niersi of Wisconsin Milwakee WM Digial Commons Teses and Disseraions December 1 Nmerical Analsis of Die-Casing Process in Tin Caiies sing Lbricaion Approimaion Aleandre Reiker niersi of Wisconsin-Milwakee Follow is and addiional works a: p://dc.wm.ed/ed Par of e Engineering Commons Recommended Ciaion Reiker, Aleandre, "Nmerical Analsis of Die-Casing Process in Tin Caiies sing Lbricaion Approimaion" (1). Teses and Disseraions. Paper 65. Tis Disseraion is brog o o for free and open access b WM Digial Commons. I as been acceped for inclsion in Teses and Disseraions b an aoried adminisraor of WM Digial Commons. For more informaion, please conac krisinw@wm.ed.
Nmerical Analsis of Die-Casing Process in Tin Caiies sing Lbricaion Approimaion b Aleandre Reiker A Disseraion Sbmied in Parial Flfillmen of e Reqiremens for e Degree of Docor of Pilosop In Engineering a Te niersi of Wisconsin Milwakee December 1
ABSTRACT Nmerical Analsis of Die-Casing Process in Tin Caiies sing Lbricaion Approimaion b Aleandre Reiker Te niersi of Wisconsin Milwakee, 1 nder e Sperision of Professor Krisna M. Pillai Casing of in wall pars as become a reali of e die cas indsr oda. Compaional flid dnamics analses are an inegral par of e prodcion deelopmen process. Tpicall e ree-dimensional Naier-Sokes eqaions copled wi e energ eqaion ae o be soled in order o gain an ndersanding of e flow and solidificaion paerns, posiion of e flow fron, as well as locaion of e solid-liqid inerface as a fncion of ime dring e process of cai filling and solidificaion. A pical solion of e goerning eqaions for a in-wall casing reqires large nmber of compaional cells, and as a resl, akes impracicall long ime o generae a solion. sing e Hele- Saw flow modelling approac, solion of e flow problem in a in cai can be simplified b neglecing e o-of-plane flow. As a frer benefi, e problem is redced from a ree-dimensional problem o a wo-dimensional one. B e Hele-Saw approimaion reqires a iscos forces in e flow are mc iger an is ineria forces, and in sc a case, e Naier-Sokes eqaion redces o e Renolds s lbricaion eqaion. Howeer, owing o e fas injecion speed of e die-cas process, e inerial forces canno be negleced. Terefore e lbricaion eqaion as o be modified o inclde e inerial effecs of e flow. In is PD esis, a fas nmerical algorim is deeloped for modeling e sead-sae and ransien flows of liqid meal accompanied b solidificaion in a in cai. Te described problem is closel relaed o e cold-camber, ig-pressre die-cas process and in pariclar o e meal flow penomenon obsered in in enilaion cannels. ii
sing e fac a e rae of meal flow in e cannel is mc iger an e solid-liqid inerface eloci, a noel nmerical algorim is deeloped b reaing e meal flow as sead a a gien ime-sep wile reaing e ea ransfer along e ickness direcion as ransien. Te flow in e in cai is reaed as wo- dimensional afer inegraing e momenm and conini eqaions oer e ickness of e cannel, wile e ea ransfer is modelled as a one-dimensional penomenon in e ickness direcion. Te saggered grid arrangemen is sed o discreie e flow goerning eqaions and e resling se of parial differenial eqaions is soled sing e SIMPLE(Semi-Implici Meod for Pressre- Linked Eqaions) algorim. Te ickness direcion ea-ransfer problem accompanied b pase cange is soled sing a conrol olme formlaion. Te locaion and sape of e solid-liqid inerface are fond sing e Sefan condiion as a par of e solion. Te simlaions resls are fond o compare well wi e predicions of e commercial sofware FLOWD a soles e fll ree-dimensional se of flow and ea ransfer eqaions accompanied wi solidificaion. Te proposed nmerical algorim was also applied o sole a ransien meal-filling and