FOUNDRY IGINEERING -I

Published quarterly as the ogan of Ihe Foundry Commission ol !he Polkh Amdemy of Suences

Simplified model of metal solidification in the thin plane cavity of the casting mould

L. Sowa "*, N. Sczygiol " T, Domoilski ", A. Bokota tr

Vnslitute of Mechanics and Machine Desipn, "mtitute of Computer and lnfosrnation Scicnccs, Czptochowa University of Technology, 73 Dqbrowskiego str. 42-200 Czqstocliowa, Poland

*Corresponding author. E-mail: sowa@imipkrn.pcz.czcst.pl

Rcccivcd 07.03.2008; acceptcd in rcviscd form 01.04.2008

Abstract

In thc papcr, a innthcmatical inodcl of thc solidification of a thin-wallcd casting, which takcs into account thc proccss or filling thc mould cavity wilh moltcn tnetal. has hccn proposed. Prcssurc and vcloci~y ficlds wcrc obtaincd by solving thc ino~ncntl~m ctliiations and rhc continuity cquation. whilc [he thcrtnal hclds were ubtaincd by solving thc hcat conduction equation containing rhc convcciion tcrm. Making assumptions rcla~ing to both thc rnatcrial and Ihc gcornctry e l the rcgion. thc gcncral equations for continuity and rnomcntiun havc bccn rcduccd to single cquation for prcssurc. This approach Icads as to furlhcr si~npliry of ~lic fluid flow cnlculations, I11 thc tnodcl onc takes into account intcrdcpcndcncc Ihc hcat Iransfcr and fluid flow phcnomcna. Coupling of zhc thcrmnl and fluid flow plicnomcna has bccn takcn into consideration by the changcs of thc fluidity fitnction and thcnnophysical paramctcrs of alloy with rcspcct lo thc rcrnperaLure. Thc problem has bccn solvcd by ihc finilc clcmcnt mcrhod.

Keywords: Solidification; Moltcn Mctal Flow; Mourd Filling; Mathcrnat icat Modcling

1. Introduction

Casting proccsscs arc widcly used to producc inetal components. Much rcscarch has been devotcd toward pmccss devclopmcnt for thc production of high quality casling goods at low costs. From a macroscopic point of vicw, casting proccsscs involve rhc coupling of solidification heat transfcr and fluid flow. Threc dirferenl flow mechanisms may bc idcnlificd: mould filling through the gating system, rcsidual flow due to the incoming momcntilm and thc natl~ral-convection-drivcn flow En the mould can he considered during the casting pmccss. From the hcat tsanskr point or vicw. thc solidificarion and thermal strcsscs duc to shrinkage are important physical phcnornena. Thesc cornplicatcd physical phcnomcna during a casting proccss requirc cornpf icatcd nurncrical mcthodology [ I]. As a rcsult, unrcalist ic restrictions are often imposcd in order to rendcr a modcl tractable.

The most widcly slr~died arcn of casting is phnsc-chnngc hcat transfcr 121: natural convccrion cffccis and macro/micm segregation [3.4] havc only rcccnily hccn siudicd in somc dctail. Fluid flow analysis during thc mold filling proccss has bccn studied vigorously in rcccnt decadcs duc to ~ h c advcnt of computcr hardwarc systcrns. Thc lilling of simplc mold gcomctry has bccn prcviousIy rnadcIcd 151 and stitdics Inr ;in crl'cctivc filling algorithm havc also hccn rcponcd [G, 71.

Liquid mctal cools as it flows into thc mold. In ihc casc of ii

thin cast part or a casting of rnoltcn mctnl with low supcrhcnt, thc moItcn mcrnl is coolcd during iillinc and thc moItcn ~nctal is locally fully solidified bcforc thc moId is cornpIctcly fiIlcd. Hcncc. i t is ncccssary to simi~ltnncousl y xnalyzc thc mold filling and solidificati~i~ proccsscs. Most of Zhc prcvioits works havc considered only the solidification proccss alicr filling 121. Rcccnily, thc simultaneous analysis or mold filling and soIidification phcnotncna has hccn rcporlcd [ I , 8, 91. Wang and

