Time dependant Thermo-mechanical Modeling including Phase Changes in Direct Drive Inertial Fusion Energy Targets

Transcription

1 Uvesty of Calfoa, Sa Dego UCSD-CER-06-0 me depedat hemo-mechacal Modelg cludg Phase Chages Dect Dve Ietal Fuso Eegy agets Kut-Jula Boehm Febuay 8, 006 Cete fo Eegy Reseach Uvesty of Calfoa, Sa Dego 9500 Glma Dve La Jolla, CA

2 UNIVERSIY OF CALIFORNIA, SAN DIEGO me depedat hemo-mechacal Modelg cludg Phase Chages Dect Dve Ietal Fuso Eegy agets A thess submtted patal satsfacto of the equemets fo the degee of Maste of Scece Egeeg Sceces (Mechacal Egeeg) by Kut-Jula Boehm Commttee chage: A. Ree Raffay, Cha Mak S. llack Geoge R. ya 006

3

4 he thess of Kut-Jula Boehm s appoved: Cha Uvesty of Calfoa, Sa Dego 006

5 able of Cotets Sgatue Page able of Cotets.... v Lst of Symbols... v Lst of Fgues. x Lst of ables.. x Abstact.. xv. Itoducto: Whee do we stad? hemal Loadg He Bubble Fomato tum Decay Dffuso Nucleus Fomato he Nucleus Fomato Model 3.5. Results fom the Nucleus Fomato Model Relevace to the Bubble Nucleato Model 0 4. he Model Ovevew Setup ad Results fom LANL Expemets Fom - D to - D Modelg a Bubble. 30 v

6 4.5. Appoxmatos to the Bubble ucleato model he Sphecal Bubble Model estg the Model agast Aalytcal Solutos Lqud to Sold Phase Chage Meltg ad Soldfcato Lqud to Vapo Phase Chage Bubble Nucleato ad Gowth Equatos Results fo the Cyldcal Model (pe ut heght) Results fo the Sphecal Model Compag the Results fom the Compute Model to the Expemetal Results fom LANL he Sold-Lqud Phase Chage he Bubble Nucleato Smulato Vayg dffeet Iput Paametes Summay Applcato of the Bubble Gowth Smulato to the Sphecal aget Geomety Icease Pessue due to Melt Laye Gowth Results fom the Bubble Gowth Model Coclusos Coclusos fom ths wok 80 v

7 Appedces A. he effects of Cyocotamats o the aget Reflectvty B. Mmum Allowable Iecto Veloctes 84 C. Paametc Studes o Lage aget 86 D. 3-D Model Set Up 89 E. he Multgd Algothm.. 90 F. he Heat Coducto Code cludg Sold- Lqud Phase Chage ad Bubble Nucleato G. he Dffuso Model 6 H. Aalyzg the LANL Melt Laye hckess 8 I. Plottg MatLab 9 Refeeces 0 Bblogaphy.... v

8 Lst of Symbols SYMBOL MEANING UNIS A Chage Aea m c Specfc heat of the coespodg mateal J p at costat pessue Kg K d Damete M d Melt laye thckess M D melt D o Dffuso coeffcet Dffuso costat s E Actvato Eegy (Chapte 3) J E Youg s Modulus (Chapte 7) Pa Vaable epesetg adal posto No dmeso fg Latet heat of vapozato J mole Ja Jacobs Numbe No dmeso k Boltzma Costat (Chapte 3) J K k hemal coductvty (Chapte 4,5,6,7) s L Latet heat of Fuso J mole M Molecula mass Kg mole moles melt Numbe of moles melted moles /m Posso s Rato No dmeso p g Pessue the gas Pa p f Pessue flud Pa p Pessue the lqud befoe meltg occus o melt (Chapte 7) Pa q & Icomg heat flux Q otal heat (Chapte 4) otal heat pe ut heght (Chapte 5) m s m m W m W W, m v

9 Radus of the sphee / cylde m R Uvesal gas costat m * Nucleus sze m Ie Radus of the aget m o Oute adus of the taget (Chapte 7) m out b Radus of the bubble m Chage adus m s(t) Locato of the sold lqud teface (Chapte 5) m t hckess of the shell (Chapte 7) m t me s empeatue K Satuato empeatue at pessue SA lqud K empeatue sde the bubble K vapo w empeatue of the eacto wall (Chapte ) K t Chage tme s Supeheat tempeatue K SA V Volume 3 m V Volume melted 3 m m v D lqud Molecula volume of D m 3 mole m 3 V Chage volume V Chage mola volume due to the phase 3 mola, phase chage m chage sold to lqud at P α themal dffusvty s κ Costat used Chapte 7 to elate the deflecto of the plastc ad the D shell No dmeso λ Coeffcet used equato No dmeso µ Reflectvty (Chapte ) No dmeso ρ Desty Kg 3 m θ Agle the gd No dmeso θ Chage agle No dmeso σ Stefa Boltzma Costat (Chapte ) W 4 m K σ Suface teso (Chapte 3-7) J m m v

10 SUBSCRIPS: Used o D Refeg to a D popety V, V, t, E, d, /m Radal posto the gd, k, t Ital tempeatue (Chapte 5) Agula posto the gd, k, θ l Lqud (Chapte 5),α, k melt Meltg tempeatue (Chapte 5) plastc Refeg to a plastc popety V, V, t, E, d, /m s Sold (Chapte 5),α, k SUPERSCRPIS: me step x

11 Lst of Fgues Fgue 3.: he tme depedat cocetato of 3 He due to ttum decay. Fgue 3.: he 3 He-cocetato as a fucto of the adal dstace fom the tap.. 6 Fgue 3.3: he pofles computed by the -D dffuso code fo dffeet ad of fluece Fgue 3.4: he 3 He accumulato a sgle evesble tap.. 9 Fgue 3.5: he elato betwee the umbe of 3 He atoms peset a tap ad a ucleus adus the lqud phase at kpa. 0 Fgue 4.: he expemetal setup used fo the LANL heatg expemets. 5 Fgue 4.: Dffeet modes of bubble gowth Fgue 4.3: Schematc appoxmato the bubble epesetato the oveall doma.. 34 Fgue 4.4: he smplfed bubble gowth model Fgue 4.5: As the bubble gows, the tempeatue feld eeds adustmet 38 Fgue 4.6: Modelg a bubble -D sphecal coodates allows fo a 3-D bubble 39 Fgue 5.: empeatue pofles fo two selected tmes the soldfcato pocess. 43 Fgue 5.: me depedet thckess of the sold laye. 43 Fgue 5.3: Bubble gowth s plotted compag a aalytcal soluto wth the umecal soluto cyldcal coodates 47 x

12 Fgue 5.4: he evoluto of the tempeatue feld aoud the bubble a cyldcal doma as a slde show Fgue 5.5: Bubble gowth s plotted compag a aalytcal soluto wth the umecal soluto sphecal coodates 5 Fgue 6.: he melt laye thckess calculated usg dffeet models supemposed wth the LANL measuemets Fgue 6.: he melt laye thckess s plotted fo a hghe heat flux. 55 Fgue 6.3: Dffeet sceaos fo dffeet tal taget tempeatues ad heat fluxes ae supemposed.. 56 Fgue 6.4: he tempeatue felds fo the two dffeet stages of bubble gowth 58 Fgue 6.5: Pctue take dug the LANL heatg expemet 59 Fgue 6.6: Compag the bubble gowths (LANL expemets ad umecal smulato) 60 Fgue 6.7: he tempeatue felds aoud the bubble at specfed tmes.. 6 Fgue 6.8: he supeheat equed fo dffeet sze ucle to gow to bubbles. 6 Fgue 6.9: Hghe lqud pessue flueces the oset of bubble gowth ad the gowth ate at late stages Fgue 6.0: he tempeatue feld aoud the bubble at hgh ad low lqud pessues. 64 Fgue 6.: he fluece of the tal tempeatue o bubble gowth 65 Fgue 7.: he pessue the taget due to the deflecto of the plastc shell dug melt laye gowth fo low ad hgh value of D Youg s modulus. 73 x

13 Fgue 7.: empeatue hstoes fo the thee oute most odes supemposed wth the step wse sg of satuato tempeatue 75 Fgue 7.3: Afte a shot tme, whch the bubble gows to a damete of about 8 µm, the fast gowth comes to a stop 76 Fgue 7.4: Allowg fo a sold to lqud phase chage, but ot allowg fo bubble gowth, povdes a lage mag to allow ethe lage heat flux of loge flght tme.. 78 Fgue A.: he eflectvty of a 00mco Au laye ad cyodeposts of H O ad CO. 83 Fgue B.: Ital taget ecto tempeatue as a fucto of ecto velocty. 85 Fgue C.: A lage adus ad thcke layes have bee poposed as a ew taget desg 86 Fgue C.: he tempeatue pofles fo the taget wth chaged geomety 87 Fgue C.3: he maxmum allowable heat flux fo tal taget tempeatue of 7.3 ad 6 K 88 x

14 Lst of ables able.: he heat load ad the dag foce Helum 4, tum, Deuteum ad Xeo at tempeatues of 000K ad 4000K 5 able.: Dffeet wall tempeatues wll mpose dffeet adato heat fluxes oto the taget... 6 able 3.: Dffeet values fo the tempeatue depedat dffuso coeffcet. 4 able 3.: Numecal values fo dffeet dffuso tmes ad ad of fluece. 8 able 6.: Numecal put paametes fo the show cases. 68 able 7.: Geometc paametes ad mateal popetes used the pessue buldup computatos.. 7 able B.: he heat load o the taget mposed by deuteum backgoud gas usg DSV able B.: Assumg a chambe pessue of mo at S, the followg mmum tal ecto tempeatues fo dffeet ecto veloctes ae detemed 84 x

15 ABSRAC OF HE HESIS me depedat hemo-mechacal Modelg cludg Phase Chages Dect Dve Ietal Fuso Eegy agets by Kut-Jula Boehm Maste of Scece Egeeg Sceces (Mechacal Egeeg) Uvesty of Calfoa, Sa Dego, 006 A. Ree Raffay, Cha A two dmesoal bubble ucleato mode was added to the pevously peseted themo-mechacal model used to defe the desg mag fo dect dve (DD) etal fuso eegy (IFE) tagets. ested o aalytcal solutos, the ew model successfully smulates heatg expemets o D tagets coducted at LANL. he 3 He peset the D due to the ttum decay gets tapped lattce stes evolvg to ucle bg eough to seve as ucleato stes fo heteogeeous bubble ucleato. Depedg o the sze of these ucle, a ceta lqud supeheat tempeatue s equed to ucleate bubbles. he lqud supeheat tempeatue futhe ceases as the pessue wth the taget duced by the plastc shell ses. he pevous equemet fo taget suvval was fo the tempeatue of the D to ema below tple pot of D (9.79K). If the exstece of a melt laye does ot xv

16 volate the symmety equemets o the taget fo successful mploso, whle the exstece of a bubble does, the pevous estcto ca be lfted allowg fo a melt laye to gow as log as the occuece of bubble ucleato ca be avoded. hs study shows that meltg ad bubble ucleato ca be tmely sepaated. Depedg o the 3 He ucleus sze, the pessue the taget ad the tal tempeatue of the taget, the maxmum allowable heat flux fo a gve suvval tme ca be ceased, allowg a less estcted desg mag fo the chambe desg. xv