solidificaion problem in in cannels. Te presence of a moing solid-liqid inerface inrodces a non-lineari in e resling se of flow eqaions, wic are now soled ieraiel. Once again, a good mac wi e predicions of FLOWD was obsered. Tese wo sdies indicae a e proposed ineria-modified Renolds s lbricaion eqaions accompanied b rog-e-ickness ea loss and solidificaion models can be sccessfll implemened o proide a qick analsis of flow and solidificaion of liqid meals in in cannel dring e die cas process. Sc simlaion resls, obained wi remendos saings in CP ime, can be sed o proide a qick, iniial analsis dring e design of e enilaion cannels of a die-cas die. iii
Table of Conens Absrac Table of Conens Lis of Figres and Tables ii i ii 1. Inrodcion 1.1 Casing processes 1 1. Sand casing process 1 1. Permanen mold 1..1 Grai casing 1.. Low-pressre permanen mold casing 1.. Coner-pressre casing 1. Die casing 5 1.5 Die cas process 6 1.5.1 Ho camber process 7 1.5. Cold camber process 9 1.6 Callenges of meal flow in in caiies 1. Lierare oeriew 1 i
.1 Hisorical oeriew of nmerical meods 1. Te finie difference meods 1. Conrol olme Meod (CM) 1. Free srface approimaion 1..1 Marker and cell (MAC) meod 1.. Srface marker meod 15.. Free srface approimaion sing olme of Flid (OF) 15 Meod.5 Renolds s lbricaion approimaion 16.6 Scope and objecie 17. A fas nmerical simlaion for modeling simlaneos meal flow and solidificaion in in caiies sing e lbricaion approimaion 19.1 Inrodcion. Deelopmen of sead sae solion of flow in in cai 5. Goerning eqaions 9. Solion procedre 7.5 Resls 1.6 Conclsion 6
. A fas simlaion of ransien meal flow and solidificaion in a narrow cannel 7.1 Transien model deelopmen 7. Main cai model 9. Tin cannel model 5. Copling of e main cai and in cannel flows 5.5 Discreiaion of goerning eqaions 55.5.1 Main cai 55.5. Tin cannel 56.6 Solion procedre 57.6.1 Solion procedre in main cai 57.6. Solion procedre in in cannel 6.7 Smmar of nmerical meods 6.8 alidaion of e Proposed Nmerical Algorim 6.9 alidaing eloci disribion in e cannel 68.1 alidaing solidificaion analsis 7.11 alidaing locaions of liqid meal-air and solid-liqid inerfaces 71.1 erificaion of solidificaion rae sing measred secondar dendrie arm spacing 78.1 Significan Improemen in Compaional Speed 81.1 Frer alidaion rog a Parameric Sd 8 i
.1.1 Effec of canges in enilaion-cannel ickness 8.1. Effec of canges in wall emperare of e enilaion cannel 8.1. Effec of canges in e iniial meal eloci 8.15. Smmar and conclsions 86 5. Some Concerns and Fre Researc Direcions 88 6. References 9 Appendi A Renolds lbricaion eqaion afer inclding e effec of ineria 98 Appendi B Discreiaion of momenm eqaions 17 Appendi C Esimaion of e locaion of solid-liqid inerface 111 CRRICLM ITAE 11 ii
Lis of Figres No Descripion Page Fig. 1.1 Scemaic represenaion of sand casing mold Fig. 1. Scemaic represenaion of o camber die cas macine 8 Fig. 1. Scemaic represenaion of seps in e o-camber die-cas process 8 Fig. 1. Scemaic represenaion of cold camber die-cas macine 9 Fig. 1.5 Scemaic represenaion of seps in e cold camber die-cas process 1 Fig..1 Conrol olme meod 1 Fig.. olme of flid meod 16 Fig..1 Fig.. Sraig cannel wi a recanglar cross-secion: e liqid meal eners from e lef-mos secion in e - plane, flows along e direcion, and en eis from e oer end. A pical conrol olme, defined arond e nodes of e mid-leel - plane, is sed o model e -direcion ea loss and sbseqen solidificaion, in e in cai. 6 6 Fig.. Flow car for e sead sae solion algorim Fig.. Fig..5 Grid independence sd condced a =.5 plane. [Te cai wid in direcion was non-dimensionalied as /L, Eq(.8), afer sing e leng of e cai as L=.1(m). Te eloci was rendered dimensionless as /, Eq(.8), afer emploing e caracerisic eloci ale of =1(m/s).] eloci desribion; a) eloci disribion a =., b) eloci disribion a =.5, c) eloci disribion a =.9 [Te cai iii