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Pcrry [ IO] stttdicd t lic ljllirlg characrcrist ics in thin-wall invcstmcnt cnsfings ; is ir runctinn oi pnrnmcrcrs such as pnlc wlocity. supcrltc;~~ o l inol~cn ~iictnl and mold prc-hcat condition. 3 l i t c! :I!. [ I I ] proposcd a modcl for filling and solidifica~ion in n prcsszirc rlic cnrliny proccss. F~11 undcrstnnding of filling md heat rransrcr phalomcna in caqaing proccsscs rctluircs an analysis that includcs thc cffccl of tnold filling. rcsiditnl flow aftcr filling and tlrc crrcct nT ~ l i c flow, duc to natur;il convection. oir rhc soEidificnt ion prnccss.

Thcrcforc n! 111c prcscli! t imc. ~ h c numcrical simulnt ion of cnstiny solidification wit11 taking into account tlic mnrtcn mctnl rnntiorl and thc rnnl~ld cnvily f i t l i t~g prnccss is \.cry oftcn carricd orat. 'The mnthcmntical mtrdcl takcs inro considcra~inn isltcrtlcpc1rdc1tcc (IT !hc tl1crm;tl and dynamical phcnomcnn. Vclocit y ficlrls arc oht aincd u<u;~lly hy solving the Navicr-Stokcs cq\~atio~ir illid thc canlinuily cquatiou. whcrcns tFrc thcrlnal ficlds arc c;ilci~l;~tctl hy snlvin_r thc Fourier-KErclilioTT ctlirnfinn with tlic convcc~irln tcrm. This is a complcx antl rliTi1cul1 prohlcm to solvc ~lurncrically I I. h, K, 01. Shc analysis nT thcsc phcnnmcna is li~nilcd oCtcn cmry tn thcir pmcccdings during thc filling proccss nC tIlc c!.lindric;tl irilct cti;inncl [ 12.131, tlic slcndcr rnot~ld oi fluidity trsr 1 14 I or t l l i t l p1.111~ ~':ivity nT cnstiiig riioi~ld 1 13. 15, Ihl. Just its k)r thc CiESC of' tlic cnvitics. pcomciric considcrntions ;~llou. hr ! hcr simp1 ific:rticrr~ nT the pvcrning cqrlnlions. Making ;tssl~rnplinns rclatiny to ho!h thc iiia!crinI and thc gcomct ry nT llrc region, thc gcncral cqt~ario~i.; lilr continuily and moincniitrn have hccn rcilucctl to sitlglc cqu:htitlri for prcssurc. This npprnach Icndq ;is to f~~i-~t icr sirnplify aT tllc nulncrical calcuI;ltians [ 12. 13. 15. 161.

It) this sturly. ;I simultnncous analysis mcrrlcl a f ~ b c mclal so1idific:itioii pnwcw in thc thin planc cavil y or thc ca5ting mould du r in~ its filriii? has hccn propscd. Mnthcmatical modcl takcs into considcc~llnn intcrdcpcndcncc or t l icr~ni~l and dynamicnI phcnomcn;!. Couplirlg oI' the ~ItcrrnaI and fluid flow plicnorncnn ti;^< bccn takc~t into cunsirIrr:itinn hy ihe chanycs or ttic l l i~it l i ty i ~~n r t i on and thc t l~crrnnpliysicnl pnrilmczcrs of nlloy wit I1 rcspcct tn Iliv tc~npcr;~turc. Tlic \rliolc task. that is hollr tlic hcnt c~~ntluctior~ ctlui~linn nntl thc prcswrc crluation. has hccn snlvcd usirlg thc i i n i ~ c clc~ncnt mcthntl j 2. b-X. 1.1, I5].