17 . Itoducto Whee do we stad? hs study exteds the eseach wok o dect-dve (DD) etal fuso eegy (IFE) taget suvval stated at UCSD wth the wok of Ba Chstase ude supevso of D. Ree Raffay [], [], ad [3]. he key bass of ths eseach s the stct symmety equemets mposed by taget physcs fo compesso ad gto of the D fuel pellets usg multple lase beams. Wall adato ad eegy exchage fom the chambe gas ca sgfcatly mpact these symmety equemets patcula f phase chage occus. he phase chage behavo of D tagets s qute complex (e.g. the pesece of 3 He fom ttum decay ca fluece the oset of ucleato) ad eeds to be bette chaactezed though a combato of umecal modelg ad expemetal wok. he ma focus of the peset wok s the expaso of the pevous oedmesoal themo-mechacal model to a -D veso ode to moe accuately smulate bubble ucleato ad gowth, ad pedct ts effects. A moe detaled estmate of the mpact of phase chage (sold to lqud ad also ucleato) o the taget symmety would povde a bette bass to deteme whethe the pevously appled cosevatve estcto of matag the tempeatue of the taget below the D tple pot (9.79 K) could be elaxed whle stll satsfyg the taget physcs equemets. Allowg a hghe outsde tempeatue the D would ustfy the assumpto of the taget to be able to accommodate hghe heat fluxes, ad to be moe

18 themally obust to chages the chambe evomet dug ecto. I paallel to the umecal smulato of bubble gowth, the heat loads o the taget eed also to be chaactezed fo dffeet chambe desgs. Futhe, the effects of the ttum decay to 3 He eed to be assessed, as t s suspected that the 3 He ucle mght ehace D bubble fomato the fst place. Sce the behavo of cyogec D ude heat loadg has geeally bee uexploed the past, t s mpotat that umecal modelg of phase chages as well as themal ad mechacal esposes of the taget s compaed to expemetal esults to establsh accuacy ad elablty of the model. he peset wok cludes a compaso of umecal esults to aalytcal solutos fo cotolled cases fst, followed by a compaso of umecal esults to expemetal esults fom Los Alamos Natoal Laboatoy (LANL) o D heatg. hese expemets use cyldcal tagets: so, the smulato code was fst wtte cyldcal coodates, ad the tasfomed to sphecal coodates to model IFE tagets dug ecto a chambe. Wth ths wok, we ted to bette defe the desg age fo IFE tagets ad gve gudeles as to what pellet desg wll be equed depedg o the ecto velocty, backgoud gas pessue ad wall tempeatue.

19 . hemal Loadg I ths secto the heat load o the taget dug ecto the chambe s calculated fo dffeet sceaos ad codtos assocated wth vaous chambe desgs. Befoe the effects of the heat flux ca be modeled, ts magtude has to be foud. As ecommeded by Chstase, DSV, a commecal Mote Calo gas flow smulato code [4], s used to estmate the eegy exchage due to the teacto of the taget wth the chambe gas []. Pevous esults teded to focus o elatvely hgh chambe gas destes (e.g. coespodg to ~50 mto at a S=300K) equed fo wall potecto compact chambes (~6 m adus). Recetly, the HAPL pogam has bee lookg at the possblty of avodg the use of a potectve chambe gas by cosdeg lage chambes (~0- m adus). he absece of a potectve gas elaxes the costats fom taget heatg ad placemet, ad avods ay potetal mpact o lase popagato. he chambe evomet dug ecto would the cosst of the taget bu emats cosstg mostly of He, D ad. he desty of these chambe costtuets dug ecto would deped o the vacuum pumpg pefomace, but s estmated as less tha ~-0 mto at S. able. summazes the esults fo a uppe boud desty case of 0 mto at S (coespodg to a umbe desty of 3.4x0 0 m -3 ) fo vaous costtuet gases at tempeatues of 000K ad 4000K, espectvely, ad fo ecto veloctes of 00 m/s ad 400 m/s, espectvely. Clealy, the ages elevat to IFE smulatos, the ase tempeatue fom 000 K to 4000 K has a hghe effect o the heat flux tha asg the ecto velocty of the taget fom 00 m/s to 400 m/s. We ca also see that a 3

20 4 heave gas wth lage molecules wll ceate a smalle heat flux tha a lghte gas wth small molecules. he adato heat tasfe ca be foud by equato (.): q = (.) ad 4 ( µ ) σw hs heat load would be vey sgfcat f the eflectvty of the taget suface was low. Usg the poposed Au-Pd laye, the eflectvty of the taget ca be ceased to as much as 96%. ( µ = 0.96) [Appedx A] [].

21 5 able.: he heat load ad the dag foce ae lsted ths table fo Helum 4, tum, Deuteum ad Xeo at tempeatues of 000K ad 4000K ad a pessue of 0 mo at S. he small effect of the heat load whe ceasg the speed s show by gvg the heat flux at veloctes of 00 m/s ad at 400 m/s Heat Load Chat fo Dffeet Gases, empeatues, ad aget Speeds Gas emp. Speed Head Load He 000K 4000K 00m/s 0.55 W/cm F - Dag E-4 N 400m/s 0.75 W/cm 0.35 E-4 N 00m/s 4.3 W/cm 0.8 E-4 N 400m/s 5.0 W/cm 0.66 E-4 N Paamete M= 4 g/mol m = E-6Kg D=.86 E-0 µ = K 00m/s 0.65 W/cm E-4 N 400m/s 0.80 W/cm 0.30 E-4 N M= 3 g/mol m = E-6Kg 4000K 00m/s 4.9 W/cm 0.6 E-4 N 400m/s 5.4 W/cm 0.59 E-4 N D=.86 E-0 µ = 0.8 D 000K 4000K 00m/s 0.75 W/cm E-4 N 400m/s.0 W/cm 0.4 E-4 N 00m/s 6.0 W/cm 0. E-4 N 400m/s 6.5 W/cm 0.48 E-4 N M= g/mol m = 0.33 E-6 Kg D=.86 E-0 µ = K 5m/s 0.75 W/cm 4000K 5m/s 4.8 W/cm 000K 00m/s 0.4 W/cm 0.49 E-4 N 400m/s 0.6 W/cm.3 E-4 N M= 3 g/mol m = 0.8 E-4 Kg Xe 4000K 00m/s 0.9 W/cm 0.97 E-4 N 400m/s.0 W/cm 4.0 E-4 N D= 3.8 E-0 µ = 0.8

22 6 he effects o the eflectvty due to cyogec dust patcles ae dscussed Appedx A. he coce addessed Appedx A deves fom the hgh adsoptvty value of cyogec gas (CO ad HO), whch mght accumulate o the taget suface dug hadlg ad could lead to a lowe eflectvty value of the taget suface. able.: Dffeet wall tempeatues wll mpose dffeet adato heat fluxes oto the taget. Fo the two lmtg cases of 000 ad 500 K wall tempeatue the espectve heat flux due to adato s gve. Also, the effect of a educed eflectvty of the taget suface s llustated. Reacto wall Radato heat flux µ = 0.96 Radato heat flux µ = 0.9 tempeatue 000 K 0. W/cm 0.4 W/cm 500 K. W/cm.4 W/cm he heat fluxes fom both adatve ad covectve effects must be added to deteme the total heat flux o the taget. he absece of a potectve gas mght eable the taget placemet equemets to be met wth lowe ecto veloctes, thus allowg fo smple mechacal ecto systems [5]. I suppot of ths, a paametc study was doe to deteme the mmum ecto velocty equed fo the D to stay below the tple pot tempeatue (9.79 K) fo dffeet codtos. he esults ae descbed Appedx B ad the key fdgs summazed below. Usg a chambe pessue of mo at S ad a wall tempeatue of 000K two lmtg cases (backgoud gas tempeatue of 000K ad 4000K) wee vestgated to coelate the ecto velocty ad the tal taget tempeatue equed fo the taget tempeatue to ema ude D tple pot at the ed of the flght though a 6.5m

23 7 adus chambe. Fo the lowe heat flux case (000 K backgoud gas) t was foud that the ecto velocty ca be educed to below 50 m/s, whle keepg the tal taget tempeatue easoably low (6K). Iceasg the velocty to values hghe tha 00 m/s shows lttle effect o the ecessay tal tempeatue. I the hghe heat flux case (4000 K backgoud gas), tal tempeatues below K wee computed ecessay to educe the ecto velocty to 50m/s, whch s a vey sgfcat educto compaed to the 7.5K equed f the taget fles at 400 m/s. Geeally, a hgh tempeatue backgoud gas wll equed fast ecto speeds whle lowe backgoud gas tempeatues wll allow fo slowe ecto speeds. Sce the ecto velocty has oly lttle fluece o the heat flux, the backgoud gas tempeatue s the most elevat paamete fo the heat flux ad such fo taget suvval.

24 3. 3 He - Bubble Fomato he effect of Helum solds, especally metals has bee aalyzed by may scetsts the past two decades. he peset wok focuses o the effect of the 3 He, mplated by the tum decay, heatg expemets of D ad ts elevace to IFE taget suvval. Obsevatos at the Los Alamos Natoal Laboatoes [6] suggest a close elatoshp betwee the cocetato of 3 He the fuel pellet ad the appeaace of bubbles dug heatg expemets. he peset chapte explas the physcs behd these obsevatos. 3. tum Decay tum decays to 3 He followg equato (3.) wth a half lfe of.3 yeas. As a esult of ths compaatvely low half tme, a cosdeable amout of 3 He accumulates the D sold lattce wth a few hous afte layeg the tagets. 3 3 H He electo (3.) Fo a powe plat desg t s estmated that a tme of 4 0 hous wll be equed fo taget hadlg betwee layeg ad ecto to the chambe [7]. Dug ths tme the decay of ttum evtably causes two kds of defects the lattce: atomc dsplacemet damage ceatg a vacacy ad/o a testtal atom, ad the ceato of a foeg elemet ( 3 He) whch has the tedecy to pecptate to bubbles [8]. I the lteatue, these defects ae eseached maly fo 4 He dffeet metals; the behavo of 3 He a D lattce has ot bee aalyzed. 8

25 9 Nevetheless we beleve that the physcs behd the dffuso mechasms, the ketcs of bubble ucleato, the chages of mechacal popetes ad the atomc popetes of the 4 He the lattce ca be appled to 3 He a D lattce accodgly. hs thess cocetates o the dffuso mechasms ad ucleus fomato because of the elevace to bubble ucleato heatg expemets. he fst physcal obsevato s that wth evey sgle ttum atom decayg, oe 3 He atom s ceated whle the eegy eleased by the decay (= 8.6 kev) would be eough to ceate a testtal ste. At ths pot we leave t ope whethe the et esult s the ceato of a vacacy ad a testtal o whethe the 3 He atom oly occupes the spot of the decayed ttum atom the lattce. 3. Dffuso As soo as oe 3 He atom s peset the lattce, t wll stat movg aoud by adom umps to the eghbog lattce stes. hese adom umps esult a ceta dstace that the atom wll move the lattce afte a ceta tme (o a ceta umbe of umps). hs movemet s called sold state dffuso [9], [0]; t ca happe to a testtal atom as well as a substtutoal atom, ad t s hghly tempeatue depedet. It s mpotat to otce that testtal dffuso s geeally vey fast compaed to the vacacy o substtutoal dffuso. khaus [8] pots out the stog bdg of the helum atoms to vacaces. hs meas that a testtal 3 He atom wll move aoud feely the lattce, utl t falls to a vacacy. Oce the atom occupes the vacacy t wll eque a much hghe