No Descripion Page wid in direcion was non-dimensionalied as /L, Eq(.8), afer sing e leng of e cai as L=.1(m). Te eloci was rendered dimensionless as /, Eq(.8), afer emploing e caracerisic eloci ale of =1(m/s).]. Fig..6 Temperare disribion along e cai ickness a ime=1s: a) Fig..7 Fig..1 Fig.. Fig.. Fig.. Temperare disribion a =.; b) Temperare disribion a =.5; c) Temperare disribion a =.9. Coordinae in e direcion was non-dimensionalied as / H o, Eq(.8), wile sing H o =.1(m) as e cai ickness. Eolion of e solid liqid inerface wi ime for =1: a) Te inerface locaion a =.5s, b) Te inerface locaion a =1s. [Te cai leng in direcion was non-dimensionalied as /L, Eq(.8), afer sing e leng of e cai as L=.1(m). Coordinae in e direcion was non-dimensionalied as / H o, Eq(.8), wile sing H o =.1(m) as e cai ickness.] A scemaic sowing e ick and in secion of a pical die-cas par A scemaic sowing e main cai and e in cannel of a diecas die conneced rog a c-off plane. Te in-cannel geomer sed for model alidaion: e lef- and rig-side segmens are 1 mm and.5 mm ick, respeciel. Te saggered grid arrangemen (based on SIMPLE algorim) was emploed for soling e in-plane flow ariables. 5 8 9 5 56 Fig..5 Te proposed solion algorim for soling flow and emperare 58 i
No Descripion Page ariables in e in cai. Fig..6 Fig..7 Fig..8 Fig..9 Fig..1 Fig..11 Fig..1 Fig.1 A scemaic of a die-cas die wi so sleee and plnger: 1) So sleee, ) Plnger, ) Saionar alf of e die-cas die, ) Ejecor alf of e die-cas die, 5) Mold cai, 6) enilaion cannel. Tpical plnger eloci profile (IPS = inces per second) posiion is e disance plnger raels dring e die cas process. A picre (a fll so ) of a par made sing e die-cas process. Te oerflows are creaed wen e meal fron, afer filling e main cai, fills p e macined oerflow pockes in e diecas mold. enilaion cannel is las o fill-p. Flow analsis resls sing FLOWD of e meal flow and solidificaion in e main cai. Temperare disribion in e considered cai of e die-cas die, filled wi liqid meal a e end of e fill process. A scemaic of e considered enilaion cannel: e sepped profile is necessar o solidif and conain e oerflowing meal. Te dimensions a, b and c are lised in Table.1. A scemaic sowing a pical cross-secion of e sepped enilaion cannel Comparison of eloci disribions in e enilaion cannel: (Lef a.5s) secion I and (Rig a.5s) secion II of Figre.1. Noe a e plo gies elociies aeraged along e ickness direcion. 59 6 65 66 66 67 69 69
No Descripion Page Fig.1 Fig.15 Comparison of emperare disribions in e enilaion cannel: (Lef a.5s) secion I and (Rig a.5s) secion II of Figre.1. Locaions of e free srface: Proposed nmerical algorim (op), FLOWD (boom) a a).1s, b).18s, c).9s, d).5s, e).6s [In all iews, e orional ais along direcion is along e cannel leng, wile e erical ais is e wid of e cannel (m) in direcion] 71 7 Fig.16 Eperimenall obsered solidified meal in e enilaion cannel 7 Fig.17 Locaion of e solid-liqid inerface prediced b simlaions: e proposed algorim (op), commercial sofware FLOWD (boom) a a).6s, b).55s, c).6s. Te ellow (lig) color represens e liqid meal wile e green (dark) color signifies solidified meal. 78 Fig.18 Cross secion of e casing sed o measre SDAS (X magnificaion) Fig.19 Temperare isor a e cenreline of locaion I in Figre.1; 79 81 Fig. Fig.1 e solidificaion rae, ν, is obained from e slope of e cre. Meal flow-leng s. cai ickness resl of e parameric sd. Meal flow-leng s. wall emperare resl of e parameric sd. 8 8 Fig. Meal flow-leng s. meal eloci a e enrance of e 85 i
