Voronoi Tessellations Description of Cast Composite Solidification Processes

Transcription

1 Procdings o th World Congrss on Enginring 9 Vol II WCE 9, July - 3, 9, ondon, U.K. Voronoi ssllations Dscription o Cast Composit Solidiication Procsss Zbigniw Domański, Mariusz Cisilski, and Bohdan Mochnacki Abstract h papr prsnts a mthod to analys th thrmal procsss occurring in th cast composit solidiication. h domain o cast is ormd by paralll ibrs bundl randomly immrsd in a host mtal matrix. h hat is transrrd rom th mtal matrix and absorbd by ibrs. h objctiv o this papr is to valuat th volumtric raction o ibrs or which th solidiication o mtal matrix occurs. Our mthod is to comput Voronoi diagrams with Voronoi rgions rprsnting th gomtry location o th ibrs in th mtal matrix and to us ths rgions as control volum within a variant o Control Volum Mthod. Indx rms cast composition solidiication, hat transr, Voronoi diagrams. I. INRODUCION Hat conduction is a rlvant topic in many industrial procsss. Knowldg o th thrmal conduction phnomna and its ctiv dscription ar ssntial or th analysis o th hat transport procsss occurring in htrognous mdia. An xampl o industrial importanc is th solidiication procss o mtal matrix composit (MMC) [] [7]. h ormation o microstructurs can b altrd during th solidiication procss as is th cas o ibr rinorcd aluminium composit with th xtndd nds o th ibrs coold by a hat sink. MMC solidiication has bn invstigatd in various ibr arrangmnt scnarios including ixd inlin or staggrd ibr in a varity o spacing distributions. h prsnc o ibrs has a rinorcing ct on mchanical proprtis o MMCs compard to monolithic mtals. Suprior mchanical, lctrical and thrmal proprtis o MMCs dpnd on th appropriat choic o th matrix and ibr matrials, thir shap and abrication mthod. h problm w considr is th solidiication o th mtal (th matrix) in th prsnc o ibrs. h aim o our invstigation is to dtrmin th volumtric raction o ibrs or which th solidiication o mtal matrix procds in a natural way du to th hat xchang btwn th matrix and ibrs. II. MAHEMAICA MODE W considr a hypothtic procss o th synthsis o MMC Manuscript rcivd Fbruary, 9. his work was supportd in part by th Ministry o Scinc and Highr Education (Poland) undr Grant No. BS--5-3/99/S. Z. Domanski, M. Cisilski, and B. Mochnacki ar with th Czstochowa Univrsity o chnolog Dabrowskigo 69, P-4 Czstochowa, Poland. (corrsponding author: Z.D; -mail: zdomanski@imi.pcz.pl). by casting. During this procss th ibrs do not undrgo phas chang bcaus thir mlting tmpratur is much highr than that o th matrix. h solidiying matrix and ibrs ar containd within a cavity with adiabatic walls and th only possibl nrgy xchang procss is th hat transr orm th matrix to th ibrs so th ibrs srv as a hat sink. h dominant mchanism o nrgy transport in th cavity is diusion. h govrning quations or th consrvation o nrgy [8], including th prsnc o th ibrs bundl ar: m cm ( ) [ λ m ( ) m ] + qv, m, () t whr c m () is th spciic hat pr unit o volum, λ m () is thrmal conductivit, x, t dnot th tmpratur, gomtrical co-ordinats and tim. Indx m idntiis th matrix sub-domain and m idntiis th ibrs sub-domain. h trm q V is th sourc unction and this trm can b writtn in th orm S, or m qv t (), or m whr is th latnt hat pr unit o volum, S is th solid stat raction at th nighbourhood o th considrd point (x, y). Pur mtals (.g. aluminium) solidiy in a constant tmpratur. h unction S changs rom (moltn mtal: or (x, > ) to (solid body: or (x, < ). h valu o λ m () is dtrmind by rlation λ or m and > ( S ) λ + S λ S or m and λ m ( ) (3) λ S or m and < λ ib or m whr λ, λ S ar thrmal conductivitis o th liquid and solid stat o th mtal matrix and λ ib is th thrmal conductivity o th ibrs bundl. In a similar way on can dtrmin c m (). On th contact surac btwn th matrix and ibr a continuity o tmpratur and a continuity o hat lux ar assumd λ λ n n (4) ISBN: WCE 9