26 0 eegy to dslodge the atom fo futhe dffuso. he eegy equed to move the helum atom fom the lattce spot to a testtal aga s vey lage, whch suggests that t wll ema occupyg the lattce ste. Futhemoe, thee s space fo moe tha oe 3 He atom oe lattce vacacy, whch esults the possblty of aothe testtal 3 He atom fallg to the vacacy, f the vacacy les o the helum atom s adom path []. hs esults the vacaces actg lke taps; as a esult of these taps, He-3 stats buldg clustes, as thee wll be moe ad moe He-3 atoms accumulatg the vacaces []. 3.3 Nucleus Fomato Now that we have establshed the fact that thee s a ceasg cocetato of 3 He (wth tme) the D ego ad that the 3 He atoms ca dffuse to taps, we ca assume that the taps wll fom to ucle ove a suffcetly log tme peod. Accodg to khaus [8], the sze ad umbe desty of these bubble ucle ae depedet o the tempeatue ad the helum poducto ate. o be moe pecse, the bubble paametes deped o the dffuso ate ad the helum cocetato, wth bubble fomato occug by cocuet dffuso ad clusteg of 3 He. It seems easoable to assume that the hghe the oveall cocetato of 3 He, the hghe the 3 He cocetato tapped clustes []. At the same tme, the hghe the tapped 3 He cocetato, the fewe the fomato of ew bubbles. hs sot of self lmtg mechasm s esposble fo the bubble cocetato to ted to a costat value ove a suffcetly log peod of tme. Both ths chaactestc tme ad the

27 umbe desty of clustes wll deped o the tempeatue of the medum, the eegy levels of dffeet sze clustes ad othe ukow paametes lke the stegth of the sk mposed by the taps. I addto, ga boudaes ad othe lage lattce defects ca affect the dffuso ad the tap stegth. Dffuso alog ga boudaes s extemely fast, whle lage lattce defects wll tap gas molecules wth a hghe tappg eegy tha that of the lattce vacaces [9]. N.M Ghoem, S. Shaafat, et al. developed a model cludg all the dffeet adsopto ad emsso eeges of atom ad atom clustes to a ucleato model fo helum dffuso dffeet metals []; esults fo 3 He a D lattce ae ot avalable yet. A model smla to the oe developed by Ghoem ad Shaafat fo 3 He dffuso D would go beyod the scope of ths thess. A smplfed model has bee developed to gve a ough dea of the bubble szes occug ou case. he assumptos ad smplfcatos ae peseted the ext secto. 3.4 he Nucleus Fomato Model A pecse calculato of the umbe desty ad ucleus sze of 3 He gas a D lattce would eque a tme-depedet umecal model wth all statstcal possbltes, fa too extesve to be wth the scope of ths thess. Istead, the followg model s poposed to povde a ough fst estmate. he smplfyg assumptos ad physcal ustfcatos ae explaed hee.

28 he tme-depedet umbe desty of 3 He the doma ca be calculated fom the decay of the ttum the doma. he poducto of the gas wth the lattce ca be assumed to be ufom, sce the decay happes at adom. Fg. 3.: he tme-depedet cocetato of 3 He due to ttum decay s plotted hee. Notce that fo the tme fame we ae teested (up to a few days) the behavo s almost pefectly lea (slope:.6069 x 0-6 moles/hou). A ewly fomed 3 He atom wll mgate though the sold utl t falls to a tap (a vacacy fo example). he calculatos assume a homogeeous dstbuto of taps ad focuses o the sphecal doma suoudg the tappg ste. Sce we focus o a small pat of the taget doma (a sphecal poto of a few mcos adus), the shape of the oveall doma ( the taget case the doma would be a hollow cylde, the tap would be off- ceteed) s elevat, ad we ca

29 3 apply sphecal coodates to the small poto we ae spectg. he cocetato o the emost pot of the doma s set to be zeo, sce we assume that the atom umps to the vacacy, ad does t come back out. he assumpto of havg a sk o evesble tap at a adom pot the doma ca be ustfed followg the deas peseted the sectos above. Ref.[], [3], ustfes the possblty of a cotuously gowg cluste by epotg that the closest lattce atom gets pushed to a testtal posto, f the umbe of 3 He atoms the cluste becomes too lage. hs ceates a d vacacy whch povdes moe oom fo moe 3 He atoms. W.D. Wlso ad, C.L. Bsso ad M.I. Baskes [3] cofm that afte the ceato of a lattce vacacy 5-8 atoms ca coglomeate the lattce. As moe gas atoms fall to the vacacy tap, adacet lattce atoms ae pushed to a testtal posto povdg space fo the gowg cluste. hey also pedcted a cotuous gowth of the ucleus as moe ad moe atoms dffuse to the vacacy ad call the tap satuable. N. Kawamua et alt. [4] cofm that most to the 3 He ceated by the ttum decay wll be tapped the lattce of the foze D whch suppots the above descbed dea of satuable taps. Ref. [] also suggests that the assumpto of a mmoble vacacy captug testtal 3 He atoms as they dffuse though the lattce ca be defeded. It s also metoed by the autho, that the tap effectveess ceases as the bubble ucleus cotues gowg. Ref. [0] cofms a hgh moblty of He atoms eve at low tempeatues (although that case coppe s used as a lattce mateal).

30 4 Based o these statemets, t seems easoable to assume a smple model wth a tap the cete of the doma actg as a evesble sk tem. Oce we have a estmate of the umbe of 3 He atoms whch have mgated the tap, a estmate of the adus of the fomed 3 He ucleus ca be made. A key paamete estmatg the tme fo 3 He to coglomeates the ucleus s the dffuso coeffcet of 3 He the sold D lattce. I the lteatue, the followg values wee publshed fo the dffuso coeffcet: able 3.: Dffeet values fo the tempeatue depedat dffuso coeffcet E D = D0 exp k (3.) Slvea [5] (H) Soues [6] (H) Soues [6] (D) E, [K] 00 ± k m D 0 [ ] 3E-3.4E-3 3E-4 s D (@ 8K),.a..a 9.95 E- m [ ] s Fo the smple -D model, zeo cocetato s assumed as bouday codto the cete (tap locato =0) ad zeo cocetato gadet as assumed at the oute bouday assumg equdstat taps.

31 5 Oce the model etus the cocetato pofles at each tme step, we ca compute the umbe of 3 He atoms the tap by tegatg the cocetato the whole doma. 3.5 Results fom the Nucleus Fomato Model Followg the smplfcatos peseted above, the -D sold state dffuso code was mplemeted. Fgue 3. shows the 3 He cocetato pofle as t chages evey 5 mutes. he oveall cocetato of 3 He ceases accodg to the ttum decay, whle the tap effectvely pulls a umbe of 3 He atoms to the vod. By tegatg the gee shaded aea ad multplyg t wth the coespodg volume, the umbe of 3 He atoms the tap ca be computed.

32 6 Fgue 3.: he 3 He-cocetato as a fucto of the adal dstace fom the tap s plotted fo 300 s tme steps. he adus of fluece fo ths plot s 0 mcos. he gee shaded aea dcates the amout of 3 He that dffused to the tap 4 hous. Next, the effects of the adus of fluece ae aalyzed. By usg dffeet szes of ad of fluece (o doma szes), ad plottg the espectve fal pofles afte 4 hous, fgue 3.3 was ceated. We ae teested estmatg, how may 3 He molecules accumulated the tap fou hous. he fst obsevato s that fo ad of fluece lage tha 6 o 7 mcos, the pofle emas uaffected ove the tme peod cosdeed (4 hous); at a dstace lage tha 7 mcos fom the cete of the tap, the model pedcts a costat cocetato of 3 He accodg to the ttum decay.

33 7 Fo smalle ad of fluece the pofles chage. Physcally, mposg a small adus of fluece meas that two sks ae close eough fo them to affect each othe. he espectve pofles ca be exteded by a mo mage to descbe the adacet tap. Fgue 3.3: he pofles computed by the -D dffuso code ae daw hee fo dffeet ad of fluece. If the taps le closely togethe, the oveall 3 He cocetato the doma wll ema low, sce the 3 He gets tapped the vods quckly afte t was poduced. If the taps ae fa apat fom each othe, they do t fluece each othe ad the amout of 3 He tapped wll appoach a costat value. he mmum dstace betwee two adacet taps fo them ot to affect each othe s elated to the chaactestc dffuso legth. able 3. lsts the values of 3 He atoms tapped a vod usg dffeet ad of fluece ad vaous dffuso tmes. Lage values of the adus of fluece esult

34 8 a ceasg umbe of 3 He atoms the taps. Fo ad of fluece lage tha a ceta value, the umbe of 3 He atoms the tap appoaches a costat value (fo each dffuso tme). he value afte whch the umbe of 3 He atoms the tap emas costat s elated to the dffuso coeffcet ad the chaactestc dffuso legth. Fgue 3.4 shows gaphcally the esults fom table 3.; the umbe of 3 He atoms pe tap s plotted as a fucto of the adus of fluece fo dffeet dffuso tmes. Notce that fo loge dffuso tmes, lage ad of fluece ae equed to each a costat value of 3 He atoms the tap. able 3.: Numecal values fo dffeet dffuso tmes ad ad of fluece. Radus (mcos) hou Radus (mcos) hous 4 hous 8 hous 8 hous.00e-06.4e05.00e E E05.57E E06.30E-06.34E05.00E-06.49E E06.0E07.73E07.70E E E-06.9E E06.86E07 8.3E07.00E E E-06.38E06.7E07 4.7E07.69E E E E-06.40E06.8E E07.66E E E E-06.4E06.30E E E E E E-06.4E06.3E07 7.0E07 4.8E E E E-06.40E06.3E07 7.E E E E E-06.40E06.3E07 7.6E07 5.0E E E05.00E-05.40E06.3E07 7.7E07 5.9E E E05.50E-05.40E06.3E07 7.7E E08.00E E05.00E-05.40E06.3E07 7.7E E08.50E E05.50E-05.39E06.3E07 7.6E E08

35 9 He-3 Accumulato a Sgle Ievesble ap usg Dffeet Rad of Ifluece ad mes Numbe of He-3 atoms the ap.e09.e08.e07.e06.e05 0.0E00 5.0E-06.0E-05.5E-05.0E-05.5E-05 Radus of Ifluece (m) Afte 8 hs Afte 8 hs Afte 4 hs Afte hs Afte hs Fgue 3.4: he 3 He accumulato a sgle evesble tap usg dffeet ad of fluece ad dffuso tmes ae plotted hee. he loge the dffuso tme, the futhe the dstace betwee two taps has to be fo the two ot to affect each othe. Based o these esults, a value fo the adus of a 3 He ucleus ca be assged to the umbe of 3 He atoms a tap afte the phase chage fom D sold to lqud has occued by applyg the deal gas law. hese esults ca be see fgue 3.5. Assgg a value the sold state s dffcult, because the pessue the tap s ukow, whle the pessue the lqud phase s easy to calculate.