No Descripion Page enilaion cannel resl of e parameric sd. Figre 5.1 Te measre solidificaion cre for alminm A8 allo 9 Figre 5. Figre 5. Te firs-deriaie cre, obained from e slope of e solidificaion cre sown in Figre 5.1, is oerlaid on e original solidificaion cre. Non-dendrie srcres seen in e micrograp of a secion of a in enilaion cannel. 9 91 Fig B.1 Saggered grid arrangemen 17 Fig B. Grid lines arrangemen near e cannel walls 19 Fig. C1 Meal solidificaion in e cannel 11 ii
Lis of ables Table 1.1 Properies of allos 6 Table.1 Properies of A8 alminm 5 Table.1 Tin cannel dimensions 68 Table. ales of ariables in Eqs.(.-.1) 8 iii
1 Caper 1: Inrodcion 1.1 Casing Processes Tere are seeral casing meods a are sed o prodce lig meal pars. Te mos widel sed are Sand casing Permanen mold casing Die casing sall economic consideraions are e driing force in deciding wic casing process can be sed. Te sand casing process reqires e leas amon of p-fron inesmen in ooling. B pars canno be prodced wi close olerances and minimm macine sock 1. I will reqire era macining operaions, wic will drie e par price p. Permanen mold reqires p-fron inesing in ooling. B pars can be cas wi mc closer olerances and less macining operaions. De o inensie cooling, pars can be prodced in a mc sorer ccle ime, compared wi sand casing. Te die-cas process reqires a large p-fron inesmen in ooling. De o ig pressre sed dring e die-cas process, pars can be prodced wi close olerances and minimm macine sock. 1. Sand Casing Sand casing is e oldes wa o prodce near ne-sape pars. Sand casing molds (Figre 1.1) are made sing green or cemicall bonded sand. Green sand molds se 1 Macine sock is a maerial added o e casing srface for sbseqen macining operaions wi e prpose o aciee beer srface finis and closer olerances an die cas process allows.
eier a mire of naral sand and cla or sneic sands. A pical sand casing mold as a gaing ssem, risers and cills. Figre 1.1 Scemaic represenaion of sand casing mold 1. Permanen Mold Casing Permanen mold casing is referred o as a meod of casing in wic e mold is no desroed dring eracion of e casing. Permanen molds are capable of prodcing large nmber of e same casing. Casings prodced in permanen molds ae generall finer grain srcre and sperior mecanical properies compared wi sand casings. Casings also ae less gas porosi, major defec of e die-casings. Permanen mold as e following major componens: 1. Gaing ssem, wic direcs liqid meal ino e cai a a seleced rae.. Feeding ssem, wic feeds liqid meal o icker areas of e par dring e solidificaion period.
. Cills, wic complemens e feeding ssem b cooling icker areas of e par.. ening ssem, wic allows gases o escape dring e cai fill process. In general, e permanen mold casing is operaionall er similar o a sand casing. I emplos grai as a feeding meod. In order o ensre proper filling of e casing, sfficien ead as o be proided. Posiion of e gaing ssem, risers, and cill as o allow direcional solidificaion, saring from e areas of e casing awa from e gae and moing ino e direcion of e gaes and feeders. Incorrecl designed and posiioned gaing ssem will resl in a sor fill and srink porosi. Misakes in e design of e feeding ssem and cills will resl in ecessie srink porosi, or longer dwell ime. Incorrecl placed and sied enilaion cannels will resl in ecessie gas porosi in e casing. Tere are ree major processes a are crrenl sed o prodce casings in permanen molds: Grai casing Low-pressre casings Conerpressre casing 1..1 Grai Casing Grai casing is a basic casing process a ses grai o fill e cai of e mold. Tis process can be sed for simpl saped pars a are no going o be sed in ig sress or leak free applicaions.