2 Procdings o th World Congrss on Enginring 9 Vol II WCE 9, July - 3, 9, ondon, U.K. whr / n dnots a normal drivativ. On th xtrnal surac o th domain th boundary condition in th orm m (5) n is givn. For tim t th initial condition is also known ), ) t : (6) III. VORONOI ESSEAION AND MESH GENERAION In a numrical modlling o th solidiication procss considrd th mtal-matrix and ibr sub-domains ar dividd into small clls (control volums), known as th Voronoi polygons (also calld th hissn or Dirichlt clls in two dimnsions) [9] [3]. h Voronoi polygons ar on o th most undamntal and usul constructs dind by irrgular lattics. For a st X {x, x,..., x N } o N distinct points in R, th Voronoi tssllation is th partition o R into N polygons. h polygon that contains point x i (cntral poin is dnotd by V i. Each rgion V i is dind as th st o points in R which ar closr to x i than to any points in X: { X : d( X, x ) < d ( X, x ), j i, j, K N} V R (7) i i j, whr d(, ) is th Euclidan distanc unction. All o th Voronoi rgions ar convx polygons. In Fig. th xampl o Voronoi polygons is shown. A singl polygon is dind by th lins that bisct th lins btwn th cntral point and its surrounding points. h biscting lins and th connction lins ar prpndicular to ach othr. Whn w us this rul or vry point in th ara, th ara will b compltly covrd by adjacnt polygons. Som o thm ar ininit ( opn ) bcaus thy hav no nighbouring points in that dirction. D(X). his triangulation is th dual structur o th Voronoi tssllation. h polygons ar dind by lins that bisct th connction lins btwn nighbouring points and th cntrs o circumcircls ar th vrtics o th Voronoi polygons. In th cas o unboundd polygons whos cntr points li on th convx hull o st X, thy ar bound by th boundaris o th domain Ω p s Fig.. Positions o points in th st X ar usually dtrmind randomly and hnc th msh o control volums is unstructurd. W additionally impos th rquirmnt that th minimal distanc btwn two points must b gratr than th pr-assignd valu. It maks th polygons mor uniorm and th cntral point lis narr th middl o th polygon. In Fig. an xampl o th structur o cast composits with 4% ibrs is shown. h domain has bn dividd into 35 control volums. Gray control volums dtrmin th ibr sub-domain. I w assum th ibrs bundl is paralll thn on th sction considrd on obtains th discrt st o circls. h position o th ibr cntr P j (x j, y j ), whr j is an indx o th ibr considrd, is dtrmind in a random way. Also th ibr radius r j is randomly dtrmind r j [r min, r max ]. h numbr o CV should assur a good approximation o ibr circular cross-sctions. h positions o cntral points in CV clos to th contact btwn ibr and mtal-matrix ar analytically dtrmind in ordr to achiv a bttr approximation o th ibrs shap. h gomtrical paramtrs o a systm matrix-ibrs ar chosn on th basis o th optical micrographs prsntd in [4]. It can b sn that th ibrs diamtrs ar dirnt, and also thir mutual positions ar rathr incidntal. h only unqustionabl inormation concrning th gomtry o th systm rsults rom th volumtric raction o th ibrs in th domain analyzd. So, th numrical procdur ralizing th msh gnration, ibrs localization, valus o ibrs radii bass on th application o random numbrs gnration. On th stag o msh gnration, th typ o sub-domain (m, ) or vry CV is assignd. Fig.. h Voronoi polygons or a st o arbitrarily distributd points Many algorithms to construct th Voronoi polygons can b ound in litratur. On popular mthod is basd on th Dlaunay triangulation [9] []. his triangulation can b ormulatd in a circl critrion. h Dlaunay triangulation D o a st X o points in th plan is such a triangulation D(X) that consists o non-ovrlapping triangls and contains no points o X insid th circumcircl o any triangl in Fig.. Exampl o msh ISBN: WCE 9