36 0 Fgue 3.5: hs fgue elates the umbe of 3 He atoms peset a tap to a 3 He gas ucleus adus the D lqud phase at kpa that ths tap would volve to. he fgue also dcates what sze ucleus to expect afte 4 ad 8 hous. 3.6 Relevace to the Bubble Nucleato Model Whe elatg the 3 He gas accumulato the sold D lattce to a ucleus sze the lqud phase, we do that atcpato of modelg a lqud to vapo phase chage. hs phase chage s expected to happe accodg to classc ucleato theoy. I ths theoy, a ucleus of a ceta sze s suouded by a lqud of a ceta pessue ad tempeatue. If the tempeatue of the lqud s ceased above a ceta value, a themodyamcally ustable equlbum s ceated, esultg a bubble to gow out

37 of a pe-exstg ucleus. If the pe-exstg ucleus s ceated by o-codesable gas bubbles held suspeso the lqud, as they do ou case, the ucleato mode s called heteogeeous ucleato [7], [8]. Colle states that the pesece of dssolved gas educes the supeheat equed to gow a bubble out of a ucleus of a ceta sze. Equato (3.3) elates the pessue sde the bubble to the pessue the lqud. We obseve a cease pessue the bubble as compaed to the lqud pessue due to suface teso. p g σ = p f (3.3) b he supeheat equed to poduce a ustable equlbum of a ucleus wth adus b a lqud of ufom tempeatue s descbed ( a smplfed mae) by equato (3.4): SA = R J fg SA M σ p f b (3.4) (Paametes epeseted by symbols ae lsted the omeclatue secto) Clealy, f we wat to avod bubbles to ucleate a system, smalle ucle ae desed, sce the maxmum allowable supeheat s ceased. I the followg pats of ths thess, the elato betwee the oset of bubble ucleato heated systems ad the pesece of ucle of a ceta sze a supeheated lqud wll be vestgated. he esults fom ths chapte wll help us estmatg what sze of tal bubble ucleus to assume.

38 We ca coclude fom ths chapte, that the decay of ttum to 3 He affects the bubble ucleato heatg expemets ad wll also affect taget suvval f phase chage s allowed.

39 4. he Model 4. Ovevew he eed fo a accuate model to descbe the mechacal ad themal espose of a IFE taget has bee pevously dscussed at legth [], []. Reseach ad modelg wok so fa focused o the cosequeces of phase chages wth the taget assumg a oe-dmesoal geomety. hs appoach leads to modelg a lqud ad vapo laye at the oute edge aoud the taget. As fa as the sold lqud phase chage s coceed ths s a vald smplfcato, sce we do expect a melt laye to fom symmetcally aoud the taget, but modelg the lqud to vapo phase chage oe dmeso oly, dsagees wth the bubble ucleato theoy ad expemetal heatg obsevatos. he Los Alamos Natoal Laboatoy (LANL) coducted a sees of heatg expemets past the tple pot o foze D tagets 004 ad peseted the esults at the HAPL meetg Jue 004 at UCLA [6]. he peset wok wll fst smulate these expemets ad pedct a smla behavo by umecal modelg the cyldcal geomety used the laboatoy ad the taslate these esults to sphecal coodates to establsh a accuate desg mag fo the D tagets. 3

40 4 4. he Setup ad Results fom LANL Expemets he sees of expemets elevat to ths study wee efeed to as dectheatg of sold D layes by the authos of the Jue 004 HAPL pesetato [6]. I that sees of tests, a cyldcal taget of 4mm damete ad 0.4mm heght was exposed to dect heatg fom the outsde usg a electcal col. he comg heat flux was teded to be W/cm, but was estmated to have a 0% eo mag due to heat loss though the outsde teface of the cols (Fg. 4.). Seveal sets of expemets wee coducted usg dffeet equlbum tempeatues (6K, 7K, 8K, ad 9K) ad equlbum tmes (4hs, 8 hs). Pctues take dug the 00ms heat pulse clealy show bubble ucleato statg at dffeet oset tmes ad wth a dffeet bubble desty o the outsde laye of the taget. Obsevatos fom that study dcate that thee ae elatos betwee the equlbum tme ad the bubble desty as well as betwee the equlbum tme ad the oset of bubble gowth. We suspect that the foudato fo these elatos les the tme depedat buldup of the 3 He cocetato. hese coectues wll be aalyzed by a -D umecal heat tasfe model ad the esults fom the -D dffuso model fom the pevous chapte.

41 5 Fgue 4.: he expemetal setup used at the LANL expemets s show hee. he dak blue ectagles epeset the coss secto of the D cylde. Clealy the heatg cols ae mouted eclosg the oute suface. he pctues show the LANL pesetato ae take alog the axs of symmety. 4.3 Fom -D to D Sce the LANL expemets mpose a costat heat flux aoud the cyldcal taget, a oe dmesoal model ca accuately be used to descbe the behavo of the taget utl bubble ucleato occus. It has bee show the pevous secto 3.6 that bubbles wll ucleate, f the lqud s suffcetly supeheated ad f a ucleus of a ceta sze s peset the lqud (see also [7], [8] ). It has also bee show that these ucle ca occu due to the ttum decay to 3 He (secto 3.5). If we

42 6 assume a ucleus ste of a ceta sze to be peset, we ca calculate the supeheat equed fo a bubble to ucleate. Afte usg a oe dmesoal model to smulate the heat tasfe utl that supeheat has bee eached at a gve dstace fom the oute edge, the effect of bubble ucleato wll be smulated by swtchg to a -D model. he heat coducto ad sold lqud phase chage wll be modeled -D, sce the expaso to the secod dmeso oly ceases the computg tme equed to acheve esults. he coducto equato the eads as follows [0]: ( ) = k k k c t p δ δ δ δ δ δ ρ δ δ (4.) Note that the oly dffeece wth the sphecal coodate system ad equato (4.) s that the last tem s multpled by two the sphecal case. o accout fo the apd chage themal popetes at tempeatues the cyogec ego, ad to popely model the phase chage, equato (4.) cludes tempeatue vaat coeffcets. Applyg ufom spatal ad tempoal dscetzato to equato (4.), t esults the followg equato: : ( ) ( ) ( ) ( ) ( ) = p k k k k k k k k k c t ρ (4.)

43 7 hs system of equatos ca be solved effcetly usg the homas algothm to secod ode accuacy space ad fst ode accuacy tme wth o stablty estctos []. he subscpts ote the vaable s locato space, whle the supescpts ote the tempoal locato. he bouday codtos mposed ths case ae a costat heat flux o the outsde ad zeo gadet at the e adus ( 0 ). Fo the zeo-gadet bouday codto at the e adus of the cylde, the tempeatue at the secod ode s smply coped ad set as the value o the fst ode. he oute edge bouday codto follows equato (4.3). δ q& = k (4.3 a) δ q& = k (4.3 b) Solvg fo : q& k = (4.3 c) Equato (4.3 c) wll be plugged to equato (4.) fo the oute ode [0]. o model the sold to lqud phase chage, the appaet c p method s adapted fom the pevous model []. he dea behd ths method les the egeeg appoxmato that the phase chage ad the ump specfc ethalpy afflated wth t happes ove a tempeatue age athe tha as a step wth a fte slope. Oce t comes to modelg bubble gowth, we eed to swtch to a two dmesoal model. hs adds complexty to the umecal smulato, sce ow the

44 8 equatos ca o loge be epeseted a t-dagoal matx, but a peta-dagoal system, whch caot be solved usg the homas algothm. Numeous methods ae avalable to solve two ad thee dmesoal systems (ADI, Spectal, ), [Pozkds] but the most effcet oe ths case s a teatve scheme followg the Gauss Red Black Algothm []. hs scheme coveges quckly sce the umecal soluto does t vay lagely fom oe tme step to aothe. he eve qucke covegg Multgd method was mplemeted the code [], but fals whe the coeffcets do ot vay smoothly, whch s the case wth phase chages. It s show Appedx C, but emas a academc execse fo ths applcato. A stetched gd s mplemeted the code to ccumvet the poblem of havg a small eough gd spacg o the oute suface of the taget (whee phase chages occu, ad bubble gowth ad steep tempeatue gadets ae obseved) whle keepg the umbe of total gd pots wth a easoable computatoal mag. A hypebolc taget fucto s used to stetch the gd adal decto. he bubble wll be modeled the cete of the pe-shaped doma; a hypebolc se fucto s used to stetch the gd agula decto. hs complcates the set of equatos magally but ceases the oveall pefomace ad accuacy temedously. o udestad the equato (4.4), t s useful to kow that the dffeet gd spacgs ad locatos ae saved a vecto at the begg of the code ad ust efeed to as follows:

45 9 Radus at the th pot dffeece adus betwee the th ad the th pot θ dffeece agle betwee the th ad th pot he set of equatos, usg a -d hollow cylde wth tempeatue-depedet coeffcets ad a stetched gd ae wtte equato (4.4 (a) ad (b)). Cyldcal Coodates (4.4a): ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) =,,,,,,,,,,,,,,,,,,,,,, ) ( ) ( 0.5 ) ( ) ( ) ( 0.5 ) ( ) ( ) ( ) ( 0.5 ) ( ) ( ) ( ) ( 0.5 ) ( ) ( ) ( 0.5 ) ( 0.5 ) ( ) ( ) ( p p k k k k k k k k k k k k k k t c k k t c θ θ θ θ θ θ θ θ θ θ ρ θ θ θ θ ρ

46 30 Sphecal Coodates (4.4b): ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) =,,,,,,,,,,,,,,,,,,,,,,,, ta ) ( ) ( ) ( 0.5 ) ( ta ) ( ) ( ) ( 0.5 ) ( ) ( ) ( ) ( 0.5 ) ( ) ( ) ( ) ( 0.5 ) ( ) ( ) ( 0.5 ) ( 0.5 ) ( ) ( ) ( p p k k k k k k k k k k k k k k k k t c k k t c θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ ρ θ θ θ θ ρ he mposed bouday codtos ema the same the adal decto. I the axal decto peodc bouday codtos ae used, so techcally we could use a spectal method the agula decto, but ths has ot bee mplemeted the peset wok. Now we have to face the challegg task of modelg a 3-dmesoal sphecal bubble a - dmesoal cyldcal doma.

47 3 4.4 Modelg a Bubble he supeheat equed fo a bubble of a ceta sze to gow ca be calculated by usg equato (4.5) [7]: vapo R sat vapo σ SA = (4.5) * M p fg f If we assume a ucleus of a ceta sze to be peset the melt laye (due to 3 He decay), the equed supeheat ca be detemed by the model as t s dscussed so fa. Howeve, modelg the bubble ucleato ad gowth s a challegg task, sce the bubble s sphecal geomety, ad t s off cete of ou oveall cyldcal doma. It has bee show though [8], that bubble gowth the tme- ad sze-fame elevat to ths study s heat flux estcted, whch s coveet fo ou heat dffuso model. Accodg to S. Va Stale [8], the bubble gows exactly as fast as the heat ca be delveed to the lqud vapo teface, whee t s used to accout fo the latet heat equed by the mass flux of vapo to the bubble. Assumg that the vapo sde the bubble s at themodyamc equlbum, we ca pedct the tempeatue sde the bubble to be the satuato tempeatue at the pessue sde the bubble. We ca elate the pessue the bubble wth the pessue of the suoudg lqud by equato (4.6). p b σ = p f (4.6) b Clealy, the tempeatue the bubble ad the supeheat equed to gow the bubble ae closely elated (see eqs (4.5) ad (4.6)).