1.. Low-pressre Permanen Mold Casing Low-pressre permanen mold casing is a process a ses pressre o feed meal in o e cai. Casings prodced b is meod ae a iger densi and lower gas and srink porosiies. Molen meal is fed from e boom of e cai rog e riser be nder some pressre (.5.8 Bar). Adanages of is meod are 1. Te process can be easil aomaed, wic allows conrol of meal eloci, redces e rblence of e meal flow and minimies air enrainmen.. A ermeicall sealed frnace minimies meal oidaion and aoids nwaned inclsions.. Meal is fed from e boom of e ba wic allows feeding cleaner meal ino e cai of e mold.. Direcional solidificaion o e riser allows feeding meal nil e casing is compleel solidified. Tis redces e amon of srink porosi. 5. Tis meod allows prodcing qali casing wi inner walls. 6. Hig casing ield 1.. Coner-pressre Casing Conerpressre casing is a meod a ses low pressre o feed meal ino e cai from e boom of e mold, similar o e low-pressre permanen mold casing meod. As e cai is filled wi liqid meal, e pressre consanl increases wic sppresses e drogen precipiaion. Coner-pressre permanen mold casing meod allows acieing e iges mecanical properies in a casing. Te pressried cai eliminaes srink porosi wio sing risers.
5 1. Die Casing Te earlies eamples of die casing b pressre injecion, as opposed o casing b grai pressre, occrred in e mid 18s. A paen was awarded o Srges in 189 for e firs manall operaed macine for casing prining pe. Te process was limied o priner s pe for e ne ears, b deelopmen of oer sapes began o increase oward e end of e cenr. B 189, commercial applicaions inclded pars for ponograps and cas regisers, and mass prodcion of man pes of pars began in e earl 19s. Te firs die-casing allos were arios composiions of in and lead, b eir se declined wi e inrodcion of inc and alminim allos in 191. Magnesim and copper allos qickl followed, and b e 19s, man of e modern allos sill in se oda became aailable [1]. Te die-casing process as eoled from e original low-pressre injecion meod o ecniqes inclding ig-pressre casing (a pressres eceeding 5 ponds per sqare inc), sqeee casing, and semisolid die casing. Tese modern processes are capable of prodcing ig inegri, near ne-sape casings wi ecellen srface finises. Allos of alminm, copper, magnesim, and inc are mos commonl sed for casing (see Table 1.1): Alminim is a ligweig maerial eibiing good dimensional sabili, mecanical properies, macinabili, and ermal and elecrical condcii. Copper allo is a maerial wi ig sreng and ardness. I as ig mecanical properies, dimensional sabili, and wear resisance.
6 Magnesim is e liges cas allo. I is abo imes liger an seel and 1.5 imes liger an alminim. I as a beer sreng o weig raio an some seel, iron and alminim allos. Zinc is e easies allo o cas. I can be sed o prodce casings wi.5 mm wall ickness. Table 1.1 Properies of e allos [] Alminm Magnesim Zinc Tensile sreng, psi 1 7 1 Yield sreng, psi 1 (. pc offse) Sear sreng, psi 1 8 1 Faige sreng, psi 1 1 7 Elongaion, pc in in..5. 1 Hardness (Brinell) 8 6 8 Specific grai.71 1.8 6.6 Weig, lb/c. in..98.66. Meling poin (liqid), F 11 115 78 Termal condcii, CG5..16.7 Termal epansion, in./in./ F 1 6 1.1 15. 15. 1.5 Die-cas Process Hig-pressre die casing is sed for a wide range of applicaions in all major indsries. Adanages of e alminm die casings are: 1. Hig mecanical properies in combinaion wi lig weig.. Hig ermal condcii.. Good macinabili.. Hig resisance o corrosion. 5. Pars can be prodced wi no or a limied amon of macining.