3 Procdings o th World Congrss on Enginring 9 Vol II WCE 9, July - 3, 9, ondon, U.K. IV. NUMERICA MODEING h control volum mthod (CVM) [8], [5] [7] constituts th ctiv tool or numrical computations o th hat transr procsss. h domain analyzd is dividd into N volums. h CVM algorithm allows to ind th transint tmpratur ild at th st o nods corrsponding to th cntral points o th control volums. h nodal tmpraturs can b ound on th basis o nrgy balancs or th succssiv volums. In ordr to assur th corrctnss and xactnss o th algorithm proposd w gnrat th control volums in th shap o th Voronoi polygons (s: Fig. 3). assum R (in numrical ralization.g. R ) i th surac limiting th domain V in th dirction is a part o th boundary - it assurs to zro hat lux in th dirction. h chang o nthalpy o th control volum V during th tim quals [8], [5] + ( ) c V () whr c is th volumtric spciic hat,, + dnots two succssiv tim lvls. t us writ th balanc quation in th xplicit schm + ( ) V A + V qv c () R or Fig. 3. Control volum V n + W q + V c (3) t us considr th control volum V with th cntral nod x. It is assumd hr that th thrmal capacitis and capacitis o th intrnal hat sourcs ar concntratd in th nods rprsnting lmnts, whil thrmal rsistancs ar concntratd in th sctors joining th nods. h nrgy balanc or th control volum V can b writtn in th orm Q + V qv (8) whr is a chang o control volum nthalpy during th tim intrval, Q th hat conductd at th tim rom th adjoining nods to th nod x, q V a man capacity o intrnal hat sourcs in th control volum V. I on assums that th hat luxs lowing to th lmnt V ar proportional to th tmpratur dirncs at th momnt t t, thn w shall obtain a solving systm o th typ 'xplicit schm'. So Q R whr R A (9) is th thrmal rsistanc btwn points x and x [8], A surac limiting th domain V in th dirction. I w dnot by h th distanc btwn th nods x, x thn R h + h λ λ () whr λ and λ ar th thrmal conductivitis in th control volums V and V at th momnt t t. h othr dinition o thrmal rsistanc should b introducd or th boundary volums [8]. For th boundary condition (5) w whr A W,,..., n (4) c R V n W W (5) In ordr to assur th stability o th abov xplicit schm th coicint W must b positiv. Nxt, th problm o mtal solidiication in th control volum V in a constant tmpratur will b discussd. t us assum that at th tim t t th tmpratur in nod x is > (th moltn mtal), and th calculatd + tmpratur <. h chang o nthalpy or V during th tim is as ollows + ( ) c V h nthalpy can b dividd into two componnts ( ) V c, (6) H (7) h chang o nthalpy associatd with th solidiication o th control volum V is qual to + S V (8) whr + S V is a solidiid part o th considrd volum V. So, th nrgy balanc or V in which th solidiication procss starts, can b writtn in th orm + + ( ) V c ( ) V + S c V (9) ISBN: WCE 9

4 Procdings o th World Congrss on Enginring 9 Vol II WCE 9, July - 3, 9, ondon, U.K. From th last quation on obtains + ( ) + c S () and simultanously S S, () I in th control volum V th irst portion o solid mtal is prsnt thn it is assumd that th tmpratur corrsponding to V quals and or th succssiv stps o tim th ollowing dirnc quation is rquird ( ) V () A + + t S S R without xtrnal thrmal intractions. h algorithm proposd allows to analyz th thrmal procsss in a systm matrix-ibrs or th dirnt tchnological conditions (raction o ibrs, initial tmpraturs, boundary conditions tc.). From th nthalpy balanc approach (or adiabatic systm) on can dtrmind in an analytical way ibrs raction or which th whol liquid mtal passs to th solid stat. h nthalpy balanc can b writtn as ollows ( r ) H ( ) + r H ( ) H ( ) (6) whr r is ibrs raction in th considrd domain, H and H ar nthalpis o mtal-matrix and ibr sub-domains rspctivly and H is nthalpy o th whol domain. Assuming that > thn th nthalpy H is dind by From th last quation on obtains + S S + A (3) V R H cd ( ) csdτ + + r and th nthalpy H quals to τ (7) whras +. For th on o succssiv stps o tim it + turns out that, this mans that mor than th whol S > V has solidiid. h nthalpy connctd with this actious solidiication o V abov S should b rcalculatd on th nthalpy connctd with cooling o th solid stat + + ( S ) V c ( ) V hnc + ( S ) +, (4) + S (5) c In th succssiv stps o tim th nrgy balanc or V is o th orm (3). V. EXAMPE OF COMPUAIONS Numrical simulations o a casting procss hav bn xcutd or th matrix-ibr (pur mtal Al) with thrmophysical paramtrs: c S.943 6, c J/(m 3 K), λ S 6, λ 4 W/(m K),.53 9 J/m 3, 66 C, initial tmpratur 7 C and th ibrs (Si) with thrmophysical paramtrs: c ib.63 6 J/(m 3 K), λ ib 48 W/(m K), initial tmpratur C. h considrd domain (D problm) has th dimnsion µm µm. In simulations th ibrs raction has bn assumd as 4%, 5% and 55%. In Figurs 4, 5 and 6 th kintics o th solidiication procss or dirnt ibrs raction ar prsntd. h inlunc o ibrs raction, initial tmpraturs o sub-domains considrd on solidiication tim is vry ssntial. In th cas o th simulation with th 55% o ibrs raction, th whol liquid mtal passs to th solid stat ( ) c dτ H (8) r ib Assuming that th whol liquid mtal passs to th solid stat and inal stablishd tmpratur o th whol domain (th mtal matrix sub-domain and ibrs sub-domain) is qual to, thn th nthalpy H is dind by ( ) ( r) csdτ + r H c dτ (9) r r Substituting (7), (8) and (9) into (6) and assuming that c, c S, c ib ar constant, on can calculat th valu r ib + c ( ) ( ) + c ( ) r (3) + c ib For th abov thrmophysical paramtrs w obtain th valu r 5,98%. his analytical rsult conirms th corrctnss o numrical rsults. VI. CONCUSION In th prsnt work, th Voronoi diagrams hav bn xploitd to gnrat a amily o mshs or th control volum mthod algorithm. h proprtis o Voronoi diagram prmit to ind th nodal tmpraturs in an ctiv mannr. Whn th cast solidiis th shaps o solidiid domains changs in a pculiar mannr around th cooling ibrs. hus, th random Voronoi tssllations allow to achiv highr corrspondnc with th tmpratur ild than widly usd rgular tssllations. Whil th mthod is prsntly limitd to rathr simpl, circular-ibr-shaps, work ISBN: WCE 9

