflux_t_cc','n_cccflux_t cc+nz_cccflux_t_z_cc',... 'ntflux_t_cc','n_cctflux_t cc+nz_cctflux_t_z_cc','u0_ns',0,... 'v0_ns',0,... 'w0_ns',0,... 'p0_ns',0,... 'f0_ns',0,... 'U0in_ns',1,... 'U0out_ns',0,... 'uvw_ns',0,... 'uwall_ns',0,... 'vwall_ns',0,... 'ww_ns',0,... 'k0_ns',0.005,... 'd0_ns',0.005,

150 150 'omega0_ns',10,... 'Uef_ns',1,... 'LT_ns',0.01,... 'IT_ns',0.05,... 'dw_ns','0.5h',... 'dwplus_ns',100,... 'U0_ns',0,... 'V0_ns',0,... 'p0_ent_ns',0,... 'Lent_ns',1,... 'p0_exit_ns',0,... 'Lexit_ns',1,... 'phi0_ns',0,... 'Vf0_ns',0,... 'theta_ns','0.5pi',... 'beta_ns','h',... 'n_ns','n',... 'nz_ns','nz',... 'Fbnd_ns',0,... 'Fbndz_ns',0,... 'd_ls',1,... 'n_ls','n',... 'nz_ls','nz',... 'N_phi_ls',0,... 'c0_phi_ls',0,... 'Dbnd_phi_ls',0,... 'q0_cc',0,... 'T0_cc',{0,'Tw0','Tsat0',0},... 'kbnd_cc',0,... 'd_cc',1,... 'n_cc','n',... 'nz_cc','nz',... 'N_T_cc',0,... 'c0_t_cc',{0,'tw0','tsat0',0}}; fem.bnd = bnd; % Scala expessions fem.exp = {'gphi','sqt(phi^2+phiz^2+eps)',... 'DELTA','gphi',... 'ho','ho_v+(ho_l-ho_v)phi',... 'miu','miu_v+(miu_l-miu_v)phi',... 'Fz','-ho0/F0^2',... 'nom','phi/(gphi+eps)',... 'nomz','phiz/(gphi+eps)',... 'int0','((^2/a^2)+((z-0)^2/b^2)-1)',... 'phi0','0.5(tanh(0.5int0/ee)+1)',... 'ho0','ho/ho_',... 'miu0','miu/miu_',... 'p1','pu0u0ho_',... 'gamma','gam',... 'ub','ub1/a_v0',... 'uint','mdot/(ho_vu0)',... 'k','k_v+(k_l-k_v)(phi>cc)',

151 151 'cp','cp_v+(cp_l-cp_v)phi',... 'k0','k/k_',... 'cp0','cp/cp_',... 'T1','TdT+Tsat',... 'mdot','delk_ldtgt/(hll0)',... 'usouce','uintgho/ho_ns',... 'gho','sqt(d(ho_ns,)^2+d(ho_ns,z)^2+eps)',... 'DEL','gphidee((gphidee)>0.1)',... 'Qs','-T(phi<cc)300',... 'gt','sqt(t^2+tz^2+eps)',... 'sigma','sigm(z<2)',... 'eta','miu0/nojac(e0)',... 'ho1','ho_nsho_',... 'miu1','miu_nsmiu_',... 'kappa','k0/nojac(pe0)',... 'theta','nojac(theta0)'}; % Desciptions clea desc desc.exp= {'eta','miu0/nojac(e0)','miu0','nondimensional Dynamic Viscosity','p1','Dimensional pessue','usouce','mdot(ho_lho_v)(phi+phiz)/(ho_vho_ns)','ho0','nondimensional Density','Fz','- (ho0-betat(t1- Tsat))/F0^2','phi0','flc2hs(int0,4ee)','k','0+k_l(phi>cc)','uint','0.5 (t>2.75)(0.25-z)(z<0.25)delta'}; fem.desc = desc; % Coupling vaiable elements clea elemcpl % Integation coupling vaiables clea elem elem.elem = 'elcplscala'; elem.g = {'1'}; sc = cell(1,1); clea bnd bnd.exp = {{{},'2pi(phi<=cc)',{}},{{},'2pisigDEL',{}},{{},... 'tflux_t_z_cc',{}},{{},{},'tflux_t_z_cc'},{},{},{},{},{},{},{},{},{},{},{},... {},{}}; bnd.ipoints = {{{},'4',{}},{{},'4',{}},{{},'4',{}},{{},{},'4'},{},{},{},{},{},{},... {},{},{},{},{},{},{}}; bnd.fame = {{{},'ef',{}},{{},'ef',{}},{{},'ef',{}},{{},{},'ef'},{},{},{},... {},{},{},{},{},{},{},{},{},{}}; bnd.ind = {{'1','4'},{'2'},{'3'}}; clea equ equ.exp = {{},{},{},{},{'1(phi<=cc)'},{'p1(phi<cc)'},{'1(phi>cc)'},{... 'p1(phi>cc)'},{'u_ns(phi<=cc)'},{'2pi(phi<cc)'},{... '1(phi<=cc)(gphi>1)(z<2.25)'},{'es_T_cc'},{'Qs'},{... '2pi(u+uintnom)(uint>0)ho_v'},{