7 6. Pars can be cas wi reprodcable and close dimensional olerances. 7. Low scrap rae. 8. Hig prodcion rae 9. Tin cross secions Die casing is a precision manfacring process in wic molen meal is injeced a ig pressre and eloci ino a permanen meal mold. Tere are wo basic die-casing processes: 1. Ho camber process.. Cold camber process. 1.5.1 Ho Camber Process In a o camber die-cas macine (Figre 1.), a meal injecion ssem is immersed in e molen meal. Adanages of o camber die-cas process are 1. Ccle ime kep o a minimm.. Molen meal ms rael onl a sor disance, wic ensres minimm emperare loss dring ccle ime. Te o camber process can be sed onl for allos wi a low meling poin (lead, inc). Allos wi a iger meling poin will case degradaion of e meal injecion ssem. Te o camber die-cas process as e following seps: 1. Hdralic clinder applies pressre on plnger (Figre 1.).. Plnger pses meal from e sleee rog e gaing ssem ino e cai (Figre 1.a).
8. Hig pressre is mainained dring e solidificaion process.. Afer solidificaion is complee, e die opens (Figre 1.b). 5. Te par is ejeced from e cai (Figre 1.c). Figre 1.. Scemaic represenaion of o camber die-cas macine a b c Figre 1.. Scemaic represenaion of seps in e o camber die-cas process: a. plnger pses meal from e sleee rog e gaing ssem ino e cai; b. afer solidificaion process is complee, e die opens; c. e par is ejeced from e cai.
9 1.5. Cold Camber Process Te cold camber die-cas process is sed for allos wi a iger meling poin (alminim, magnesim, brass). In a cold camber die-casing macine (Figre 1.), e meal is in conac wi e macine injecion ssem onl for a sor period of ime. Figre 1.. Scemaic represenaion of cold camber die-cas macine A pical process consiss of seeral seps (Figre 1.5): 1. Molen meal is ladled ino e so sleee (Figre 1.5a).. Hdralic clinder applies pressre on e plnger (Figre 1.5b).. Te plnger pses meal from e sleee rog e gaing ssem ino e cai (Figre 1.5c).. Hig pressre is mainained dring e solidificaion process (Figre 1.5d). 5. Afer solidificaion is complee, e die opens (Figre 1.5e). 6. Te par is ejeced from e cai (Figre 1.5f).
1 a b c d e Figre 1.5. Scemaic represenaion of seps in e cold camber die-cas process: a. molen meal is ladled ino e so sleee; b. dralic clinder applies pressre on plnger; c. plnger pses meal from e sleee rog e gaing ssem ino e cai; d. ig pressre is mainained dring solidificaion; e. afer solidificaion is complee, e die opens; f. e par is ejeced from e cai. f Disadanages of e die cas process are: 1. Porosi is e major defec of e die cas process,. Hig cos of e die-cas die. 1.6 Callenges of meal flow in in caiies Recen rends in e indsr o redce energ consmpion, redce mass of componens, and aciee greaer efficienc of e end ser prodcs resled in more comple die-cas pars. Tin wall casings, in combinaion wi new maerials, offer weig redcion wi
11 increased sreng. Secondar operaions like welding and ea reamen ae raised qali reqiremens for ese igl engineered casings. In order o aciee e greaer srcral niformi, ig-efficienc acm ssems are roinel sed on die cas dies. Major problems a increase cos and limi e se of die-cas pars are porosi and blisers. Gas porosi sall resls from improperl design flow paern or cooling ssem, inadeqae sie of e enilaion ssem. Hig emperare gradiens beween e molen meal and die-cas die cai seel reqire more carefl consideraion a e process design sage. Nmerical simlaions are reglarl sed nowadas o predic problems dring molding and opimie mold design. Flow and ermal analses in die casing are condced b commercial sofware inclding FLOWD, EKK and MAGMASOFT. Sc nmerical analses elp o predic defecs of e die cas process sc as gas and ermall indced porosiies, cold flow, and premare solidificaion. Prpose of e presen work is o simplif nmerical analsis of e liqid meal flow and solidificaion in a in cai b emploing e Renolds s lbricaion approimaion. Redcing e ree-dimensional Naier-Sokes eqaions o a wodimensional flow eqaions based on e Renolds lbricaion approimaion allows one o eliminae eloci calclaions in e raerse direcion. I simplifies compaional domain from ree dimensions o wo dimensions, redce compaional ime significanl, and allows one o aciee e solion of e flow and solidificaion problem mc faser wi reasonable accrac.