48 3 hs suggests that the bubble ca be epeseted by a heat sk mposed to the doma. he tempeatue the bubble s lowe tha the tempeatue the suoudg lqud, dawg themal eegy to the suface. Numecally, we ca add a heat sk tem by assgg a ceta tempeatue to a abtay pot the doma, whch s lowe tha the tempeatue of the eghbog pots. Physcally, by choosg the locato of such a pot the gd, the locato of the ucleus ad as such the pot fom whch the bubble wll stat gowg s detemed. Oce the ecessay supeheat to gow a bubble out of a ucleus of a ceta sze s eached, the tempeatue at that pot gets dopped, statg the heat sk. hs s ustfable sce we assume a ucleus flled wth 3 He to be peset the supeheated lqud. Oce the tempeatue the suoudg lqud s hghe tha the satuato tempeatue at the pessue the bubble, D stats evapoatg to the bubble, makg t gow. Oce the bubble gows the pessue the bubble dops eve futhe (see eq 4.6) loweg the tempeatue as well. he eegy whch the heat sk absobs wll be coseved the system, as eegy fom of latet heat s equed to evapoate some D to make the bubble gow. As such, the eegy flowg to the heat sk must be tacked ad equated to the eegy equed evapoate eough D to gow the bubble fom oe tme step to aothe. hs eegy ca be calculated usg the heat flux equato (4.7a) ad the aea ove whch ths heat flux acts (4.7b). Whe applyg ths equato to the gd, we eed to ealze, that the fst tem the backets of eq. (4.7b) s the heat tasfe the adal decto, actg ove the chage agula decto whle the secod tem epesets the heat tasfeed agula decto ove the chage adus.

49 33 = δθ δ δ δ k q & (4.7 a) ( ) ( ) = R R t k legth ut Q θ θ,,,, (4.7b) ( ) fg bubble g fg R p legth ut legth ut A legth ut Q = = π ρ (4.8) Fgue 4.: Dffeet modes of bubble gowth ae show ths fgue. O the left, the bubble gows outwad adally symmetc fom a pot the mddle of the doma. O the ght, we assume the bubble ucleus to be close to the oute edge of the cylde, ad the bubble gowth happeg wad. he obvous physcal obsevato suggests that the eegy s used by the system fo the lqud to vapo phase chage at the teface of the bubble. Because of the dscetzed atue of ou doma, bubble gowth ca oly be modeled step wse. We popose two modes of bubble gowth, show Fgue 4.. he bubble gowth follows the gd ethe adally symmetc outwad (left) o wad fom the oute edge

50 34 of the doma as show o the ght. Moe volved modes could be mplemeted to model a moe sphecal shape, but the peseted esults used these two modes. We ae ow able to calculate the aea that each bubble coves o the gd at each step. Notg that the aea of the bubble o the gd s elated to the volume by ut heght of the cylde, we ca calculate the amout of heat ecessay (pe ut heght) to gow the bubble fom oe step to the ext. Kowg the heat equed to gow the bubble o the oe had ad the heat flux to the bubble fom the heat sk model o the othe (see Fgue 4.3), we ae ow able to model bubble gowth by matchg the two. he oly fee paamete left fo adustmet s the tme step. Fgue 4.3: ths fgue shows the schematc appoxmato of how a bubble s epeseted the oveall doma. Notce that the heat tasfe to the bubble comes fom the tempeatue dffeece adal ad agula decto.

51 35 he umecal smulato cossts of the followg steps: Fst, the gd s ceated. Sce we have a fxed mesh, the bubble szes that wll be epeseted by the model ae detemed by that. Secod, the model us -D fo ufom heat fluxes utl the supeheat at the pot specfed fo bubble ucleato s eached. he, we swtch to a -D model. hs model fst calculates the heat equed fo bubble gowth at that tme step. Afte that a teatve scheme s used to fd the ght sze of the tme step to coseve the eegy of the system. Oce the tme step s foud, the bubble gows fom oe step to the ext; the tempeatue of the pots of the ew bubble s adusted to the satuato tempeatue of the pessue the bubble, ceatg a lage heat sk. Seveal pots must be cosdeed usg ths appoach to accuately compae to the eal physcs of bubble ucleato. 4.5 Appoxmatos to the Bubble ucleato model he fst poblem we ecoute deves fom the dscetzed (step wse) gowth of the bubble we model as compaed to the smooth, gadual gowth eal physcs. I the model, a heat sk s appled o the doma. he pofle elaxes ove the tme step accodg to the heat dffuso equato. he gadet flattes ad subsequetly the heat tasfe to the bubble slows dow as the pofle elaxes. he poblem we ae ow facg s whch pofle to use fo the heat flux to the bubble, the oe mmedately afte the heat sk s defed at the begg of the tme step, o the

52 36 flatte oe at the ed of the tme step. Both pofles ae atfcal ceatos ogatg fom the step wse gowth of the bubble. I ths model we popose to use half the heat flow to the bubble accodg to the tal pofle, ad half of the heat flow accodg to the fal pofle. he secod poblem to be hadled also esults fom the step wse gowth ad coces the cosevato of mass the system. he desty of the vapo the bubble s about 000 tmes lowe tha the desty of the lqud. By ceatg the bubble, the volume of the whole cyldcal doma ceases. Fo small bubbles a lage doma, ths cease volume s eglgble, but ths mposes some lmtatos o the model fo lage bubble szes. Oe aspect mpactg the bubble gowth s the way the tempeatue feld aoud the bubble chages as the bubble gows. Bubble gowth pushes the lqud mass suoudg the bubble out. he mass the dstbuted the ove a lage adus. hs meas that a pot whee the tempeatue has bee computed at the pevous tme step moves to a dffeet locato as the bubble gows. We have to calculate the ew tempeatue of the pot epeseted the gd (Fgue 4.4). o get the exact place tempeatue ( ),,3,4,5 cosevato of volume (Fgue 4.5). **,,3,4,5 at whch to compute the, we eed to accout fo ths adal effect ad fo the

53 37 Fgue 4.4: he smplfed bubble gowth modeled the code makes the bubble sze ump fom oe gd sze to aothe. I dog so, we eed to move the pofle adally outwad. he volume suoudg the bubble o the left betwee the gd spacg s dffeet tha the volume betwee the gd spacg o the ght due to adal effects. Volume betwee obseve : 0 ad Volume betwee 0 ad 0 = = tempeatues. We the use lea tepolato betwee the espectve pots to get the **,,3,4,5,3,4,5,6 0 = (4.9) ** (,,3,4,5 ) (,,3,4,5 ),,3,4,5,,3,4,5 = ( ) ( ),3,4,5,6,,3,4, 5,3,4,5,6,,3,4,5 (4.0)

54 38 he empeatue Adustmet fo the Gowg Bubble Pofle at th bubble step Pofle at th bubble step ** ( ) = ( ).5.5 ** ( ) = ( ) empeatue Radus Radus ** ** 0 Fgue 4.5: As the bubble gows, the tempeatue feld eeds adustmet. he tempeatue at a specfc adus away fom the bubble s computed by tepolato betwee the eghbog pots at the pevous tme step. Whe modelg the bubble ucleato by ust assgg a lowe tempeatue value at the gd pot detemed to epeset the bubble, the model wll be uable to esolve ay heat tasfe though the bubble. Nethe wll the model be able to expess a tempeatue gadet though the bubble. By lookg at the physcs, t ca be easoed, that the themal esstace though the bubble s much hghe tha though the lqud suoudg t, suggestg, that the heat flowg though the bubble wll have o effect compaed to the tempeatue dstbuto aoud the bubble. Lookg at the mea fee path of the gas molecules sde the bubble, we ca ustfy the costat tempeatue assumpto wth the bubble.

55 he Sphecal Bubble Model Whe applyg the above model to sphecal coodates, a 3-D bubble ca be modeled wth a -D code. Symmety aoud oe axs makes ths possble as wll be show below: Fgue 4.6: Whe modelg a bubble -D sphecal coodates, physcs allows us to model a 3-D shape, by otatg the (blue) shape aoud o axs of symmety. he bouday codtos at the otatg axs must be symmetcal, whle we mpose (zeogadet) Newma bouday codtos at the oute edge of the otatg agle.

56 40 he model follows the same scheme descbed the pecedg secto, the oly dffeece beg the followg dffeet fomulas fo heat dffuso ad heat flux: = δθ δ δ δ k q & (4.a) ( ) ( ) ( ) ( ) = R R R t k Q θ π θ θ θ π,,,, (4.b) Note that equato (4.0) we tegate the heat flux ove a complete aea of the bubble as opposed to tegatg ove a legth ad leavg the othe dmeso to be ut legth. I dog so, we have to compute the heat equed to gow the bubble also as the total heat stead of heat pe ut legth. ( ) bubble bubble fg R p V Q = = π ρ (4.)

57 5. estg the Model agast Aalytcal Solutos Befoe we ty to aalyze the LANL expemetal esults o pedct the behavo the IFE taget case, the model eeds to be checked agast aalytcal solutos fo well-defed cases. 5. Lqud to Sold Phase Chage Meltg ad Soldfcato I ode to establsh accuacy of the sold lqud phase chage as t s modeled the code, a aalytcal soluto had to be foud, ad the umecal ad aalytcal esults eeded to be compaed. As has bee pevously poposed, the appaet c p method s used to model the phase chage. hs dea has bee mplemeted ad tested the sphecal code [], but emaed utested the cyldcal case. Ozsk [3] deved a aalytcal soluto fo the case of a heat sk the cete of a cyldcal doma. As the heat sk -suouded by tally supeheated lqud- s tued o, a cylde soldfes gowg symmetcally outwad adal decto. hs soluto s gve by the followg set of equatos: Fd the coeffcet λ usg equato (5.) ( λ ) ( ) Q kl t m λ α s exp exp λ α sρsl 4π λ α = α (5.) s l E α l Use λ to compute the thckess of the sold laye ad the tempeatue pofles the two phases usg equatos (5.) (5.4): 4

58 4 s() t ( ) / λ = (5.) t α s s Q (, t) = E E( λ ) m 4πk s 4α t s (5.3) l E αl t m (, t) = E t λ α 4α lt s (5.4) E s the expoetal tegal ad exp the expoetal fucto he followg fgues (5. ad 5.) show a plot of selected pofles followg equatos (5.3) ad (5.4) fo selected tmes at a ceta heat stegth ad compae those wth the esults of the code applyg the bouday codtos coespodg to the put paametes of the aalytcal soluto. Numecal poblems evolve o both eds of the doma, sce we eed to mpose a close to fte slope at the cete ad a ukow slope o the oute edge. he =0 bouday codto s esolved by computg the heat flux betwee two pots at a ceta dstace fom each othe (accodg to the gd) aalytcally fo a ceta sk stegth, assumg that t stays costat ove tme, ad mposg ths slope betwee the fst two pots the umecal gd. he oute bouday codto s set to be zeo heat flux, whch s acceptable fo small tmes. o be exact o that sde we would eed to compute the heat flux as a fucto of tme at the oute edge, ad mpose that heat flux the model betwee the two outemost pots.

59 43 Fgue 5.: hs fgue shows the pofles fo two selected tmes the soldfcato pocess. Note the dscepacy due to the bouday codtos whch could ot be mposed a staght fowad mae. Fgue 5.: hs fgue shows the tme depedet thckess of the sold laye the soldfcato pocess. Both solutos le wth a 5% eo mag of each othe, whch s acceptable fo ou applcato.