2. Mathematical model of heat transfer and molten metal flow

1;Ei~irl flow tlurltly riiolil filling invol\,cs frcc silrrncc mot ion. T l ~ c sitbscr!~~cnf solidi ficnlion of t hc ~nctal i s a ~nnving intcrlncc prnhlcrn sincc IIIC snlitli2?c:itintl rmnt mows wirh zimc. I T liquid mct;~l is Nc\~tonian nntl tlic ilozr. is laminar. thc governing ctli~:ttions rnr cncrgy. mrrnicntum i~nd mass conscrvatio~l arc :is follnws I2.5.X.13.l-Ij: - t l ~c tlcat cnndtlct ion cqtai~ti~li containing ~ h c convcct ioil lcrrn:

- rhc iiim~icritun~ ctlui~linns nncl thc con! inui~y ctltvarinn:

whcrc: T(x. t ) - thc tcmpcratitrc [ KI. I - rimc [sl. c - tE1c spccilic

hcat [Jl(kpK)], p = p ( ~ ) - ~ h c tlcnsity [k@m3J, 1 - thc r hcrmnl

condt~ctivity cocfficicnt [tVl(aK)], p - thc prcssurc IN/I~'].

x l . ~ , 7, 3 - t l ~c coordinates of thc vcctnr of thc considcrcd nodc's psi t ion [ ~ n l , ~ ( v , . r u , 1,:) - thc vclncity vcclor or mollcn mctal flow [rids], { I (T) - rtic dynamicnl viscosity cocflicicnt [~drn ' ] , I Q - Ihc volumetric cfficicncy of Ihc intcrnal hcat sourcc [Wlm3], g - the vcctor of thc accclcratinn of gravity I d s 2 ] .

Thc proposcd inodcl ror ~ h c numcrical simrrlation of solidification givcs considcri~tioii lo rhc morions of mnctiil l irpid phnsc during thc ~ n n i ~ l d cavity filing prnccss. I t i s hascd oli solving thc following syslc~ii 01 dil'rcrcntiaI cqtint ions in ~ l i c Cnrtcsinn caordinatc syacln 15.8.131:

I !

whcrc: Ctf(T) = pLscrs + - p s L - tlrc cllcclivc licn! capacity nf TL - T.7

thc mushy zonc [ J / (~ 'K) ] , cis - ~ h c specific hca[ n i rhc mushy

zone IJI(kgK)l. p,, p , , . p , - rhc dcnsity of solitl phasc, liqttid

phasc, and mushy zonc. rcspcctivcly I k@m31.

hcnt of solidification IlRg1. In thc applicd modcl of solid pllnsc growtl~. thc intcmal hcaz sourccs arc not mmc cvidcnt in thc crltiaion of hcat conductivity. hccauw thcy arc in ~ h c cCrcaivc licat capitcity or thc mushy ronc [ I -5 . x, 91.

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To simplify thc abovc cqtlations, wc cmploy a tcchniquc calicd dirncnsionat analysis. I3nsicnlly thc idca i s to obtain cairnatcs of thc ortlcr of magnitudes of each tcrm in thc govcrning cquations. Tcrrns o f sufficiently low ordcr havc litrlc influcncc on thc numerical simulation rcsutts and so arc nqlcctcd. Thc cquations (3-7) thcn simplify to [12, 13, 15- 171:

avcragc vclocity (,T, y-componcn~) wc obtain thc following equation [ 13, 15-1 71:

whcrc w havc dcfincd

which is oftcn callcd thc fluidity function. Equation (13) i s n singlc cquation for prcssurc that comhincs thc rnomcnrum and conrinuity cquarion.

Thc systcm of cquations (8.13) i s crrrnplctcd by thc appropriarc initin[ conditions and ~ h c boirndary conditions.

Thc initial conditions for prcssurc ant1 tcmpcraturc ficlds arc

Thc boundary conditions spccificd in ihc considcrcd problcin wcrc as follows [2-5,8.9.13- t 71: -at thc inlet gatc:

- at thc mold wall:

Fig. I .Schematic diagram of cavity filling

Funhcr simpEiIicalion is availahlc by integrating thc inoincnturn nnd continuity cquat ions. Prom ~ h c moment urn cquntions (9) wc scc that thc prcssurc i s a function OF coordinatcs onty, ['or this rcason i t i s convcnicnt tn intcgratc thc momcn!i~rn cquation across thc thin cavity with thc aim o f obtaining cxprcssion for rhc x. y-cornponcnt oTvcEacity (I:. 1:) as follows:

-at the flow Front:

- at thc cavity ccntcr linc:

whcrc: T, - thc ambicnt tcmpmtutc [K], TM - thc mould

tcmpcraturc [K]. aM - thc hcat-rnnsfcr cocfficicnt hctwccn

mould and ambicnt [w/(~'K)].

whcrc wc havc dclincd rhc constant

3. Summary whcrc h dcnotcs thc half-gap ~hickncss of thc cavity Im].

Intcgraring of thc conrinuity cquation ( 1 0) nvcr thc arca of ~ h c mclt cavity (with rcspccr to :) nlid using thc dcljnition of thc

Finally. aftcr making assumption rcparding the makcrial and using thc cllcct of gcomctry, cquations govcrnin~ ~ h c flow o f thc

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moltcn metal in thc thin planc cavity or thc casting mould may lakc lbc form 112.13.15.1TI:

Thc ahovc sysrcrn o f ctpations (20, 2 1) lios hccn cnmplctd hy thc npprnpri:ltc initial conditions and ihc boundary condi~ions ( 15- 19) ;~nrl was solvcd by thc Iinitc clc~ncnt method I2.13. t 51.

Acknowledgement Thc rcscarch was pcrli~rmctl within fmrnc$r~ork of rcscnrch

projcct tinanccd hy t hc Minislry of Scicncc and I liphcr Edt~cntion.

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[15 ] B.Y. Jiang. J. Zhong. B.Y. Iluang, X.11. Qu. Y.M. Li. Elcmcnl rnodclling of PEM on thc prcssLrrc ficltl in thc powdcr injcct ion mould filling proccss. Journal of Mazcrials Processing Tcchnolopy, vol. 137 12003) 74-77.

[16] Y.K. Shcn, C.M. Ju, Y.J. Shic. I1.W. Cliictl. Rcsin flow charac~cristics of undcr fill proccss on flip chip cncapsulalion. Iirt. Comm. I lcai Mass Transtcr., vol. 3 1, No. X (2004) 1 075- 1 084.

1171 C. Shcn. L. Wang. Q. I,i, Nitmcrical simularion or cnrnprcssiblc flow with phasc changc o f litling snpc ill injcct inn mofding. Jot~rnal of Kcininrcd Plaaics and Cnmpositcs. vol. 26. No, 4 (2007) 353-332.

clcmcnt mcihod Tor mould fitling simulation in mctal ci~stings, Nurn. Mcth. in Thcrtn. I'rohl.. YO[. 7. NO. 1 ( 1993) 273-2x3.

Uproszczony modcl klzcpniqcia metaln .rv cicnkicj plaskicj wnqcc formy odle~vniczcj

W privy sibnnulorvano modcl mntcmalyczny proccsu lirzcpniqcia odlc~vu cicnEroScicnncgo z i~wzplqdnicnicrn procesu \vypt.lninnin rvnpki l i~rniy cicklyt~i rnctaIcm. I'ota ci<nioti i ptqdkof;ci otrzynano z rozwiqznnia rtjwnah pqdu i rinvnnnin ciqglogci. nnlomisst poln rcrnpcmtury. z rozwiqzanin rritvnania pr7c\vodnic~\va cicptn ;? czlonctn kon\r.ckcsinyrn. Poprzcz znlotcnin upraszcznjqcc w rriwnaniacl~ p~du. \r yni kajqcc z przyyjc;tcgo do n~zn.n?ail plynu Icpkicgo nicSciSli\tcgo ornz smuklcpo ohsrani gcomctryczncga, otnym~rjc siq pnicd>nc/c ri~rvnanie sknlarnc. *. kt6rym jcdynq niewiadomq jcsl citnicnic. Takic podcjgcic pozr\~ntn znaczniu t~prnSciC ohticrenin nuincrycznc. Spr;rc;;scnic ornnwinn ych ~jawisk u\v7_cl~dnionn zalc%noSciq od tcmpcrnlnry hnkcji IcjnoSci nrnz pilramctr0w tcnnofi?yc7nych slopu. Zndanic n)x\vi;(xane rnctodq clcmunt~irv sskoriczonych.

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