5 Procdings o th World Congrss on Enginring 9 Vol II WCE 9, July - 3, 9, ondon, U.K. Fig. 4. h kintics o solidiication 4% ibrs raction Fig. 5. h kintics o solidiication 5% ibrs raction Fig. 6. h kintics o solidiication 55% ibrs raction is prsntly in progrss to xtnd it to ibrs with noncircular cross-sctions and to mtal matrix composits with random orintations o cooling ibrs. REFERENCES [] V.V. Vasiliv, and E. Morozov, Mchanics and Analysis o Composit Matrials, Elsvir Scinc,. [] K. Kainr, Mtal Matrix Composits, John Wily & Sons, 6. [3] V. indroos, J. Hllman, D. ou, R. Nowak, E. Pagounis, X.W. iu, and. Pnttinn, Dsigning with Mtal-Matrix Composits, in: G.E. ottn (Eds.), Handbook o Mchanical Alloy Dsign, CRC Prss, 4. [4] S. Schmaudr, and. Mishnavsk Micromchanics and Nanosimulation o Mtals and Composits, Springr-Vrlag, 9. [5] A. Cantarl, E. acost, M. Danis, and E. Arquis, Mtal matrix composit procssing: numrical study o hat transr btwn ibrs and mtal, Intrnational Journal o Numrical Mthods or Hat & Fluid Flow, Vol. 5 No. 8, 5, pp [6] S. Sursh, A. Mortnsn, and A. Ndlman, Fundamntals o Mtal-Matrix Composits, Buttrworth-Hinmann, 993. ISBN: WCE 9

6 Procdings o th World Congrss on Enginring 9 Vol II WCE 9, July - 3, 9, ondon, U.K. [7].P.D. Rajana, K. Narayan Prabhu, R.M. Pillaia, and B.C. Paia, Solidiication and casting/mould intracial hat transr charactristics o aluminium matrix composits, Composits Scinc and chnolog 67, 7, pp [8] B. Mochnacki, J.S. Such Numrical mthods in computations o oundry procsss, PFA, Cracow, 995. [9] J. O'Rourk, Computational Gomtry in C, Cambridg Univrsity Prss, 998. [] A. Bowyr, Computing Dirichlt tssllations, h Computr Journal, 4(), 98, pp [] P.J. Grn, and R. Sibson, Computing Dirichlt tssllations in th plan, h Computr Journal, (), 977, pp [] D.F. Watson, Computing th n-dimnsional Dlaunay tssllation with application to Voronoi polytops, h Computr Journal, 4(), 98, pp [3] B. Mochnacki, and M. Cisilski, Application o hissn polygons in control volum modl o solidiication, Journal o Achivmnts o Matrials and Manuacturing Enginring, 3(), 7, pp [4] M. Nolt, and E. Nussl, Procss condition microstructur-strngth- -obsrvations in cast continuous ibr matrial matrix composits, Cast Composits'95, 995, pp [5] R. Szopa, and J. Sidlcki, Modlling o solidiication using th control volum mthod, Solidiication o Mtals and Alloys,, 44,, pp [6] E. Fraś, W. Kapturkiwicz, and H.F. opz, Macro and micro modlling o th solidiication kintics o casting, AFS ransactions, 9-48, 993, pp [7] J. Orkisz, Finit dirnc mthod, in: M. Klibr (ds.), Computr mthods in solid mchanics, PWN, Warsaw, 995 (in Polish). ISBN: WCE 9