60 44 5. Lqud to Vapo Phase Chage Bubble Nucleato ad Gowth Accodg to Va Stale [8], bubble gowth ca be dvded to thee dffeet stages. I the fst stage (tal mode t 0 ), the bubble gowth s estcted by hydodyamc eta effects. I ths stage the bubble gowth s lea wth tme, accodg to the Raylegh soluto. As the bubble gows, t has to acceleate the suoudg lqud as t pushes t adally outwad. I the secod stage (asymptotc modet ), the descpto of the bubble gowth ca be smplfed to a heat dffuso model. he bubble gowth s popotoal to the squae oot of tme. he bubble gows as fast as the heat equed to gow the bubble ca be delveed to the teface, whee t s used to evapoate the lqud. he thd stage s a taset stage, whch les betwee the pevously metoed modes. Both of these estctg factos ca mpact bubble gowth. Fo the tme scale ad bubble szes we ae teested, we ca safely assume that the bubble gowth s heat flux estcted. hs meas that the bubble wll gow as fast as the heat ecessay to gow the bubble ca dffuse to the teface. As dscussed above, we ca easly model ths sceao, sce we ae aleady smulatg heat dffuso ou model; as a esult, we ca use the fomato fom the code ad apply t to bubble gowth.

61 Equatos Bubble ucleato ad gowth occu wheeve a suffcetly lage lqud supeheat ad a suffcetly lage ucleus the lqud phase [7], [8] ae peset. Let us stat by examg the behavo of vapo bubbles dug ucleate bolg: the gowth of a fee, sphecal vapo bubble a tally a ufomly supeheated lqud of homogeeous composto. Followg the devato fom the Bosakovc heoy fo Isobac Heat Dffuso Cotolled Gowth [8], the bubble behavo ca be descbed by equato (5.5) ad (5.6). 0.5 αl t b () t Ja (5.5) π Ja = ρ lqud cp, lqud ρ vapo fg SA (5.6) Usg the defto fo the themal dffusvty, we expad the equato to (5.7): α b ) lqud = k ρ c k lqud SA () t () t 0. 5 ρ gas p fg lqud α lqud π (5.7) he values used fo ths calculato ae extacted fom Soues [6]. hey ead: α ) lqud m =7.8E-8 s W SA =.00 K k lqud = 0. m K fg J = 360 mol mol m ρ gas =30 3

62 46 he followg smplfcatos have to be mplemeted the code: - he lqud tempeatue the whole doma s tally Kelv ( SA =.00K). - he tempeatue the bubble s 0.00 K at all tmes. I applyg ths bouday codto, we eglect the cease tempeatue due to the cease pessue by suface teso effects (eq. (4.6)). he effect of the pessue ad tempeatue cease ca be mplemeted the umecal code fo expemetal smulatos, but wll be omtted hee to model the aalytcal soluto. Also, the eal value of the gas tempeatue at the suoudg pessue (tple pot tempeatue 9.79 K) caot be used, sce, fo modelg easo we assume that the sold lqud phase chage occus ove a dscete tempeatue age betwee 9.79 ad 9.99 K []. I ode to always accout fo the latet heat equed fo the sold to lqud phase chage, we have to use the lowest possble value of the lqud tempeatue, whch s 9.99K. - he desty of the gas s computed by usg the deal gas law ad s appoxmated to 33 mol/m 3. I the actual case, as the pessue ad the tempeatue sde the bubble chage, the gas desty wll follow these chages accodg to the deal gas law.

63 Results fo Cyldcal Model (pe ut heght) Ude the pevously stated assumptos, the gowth ate s calculated ad plotted Fg 5.3. he fgue 5.3 also shows the gowth pedcted by the umecal model, usg the same assumptos. Cosdeg the appoxmatos the bubble model ad the heat flux causg bubble gowth, the modelg esults compae easoably well to the aalytcal esults. he model ca the be appled to smulate the LANL expemetal esults wth some degee of cofdece. he model shows good coguecy of the plots whe dffeet gd stetchg coeffcets ae used. By chagg the gd stetchg coeffcets, the bubble szes ae chaged. he moe the gd s stetched, the smalle s the dffeece bubble sze fom oe sze to aothe..0e-04 Bubble Gowth Ufom Supeheat 9.79 K -.00 K Cyldcal Coodates.00E E-05 adus (m) 6.00E E-05.00E-05 Aalytcal Soluto mddle gd coase gd fe gd 0.00E E E E E E E E E- 04 tme (s) Fgue 5.3: he gowth of the bubble s plotted ths fgue both usg the umecal appoxmato ad a aalytcal esult usg the pevously descbed bouday codtos. Clealy, the two plots ae smla shape ad value, gvg us cofdece modelg assumptos.

64 48 Smla accodace could be eached usg dffeet supeheat values the doma. he adus of fluece o the tempeatue bouday laye s foud to be suffcetly smulated by the model f the closest 0 o 0 pots ae updated afte each teato. he slde show fgue 5.4 shows the developmet of the bubble ove tme. Notce that the gadet aoud the bubble emas vey steep dcatg a apd heat tasfe to the bubble suface.

65 49 me: 4.86 E-6s Radus:.94 E-6m me: 3.5 E-5s Radus:.88 E-6m me: 8.53 E-5s Radus:3.97 E-5m me:.47e-4s Radus:5.66 E-5m me:.3e-4s Radus:8.69 E-5m me: 3.5E-4s Radus:0.5 E-5m Fgue 5.4: ths slde show pesets the evoluto of the tempeatue feld aoud the bubble a pe shaped cyldcal doma. he ufom supeheat suoudg the bubble was tally K. We see that the themal bouday laye aoud the bubble emas small leadg to steep gadets ad a hgh heat flux to the bubble.

66 50 Iput values used the code (HEADER) (fo futue efeeces): Cs=.0 (coase).0 (mddle) 3.5 (fe)! stetchg facto adal decto Cst=.3 (coase).8 (mddle) 4.0 (fe)! stetchg facto theta decto g%= 00! Gd calculated to have (close to ) quadatc g%t= 75! patches whee we wat to model the bubble = 00;! # of b teatos at vaous pots Rad= 0.00;! oute adus of the D shell Rad_e= 0.006; p= 3.45; total_agle= p/8 bubb_loc= 35! # of udes the bubble ucleates sde the! oute m of the cylde bubble_steps= 5! umbe of bubble steps suf_tes= 3.9E-3! suface teso of lqud D (temp. dep) pess_lqud= 000! pessue lqud (wll chage late, but! good fo ow) lat_heat= 350! latet heat of vapozato (temp. dep) gas_cost= 8.34! gas costat 5.5 Results fo Sphecal Model As dscussed the pevous secto, the sphecal model ca smulate a 3-d bubble usg the symmety aoud oe axs. As a esult, the oly appoxmato we have ou code s the o-sphecal shape of the bubble ad the smplfyg assumpto of utlzg a effectve heat tasfe to the bubble based o a combato of the tal ad fal tempeatue pofles at each tme step. he osphecal shape should ot have a geat mpotace as the eo duced by t s expected to be small. he sphecal model also shows a good smulato of the aalytcal bubble gowth, eve f dffeet gd szes ae chose. Lke evey umecal smulato, the fe the chose gd s, the close the soluto follows the aalytcal esult.

67 5 Fgue 5.5 shows the bubble gowth pedcted by ou model fo the same case as fgue 5.4. he K ufom supeheat ad the same geometc popetes ae used; we have smply chaged the heat dffuso equato ad the heat flux equato fom cyldcal to sphecal coodates. As a gd s chose whch models vey small bubbles, the umecal soluto follows the aalytcal plot closely (gee cuve). But f we ae teested bgge bubbles ad loge tmes, a coase gd ca be chose to speed up the code. I that case, a lage eo s obseved tally, but fo bgge tmes, the bubble adus pedcted by the model agees wth the bubble szes a aalytcal calculato would gve. Bubble Gowth Ufom Supeheat 9.79 K -.00 K Shpecal Coodades 6.00E E-05 bubble adus (metes) 4.00E E-05.00E-05.00E-05 Aalytcal Soluto fe gd mddle gd coase gd 0.00E E00.00E E E E-05.00E-04.0E-04 tme (secods) Fgue 5.5: Good accodace betwee the aalytcal ufom supeheat equato ad the bubble gowth pedcted by the model s show hee. Whee exactly the dvegece comes fom ad how t ca be eased could be futhe aalyzed, but fo ou desed accuacy ths model seems easoable.

68 6. Compag the Results fom the Compute Model to the Expemetal Results fom LANL Afte havg successfully tested the umecal smulato agast aalytcal solutos fo cases wth smplfed bouday codtos, we used the model to smulate the behavo of the LANL cyldcal tagets dug heat flux expemets. hs modelg wll help to expla the physcs behd the LANL expemetal obsevato, ad to ga sght o the themo-mechacal behavo of the sphecal IFE tagets. 6. he Sold Lqud Phase Chage Please see Secto 4. fo the schematcs ad descpto of the LANL expemetal setup. he expemetal values wee gve to us dectly fom LANL [4]. As a fst step we focused o the sold- lqud phase chage. he measued thckess of the melt laye the expemet ad the coespodg esults fom the code ae plotted fgue 6.. he expemetal esults ae based o post-test examato of pctues of the D whose age of ucetaty teds to be hghe fo the melt layes. Clealy thee s a lage dscepacy betwee the calculated (umecal) ad the obseved (expemetal esults). he appled bouday codtos ae: 5

69 53 Ital tempeatue - 8K Heat flux ( q & ) - W/cm Oute Radus - Ie Radus - mm.533 mm Fom fgue 6., t ca be see, that the modelg esults show a slope smla to that of the expemetal esults, but shfted dow by about 50 µ m o the local axs. Appedx H shows a computato of the heat equed to gow a melt laye accodg to the LANL obsevatos ad compaes the esult to the total heat duced to the taget by the heat flux. Fom ths smple eegy calculatos based o the latet heat of D, t seems clea that the melt laye caot each the epoted thckess wth the epoted tme whe a W/ cm heat flux s mposed. I addto, the model dcates that a tme of about 3 ms s equed fo the D to each ts tple pot (ths s a petty elable esult fom the pevously vefed themal coducto pat of the code). hs tme, expemetal esults dcate a ump the melt laye thckess to about 50 µm wth a addtoal ms, whch s dffcult to beleve. I seach fo possble explaatos o the appaet dscepacy the expemetal esults, the followg possble factos esulted fom dect dscusso wth LANL [4]: - he melt depth measuemets cay a ucetaty due to dffcultes detemg exactly whee the melt laye teface s actually located. he covex D ce suface, though whch the pctues leadg to these esults wee take dstots the pctue ad could lead to a mscocepto about the exact locato of the sold to lqud teface.

70 54 - Due to the complexty of the heatg appaatus, the exact heat flux mposed o the taget dug the expemet could ot be exactly detemed. Whle the electcal powe to the esstos ca be detemed qute accuately, heat losses at the back ad eds of the expemetal set-up ad the possblty of addtoal heatg due to lght souces ca esult sgfcat ucetates. hese could cause a udeestmate of the heat flux by up to a facto of (as a uppe lmt but pobably close to a facto the age of.5). - No- ufomty of the melt laye could also be a possble easo affectg the expemetal esults, but ths was thought to be athe ulkely based o LANL obsevatos. Next, we ted to fd a heat flux that would gve a smla melt laye thckess as the LANL expemets showed. he esults ae plotted fgue 6.. he esults dcate that the heat flux should be ceased by a facto of about 3 ode fo the modelg esults to epoduce the expemetal obsevatos; ths s well ove the maxmum ucetaty facto of o the heat flux ad dcates that aothe facto has to be play, whch s lkely to be the ucetaty the melt laye measuemets.