152 152 '2pi(v+uintnomz)(uint>0)ho_v'},{... 'u(z>0.01)(z<=0.1)(phi>0.46)(phi<=0.54)'},{... '(z>0.01)(z<=0.1)(phi>0.46)(phi<=0.54)'}}; equ.ipoints = {{},{},{},{},{'4'},{'4'},{'4'},{'4'},{'4'},{'4'},{'4'},{'4'},{'4'},{... '4'},{'4'},{'4'},{'4'}}; equ.fame = {{},{},{},{},{'ef'},{'ef'},{'ef'},{'ef'},{'ef'},{'ef'},{'ef'},... {'ef'},{'ef'},{'ef'},{'ef'},{'ef'},{'ef'}}; equ.ind = {{'1'}}; sc{1} = {{},bnd,equ}; elem.sc = sc; geomdim = cell(1,1); geomdim{1} = {}; elem.geomdim = geomdim; elem.va = {'a_bb','fsig','ht0','ht1','a_v0','p_v0','a_l0','p_l0','ub1',... 'vb_0','a2','dt','dht','mdt1_1_v','mdt1_1z_v','u_cl0','cl0'}; elem.global = {'1','2','3','4','5','6','7','8','9','10','11','12','13','14',... '15','16','17'}; elem.maxvas = {}; elemcpl{1} = elem; fem.elemcpl = elemcpl; % Global expessions fem.globalexp = {'dp','(p_v/a_v)-(p_l-a_l)',... 'pv','p_v/a_v',... 'dv','100vb_0/(0.93(4/3)pia^3)',... 'mass','mas0l0^3',... 'x','l0',... 'y','zl0',... 'a_v','a_v0l0l0',... 'a_l','a_l0l0l0',... 'p_v','p_v0l0l0',... 'p_l','p_l0l0l0',... 'bbd','2sqt(a_bbl0l0/pi)',... 'bd','(6vb_0l0l0l0/pi)^(1/3)',... 'Fsigma','Fsigl0sin(theta0)',... 'Fbuoy','(ho_l-ho_v)gpi(bD^3)/6',... 'dee','2(sqt(eps+a_v0/pi)-sqt(eps+abs(a_v0-a2)/pi))',... 'EB','ho_vu0(HT0+HT1+dT+dHT)',... 'mas0','vb_0ho_v',... 'mdt1_1v','mdt1_1_v+mdt1_1z_v',... 'u_cl','u0u_cl0/(cl0+eps)',... 'CA1','(u_cl<=(0-u_clmx))aca+(u_cl>(0+u_clmx))ca',... 'CA2','(u_cl>(0-u_clmx))(u_cl<=(0+u_clmx))(ca+(acaca)(u_cl+u_clmx)0.5/(u_clmx))',... 'theta1','(ca(t<=0.5)+(ca1+ca2)(t>0.5))180/pi',... 'Fsig0','Fsigl0sin(ca)',... 'Fsig1','Fsigl0sin(aca)',... 'theta0','0.5(aca+ca)+tanh((fbuoy-fsig0)/(fsig1-fsig0+eps))(acaca)0.5',

153 153 'DCA','theta0180/pi'}; % Desciptions desc = fem.desc; desc.const= {'ho_v','1','we0','webe','k_l','@ 100 C','k_','0.5(k_v+k_l)','e0','eynolds','cp_l','@ 100 C','miu_l','1.04e-3[Pas]','hl','latent heat of vapoisation 2.260e6 [J/kg]','F0','Fouds','q','inwad heat flux','cp_v','@ 100 C','ho_l','1000','gam','1/u0','k_v','@ 100 C 0.025','miu_v','1.3e- 5[Pas]','p_atm','Atm pessue','ca0','initial Contact Angle 0.5(ca+aca)','cc','Inteface iso-contou 0 o 0.5'}; desc.globalexp= {'dv','vapo phase consevation','eb','enegy Bal','DCA','DCA Fsg','theta1','DCA U_cl','a_l','Aea of liquid egion','bbd','base Dia','u_cl','CL velocity','mass','mass of vapo bubble in 3D','mas0','Bbl Mas','Fbuoy','Lift Foce','mdt1_1v','Bbl Mas incs','dee','eqv Inteface Thickness','Fsigma','Contact Line Foce','bD','Bubble Dia','pv','Bubble Pessue','dp','Pes Diff (Young Laplace)','a_v','Aea of vapo egoin'}; fem.desc = desc; % ODE Settings clea ode clea units; units.basesystem = 'SI'; ode.units = units; fem.ode=ode; % Multiphysics fem=multiphysics(fem,... 'bdl',[1,3,4],... 'sdl',[]); % Extend mesh fem.xmesh=meshextend(fem,... 'dofvesion',1); % etieve solution fem.sol=flbinay('sol1','solution',flbinayfile); % Save cuent fem stuctue fo estat puposes fem0=fem; 153