71 55 Melt Laye Compaso melt laye q = W/cm^, t = 8K, BRIAN sphee meltlaye thckess (m) melt laye q = W/cm^, t = 8K, IMM sphee melt laye q = W/cm^, t = 8K IMM cylde tme (s) Melt Laye LANL Fgue 6.: the melt laye thckess calculated by dffeet models avalable (pevous ad ew sphecal model (-d ad -d) as well as cyldcal case) ae plotted hee. Notce that thee s oly a small dffeece betwee the cyldcal ad the sphecal coodate geomety. hee s a clea dffeece betwee the umecal smulatos ad the LANL obsevatos. Matchg Melt Laye hckesses.0e-04.00e-04 melt laye thckess (m) 8.00E E E-05.00E-05 Melt Laye LANL melt laye thckess heat flux =3 W/cm^ 0.00E tme (s) Fgue 6.: he heat flux s ceased to get close to the expemetal esults. If we apply a heat flux thee tmes as bg as pevously assumed, the esults become close. If that heat flux was mposed though, we should obseve a melt laye ove tmes shote tha 0 ms whch have ot bee epoted so fa.

72 56 Fgue 6.3 supemposes the dffeet sceaos. Dffeet tal tempeatues ad heat fluxes have bee appled. he soluto look the closest fo 8 K ad 4 W/ cm o 9 K ad 3 W/ cm fo tal tempeatue ad heat flux espectvely. Futhe compaso betwee dffeet sets of expemetal data would be equed to bette udestad whee exactly the dscepaces come fom.

73 57 6. he Bubble Nucleato Smulato Fo the tal bubble ucleato aalyss, we assumed a heat flux of.00 W/ cm, cosstet wth the value of 0.9 W/ cm epoted by LANL (.e. wth o coecto fo possble ucetates the heat flux). he ma obsevato to be ecoded whe compag the bubble gowth fom the LANL heatg expemets to the ufom supeheated soluto s that the bubbles gow much slowe the expemetal setup. As we show ths chapte we obseve two dffeet stages of bubble gowth dug the heatg expemet. I the fst stage exploso-lke bubble gowth s obseved. he tempeatue the bubble s much lowe tha the lqud tempeatue all aoud t povdg a lot of heat fo vapozato ad bubble gowth. Dug ths stage the bubble gowth ate s close to the ufom supeheated case. I the secod stage, the tempeatue the lqud o the sde of the bubble opposg the wall s lowe tha the tempeatue the bubble. I ths stage some of the heat flows though the bubble to the lqud behd t leavg less eegy to gow the bubble. Fgue 6.4 llustates the tempeatue pofle aoud the bubble at the two dffeet stages.

74 58 Fgue 6.4: O the left sde the bubble s show the fast stage of bubble gowth. he lqud all aoud the bubble s at a hghe tempeatue tha the bubble, causg t to be a stog heat sk. O the ght sde, the pctue zoomed close to the bubble, showg that the heat flux to the bubble (the back -ght sde of ths pctue) ca be dffeetated fom the heat flux out of the bubble (fot-left of the pctue), as the gadet s evesed fo the two cases. Whe smulatg the bubble gowth usg the model as descbed ad tested the ufom supeheated case ad compag that data wth the obsevatos fom the LANL expemets, ceta assumptos about the codtos have to be made. hese clude the pessue the lqud dug the heatg expemet, the ucleus sze peset the lqud befoe the heat pulse s stated, stegth of the heat pulse ad the tal tempeatue. Whle the heat flux ad the tal tempeatue ca be take fom the LANL epot, educated guesses have to be made about the ucleus sze ad the lqud pessue. Based o the esults fom the smple 3 He dffuso model (chapte 3), two dffeet ucleus szes have bee tested:.6 um ad 0.4 um fo the 8 hous ad 4 hous layeg tme espectvely. he lqud pessue was assumed to be 000 Pa, slghtly hghe tha the satuato pessue at tple pot.

75 59 he LANL measuemets gve two sets of bubble dametes: oe s the bubble sze measued by lookg though the lqud D, ad the othe oe measued by lookg though the sold laye. At ths pot t s mpotat to kow that these measuemets ae qute ough ad ca oly be used to gve us a appoxmate dea of the bubble szes. Fgue 6.5 shows oe of the mages used to measue the bubble damete, fom whch the dffculty of makg a exact measuemet ca be appecated [4]. Fgue 6.5: hs pctue was take dug the LANL heatg expemet. Fames lke ths ae used to evaluate the bubble damete at each tme step. It ca be see that thee ae dffeet sze bubbles thoughout the doma, oe of whch beg selected ad measued as the bubble gows. Also vey emakable s the double vew of the bubbles, oce though the sold ad oce though the lqud (as dcated).

76 60 Fgue 6.6 shows a plot of the bubble damete as a fucto of tme fo the two dffeet ucleus szes, ad Fgue 6.7 shows the tempeatue dstbuto aoud the bubble at dffeet stages (as dcated Fgue 6.6). he esults show Fgue 6.6 suggest that the smalle ucleus sze eques a hghe supeheat tempeatue to gow to a bubble, causg t to stat gowg at a late pot. Meawhle, sce t s suouded by lqud at a hghe tempeatue, t gows vey fast utl the codtos fo slowe gowth ae establshed. he esults fom the model fo the assumed case seem to smulate easoable well the expemetal esults. Howeve, t should be oted that thee ae ucetates the bubble measuemets (pehaps of the same ode as fo the melt laye measuemets), whch ca shft the esults. Moeove, eve ths case, the slope of the modelg esults fo bubble gowth s compaable to that of the expemetal esults. Compag Bubble Gowth (LANL expemets ad umecal smulato).4e-04.e-04 Fame 4 bubble =8K, p= 000 Pa, R_m=.6E-6m bubble damete (metes).0e-04 Fame 3 8.0E-05 Fame 6.0E E-05 Fame.0E E tme (secods) meltlaye heat flux = W/cm^ LANL BUBBLE UPPER LANL BUBBLE LOWER bubble = 8K, p = 000 Pa, R_m = 4E-7m Fgue 6.6: he two dffeet stages of bubble gowth ca be see: vey fast tal bubble gowth (the les appea to be vetcal), ad slowe gowth as the tempeatue of the bubble opposg the wall s hghe tha the tempeatue of the lqud that s faces.

77 6 Bubble Gowth Bubble damete =.79 um; me = ms; Bubble damete = um; me = 38.84; empeatue (Kelv) Agula Decto (metes) Bubble damete = 75.9 um; me = 64.09ms; Fgue 6.7: the tempeatue pofles dug bubble gowth ae show at the dcated tmes ad espectve bubble dametes. Notce the coelato betwee the tempeatue feld the fst two pctues ad the coespodg fast gowth ates as opposed to the tempeatue felds the last two sldes matchg the codtos fo slow gowth. hese teds dcate that the assumptos used the model ae easoable. Notce that, accodg to the code, the bubble gows deeply to the sold laye. Aalyses of sold D aoud the P dcate ts low stegth, whch could fe that the bubbles could gow by pushg the lqud the melt laye as well as the soft sold (a ted also dcated by LANL s tal obsevatos but equg moe accuate expemetal cofmato). Radal Decto (metes) Bubble damete = 3.7 um; me = 97.87ms;

78 6 I ode to bette udestad the fluece of the ucleus adus o the oset of bubble gowth, we plotted the supeheat equed fo bubble gowth as a fucto of the ucleus adus. Fgue 6.8 shows the esults. Supeheat vs. Nucleus Sze (lea plot) Hgh supeheat ego Supeheat (Kelv) low supeheat ego Supeheat E00.00E-06.00E E E E E-06 Nucleus Radus (metes) Fgue 6.8: the supeheat equed fo dffeet sze ucle to gow to bubbles s plotted. Clealy, oce the ucleus s geate tha.5 mcos, the supeheat equed s less tha oe degee, wth oly slght chages fo lage ucle. Fom ths fgue we ca see, that a adus bgge tha.6 mco s oly gog to have lttle effect o the supeheat tempeatue (ad cosequetly the oset tme of bubble gowth assumg a ceta heat flux). If the bubbles wee smalle tha 0.4 mcos, vey hgh supeheat tempeatues wll be equed leadg to eve late oset tmes fo bubble gowth. We expect the bubble gowth such cases to be extemely fast utl bubble szes smla to the szes fom eale osets wll be eached.

79 Vayg dffeet Iput Paametes A teestg study would be to vestgate the fluece of the suoudg lqud pessue o bubble gowth. A cease of the satuated vapo tempeatue the bubble would follow a cease of the lqud pessue, causg the bubbles to ucleate late tme. Sce at pessues hghe tha tple pot pessues t s possble to have a lqud coole tha the bubble tempeatue o the sde opposg the comg heat flux, we expect the bubble to ema smalle. Fgue 6.9 shows the fluece of ceased pessue o bubble ucleato ad gowth. Fgue 6.9: Hghe lqud pessues fluece both the oset of bubble gowth as well as the sze at whch the bubble gowth slows dow ad the speed at whch the gadual bubble gowth happes.

80 64 Because of the mpotace the taget case, the tempeatue felds fo dffeet pessues ae show Fgue 6.0. Notce the hghe tempeatue the bubble allowg fo a heat flux leavg the bubble wthout the eed of peetatg the sold laye. Heat flows to the Bubble Sold Phase Heat flows out of the bubble Melt Laye Hgh pessue case Low pessue case As the heat flowg out of the bubble gets close to the heat flowg to the bubble, the explosve bubble gowth stops abuptly, ad a moe gadual gowth ate sets. Fgue 6.0: he tempeatue felds of the thee phases at dffeet pessue sceaos ae show hee. he bubble damete at whch bubble gowth stagates s smalle tha the melt laye the left case (hgh pessue), ad lage tha the melt laye o the ght sde (low pessue). As a fal step the tepetato of the data sets esultg fom the LANL expemets, we eed to aalyze the elato betwee the tal tempeatue ad the oset of bubble gowth. Fgue 6. shows the bubble gowth fo the.6 um bubbles applyg vaous tal tempeatues.

81 65 7.E-05 Bubble Gowth applyg dffeet Ital empeatues R_m =.6 E-6 m bubble =6K, p= 000 Pa, R_m =.6E-6m Bubble Damete (metes) 6.E-05 5.E-05 4.E-05 3.E-05.E-05.E-05 0.E me (secods) bubble =7K, p= 000Pa, R_m=.6E-6m bubble =8K, p= 000 Pa, R_m =.6E-6m bubble =9K, p= 000Pa, R_m=.6E-6m Fgue 6.: As the tal tempeatue s loweed, the oset of bubble gowth s delayed. he speed of gadual bubble gowth s also foud to decease wth deceasg tempeatues (fom the modelg esults). 6.4 Summay he followg obsevatos ca be made fom the aalyss of the LANL expemets. As descbed above (secto 6.) thee s a lage dscepacy betwee the esults of the model ad the expemetal esults fo the melt laye thckess. he ma easos behd these dscepaces ae beleved to og fom dffcultes measug ad/o detemg the exact melt laye locato. he dastc cease heat flux equed to match the epoted melt laye umecally s beleved to be ulkely. Fo the bubble gowth, the model delves good explaatos fo some of the expemetal obsevatos. he sees of pctues shot dug the heatg expemets show sudde occuece of bubble of a sgfcat sze at a ceta tme afte the stat

82 66 of the heat pulse. he tempoal esoluto of the pctue sees s too low (4ms betwee two shots) to esolve the behavo of the bubble befoe t eaches athe lage szes (40 mcos). Fom the model we kow, that the bubble ucleato ad tal gowth happes vey fast (wth a few mcosecods). Because the bubbles ucleate a evomet chaactezed by a steep tempeatue gadet acoss the doma, we kow that stagato bubble gowth occus whe the tempeatue gadet o the sde opposg the comg heat flux s egatve, causg most of the comg heat to flow though the bubble athe tha beg etely used to gow the bubble. As a esult, two dffeet modes of bubble gowth ca be dstgushed, fast tal ucleato ad slowe gadual gowth. he LANL data ca oly show the gadual gowth, but the sze of the suddely occug bubbles suggests fast tal ucleato ad gowth. At pessues slghtly hghe tha tple pot pessue, the model pedcts that the bubble gows deeply to the sold laye. Whethe ths s physcally possble o ot, eeds to be detemed by futhe expemets. Compag the melt laye thckess data fom LANL wth the bubble szes ad the fact that sold D aoud tple pot tempeatue s epoted to have a vey low stegth, bubble gowth to the sold laye ca be explaed. If the heat flux could be detemed wth cetaty, the model could be used to estmate the sze of the ucleus. Kowg the oset tme of bubble gowth, the supeheat of the lqud ca be quatfed; fom the supeheat, cocluso ca be daw fo the mmum ucleus adus peset the lqud. he case studed fo ths thess,

83 67 suggests a ucleus adus of aoud.6µm (assumg the heat flux to be close to W/ cm ). hs esult s good ageemet wth the estmates fom chapte 3. Addtoal paametc studes esult the followg elatos betwee put paametes ad bubble gowth: A lage ucleus adus esults a eale oset of bubble gowth; ths statemet ca be explaed by the hghe pessue ad satuato tempeatue a bubble of a smalle adus due to the suface teso (equato 4.6: p ~ p(lqud)/). he hghe the lqud pessue, the moe delayed the bubble ucleato. hs s to be expected, sce a hghe lqud pessue wll lead to a hghe satuato tempeatue of the vapo the bubble. I ode fo the bubble to gow, a hghe lqud tempeatue wll be equed. Also, as a hghe lqud pessue s appled, esultg a hghe tempeatue the bubble, the bubble wll ot gow as deeply to the sold laye. he fluece of the lowe tal tempeatues s less supsg as has bee dscussed pevously []. he ext step followg the outle of ths wok s a attempt to pedct the themal behavo wth the sphecal taget as t s ected to the chambe. Fom the LANL studes ad the coclusos daw afte umecally modelg the expemets, we kow that the pessue the lqud phase wll have a sgfcat fluece the bubble gowth dcatg that we eed to focus ot oly o the themal, but also o the mechacal behavo of the taget.

84 68 able 6.: Numecal Iput paametes fo the dffeet cases: N = 00 Nt = 75 R e =.533mm R oute = mm heta = p/6 Stetchg Radal = 0.9 Agula =.0 Lage bubble.6e-6 factos Radal =.0 Agula = 4. Small bubble 0.4E-6

85 7. Applcato of the Bubble Gowth Smulato to the Sphecal aget Geomety Followg model valdato by compaso wth aalytcal esults fo cotolled example cases ad the teestg esults fom smulato of LANL expemetal esults, the model was appled to smulate the sphecal taget behavo a IFE chambe dug ecto. he pevous model [] has take the themal fluece of the plastc shell as well as dffeet popetes fo the D foam ego to accout. Hee, fo smplcty, these coectos ae eglected, sce the fluece s small whe smulatg phase chage ad bubble gowth. We wll assume dect heatg of the D due to adato ad covectve heat tasfe as dscussed chapte, ad wll cosde the cofg ad compessg effect of the plastc shell o the expadg D melt laye. 7. Icease Pessue due to Melt Laye Gowth Whe poectg the ma coclusos fom chapte 6 oto the taget case, t becomes appaet that the fluece of volume expaso dug the sold to lqud phase chage sde the plastc shell o the pessue buldup sde the taget eeds to be studed. As the volume of the D expads dug such a phase chage, t s cofed by the oute plastc shell ad the e D sold sphee, leadg to a cease pessue sde the taget. hs cease eeds to be quatfed ad ts effects cluded the umecal smulato. I ode to do so, a addtoal loop eeds to be cluded the code to compute the pessue buldup due to the gowth of 69

86 70 the melt laye. he ceasg pessue the lqud phase of the D esults a cease of the satuato tempeatue sde a ucleus (whch we assume to be peset due to the ttum decay as dscussed chapte 3). We ca plot the melt laye thckess agast the pessue the taget eglectg the small volume chages due to chages the bubble sze. hs appoxmato s easoable cosdeg how small the chage bubble volume of oe bubble s as compaed to the volume of the oveall doma ad the volume chage due to the melt laye gowth. Of couse, as moe bubbles gow smultaeously, ths effect eeds to be cosdeed. Howeve, sce the goal of the smulato s to fd the paamete space that would avod bubble gowth, ths evet ca be cosdeed outsde the boud of ths aalyss. I ode to compute the pessue buldup, the followg easog, equatos, ad mateal popetes ae used (explaato of the symbols ca be foud the omeclatue secto) followg [5]: 4π Vm = 3 moles melt V = V 3 3 ( ( d ) ) out = v D lqud out V mola, phase chage m melt moles melt (7.) Pessue buldup follows equato (7.): p f = p omelt V V 4tE 3d m (7.) I ou case, as the pessue buldup deflects both the D shell ad the plastc shell, we eed to apply equato (7.) smultaeously fo the D ad the plastc shell. Note

87 7 that the pessue the lqud due to the two deflectos have to be equal, whle the volume chages have to be added. V V total V plastc D = V D V Vplastc = V V total plastc plastc t = t D plastc E E D plastc d d plastc D m m plastc D (7.3) he ght had sde of equato (7.3) s a costat, depedg oly o the mateal popetes of D ad plastc, whch ca be foud Soues [6] fo D ad mateal popetes hadbook fo the plastc [6]. I the followg equatos, the ght had sde of equato (7.3) s efeed to as κ. V V plastc total κ = κ (7.4) (7.5): Usg equato (7.) aga, ths leads to a pessue buldup followg equato p f = p o melt κ κ 3d 4t plastc plastc E palstc m plastc (7.5) wo dffeet values fo the Youg s modulus of sold D wee foud the lteatue [6], [], dffeg by a ode of magtude. It was decded to use them computg uppe ad lowe values fo the κ costat. Applyg the umbes, we get κ = o , usg the hgh ad low values of the D Youg s κ modulus, espectvely.

88 7 κ s oly a ato of vaous geometc ad mateal popetes of the D ad the plastc shell. Its value would also be affected by paametes such as the thckess of the plastc shell ad the choce of mateals fo the shell. he geometc paametes used the taget ths aalyss ae based o those used by Chstase [] (see table 7.). able 7.: the geometc paametes ad mateal popetes used the pessue buldup computatos ae lsted hee. Ie Radus Oute Radus hckess of the shell D shell.600 mm.000 mm mm Plastc shell Youg s Modulus 40 MPa 400 MPa Posso ato mm.00 mm 0.00 mm 3.3 GPa he aalyss wll assume that the pessue the D at the momet of ecto (o melt laye peset) s 000 Pa [6]. Followg the themodyamcs of the layeg pocess (soldfcato of the D sde the plastc shell) [5], [4] a pessue close to the tple pot pessue eeds to be peset the taget. Oce the taget s layeed, thee s o mass flux though the plastc shell ad out of the taget, demadg that the pessue emas at kpa. he gowth melt laye wll add to the kpa accodg to equato (7.5). Fgue 7. shows the pessue cease wth ceasg melt laye thckess fo dffeet κ values.

89 73 Pessue vs. meltlaye due to chage volume dug sold to lqud phase chage 0 00 Pessue (kpa) E of D = 40 MPa E of D = 400 MPa Melt Laye hckess (mcos) Fgue 7.: Depedg o the stffess (Youg s modulus) of the D shell, the pessue the taget ceases accodg to ethe of these two les. Sce the popetes of the D ae hghly tempeatue-depedet, ad a lage gadet chaactezes the tempeatue dstbuto though out the D shell, a moe complex way of computg the eal tegated value of the Youg s modulus could be appled. 7. Results fom the Bubble Gowth Model As show the cyldcal case, ceasg the lqud pessue has a lage mpact o the bubble gowth a supeheated lqud close to the tple pot pessue ad tempeatue. he cease pessue sgfcatly delays the oset of bubble gowth especally fo heat fluxes that lead to a thck melt laye. he steady cease lqud pessue due to the melt laye gowth eve afte the oset of bubble gowth also esults a cease of the tempeatue the bubble. hs ca lead to a stagato of bubble

90 74 gowth o eve collapse of the bubble f the satuato tempeatue coespodg to the pessue sde the bubble gets hghe tha the tempeatue of the suoudg lqud. he peset code has bee developed to smulate bubble gowth ad s ot capable ts peset fom to also smulate stagatg o collapsg bubbles. hs s because the code ca oly smulate bubble szes of ceta dametes (based o the mesh sze);- t would be vey challegg to keep tack of the heat flux ad out of the bubble ove seveal tme steps whch the bubble does t gow fom oe gd sze to aothe the code but yet physcally chages sze betwee the two values. he possblty of fdg tme steps log eough fo the bubble sze to equal the ext gdassged value ca lead to tme steps so log that the accuacy of the code becomes doubtful. It s possble ad accuate though, to model the themal ad mechacal behavo sde the taget up to the pot whee bubble ucleato occus. Fgue 7. shows the tempeatue pofle fo the oute thee pots the doma alog wth the satuato tempeatue equed to gow a.6 µm bubble.

91 75 Fgue 7.: I ths fgue, the tempeatue hstoes fo the thee oute most pots of the sphecal doma the adal decto ae plotted. he step wse cease of the melt laye (due to the dscetzed atue of umecal solutos) leads to a step wse cease pessue the lqud, ad accodgly a step wse cease of supeheat tempeatue equed to gow the bubble. As we set the bubble gowth cteo to whe the secod pot adally wad fom the oute adus (R=.994mm) eaches a tempeatue hghe tha satuato tempeatue, the bubbles would stat gowg at the tesecto of the lght blue le wth the step wse gowg satuato tempeatue les. Bubbles would gow the plotted case at about 50 ms assumg a lowe value fo the D Youg s modulus, ad at 58 ms assumg D to have a hgh E-value. As we let the code u to smulate bubble ucleato ad gowth t etus a vey fast gowth of bubbles (smla to the sobac cases of the pevous chapte) utl about eght mcos damete. Afte that the heat flowg to the bubble o oe sde s slghtly smalle tha the heat flowg out o the othe, leadg to a stagato of bubble gowth. Meawhle, as the heatg cotues, ad the melt laye cotues gowg, the tempeatue the bubble also ceases futhe. At some pot, the