1 Caper : Lierare Oeriew.1 A Hisorical Oeriew of Nmerical Meods Nmerical analsis is e area of maemaics a soles differenial eqaions a describe real world problems b nmerical approimaion. Hisor of nmerical approimaions can be raced back o 165 BC wen Rind Paprs of Egp sed a roo-finding meod for soling eqaions []. Arcimedes of Sracse in 87-1 BC sed nmerical meods for calclaing lengs, areas, and olmes of geomeric figres []. Man crren nmerical approimaions are based on a fndamenal work of Isaac Newon and Gofried Leibni []. Nmerical meods for roo-finding and polnomial inerpolaion firs inrodced b Newon sill find wide se in modern algorims. Conribion of famos maemaicians of 18 and 19 cenr Eler (177-178), Josep-Lois Lagrange (176-181), and Karl Friedric Gass (1777-1855) laid e fondaion for reaing nmerical meods as an independen branc of e maemaical science. Beginning of modern nmerical analsis can be aribed o work b Jon on Nemann and Herman Goldsine [5]. In or das, deelopmen of new compaional plaforms as well as deelopmen and consan eolion of programming langages allows one o implemen more sopisicaed, more powerfl nmerical algorims. Forran sill remains e mos poplar programming langage for implemening nmerical algorims. Togeer wi oer programming langages inclding C, C++, and JAA, i allows one o deelop new engineering sofware-based ools for soling raer comple engineering problems. In recen ears, programming langages a combine nmerical programming and grapical ools ae gained poplari. MATLAB is one of e mos poplar was of
1 doing nmerical compaions, wile Maple and Maemaica are e mos poplar packages sed for soling e maemaical problems analicall.. Finie Difference Meod To find solion of parial differenial eqaion (PDE), compaional domain as o be discreiied ino finie difference grid. Lines a diide compaional domain called finie difference grid. Poins of inersecion of e grid lines called grid poins. One of e nmerical procedres a sole PDE a e grid poins are called finie difference meod. Afer finie difference grid is esablised, a finie difference approimaion as o be deeloped i is ofen done b wriing Talor series a eer grid poin for e dependan ariable. Ten e solion of e PDE in erms of e solion of a ssem of algebraic eqaions can be fond [6].. Conrol olme Meod (CM) Te compaional domain is sbdiided ino a finie nmber of non-oerlapping conrol olmes b orogonal b no necessaril niforml-spaced grid lines. Conrol olme meod (see Figre.1) is based on e principal of acieing fl balance in a finie conrol olme [7]. Algebraic eqaions are se b acieing e balance of a psical qani in a conrol olme. A scalar grid poin P, locaed a e cener of eac conrol olme, is sed for soring ales of ariables sc as pressre and enalp. eloci componens are sored a conrol olme faces. Howeer sorage of eloci componens are saggered wi respec o e faces: elociies are sored a wes and eas faces, wile elociies are sored a nor and so faces. So conrol olme sorage for ecor qaniies are differen from e scalar componens is arrangemen is sed o preen e esimaion of nrealisic resls for pressre and elociies.
1 Figre.1 Conrol olme meod.. Free Srface Approimaion sing a Nmerical Tecniqe..1 Marker and Cell (MAC) Meod Marker and cell meod was firs inrodced in 1965 [8]. Te meod is based on placing a se of marker paricles wiin a flid. Paricles can moe wi e flid b ae no olme or mass. Compaional cells a conain e marker paricles are considered occpied b a flid. Compaional cells wio e marker paricles are considered emp. Free srface is considered o be in a cell a as a marker paricle and borders wi a leas one emp cell. Adecion of a free srface is comped based on moemen of e paricles wi locall-inerpolaed flid elociies. A prominen disadanage of e MAC meod is e ig sage of CP ime for racing all marker paricles. Oer disadanages inclde los abili o rack olme and impracical implemenaion for ree dimensional flows.
15... Srface Marker Meod In an aemp o redce e CP and memor reqiremens of e MAC meod, marker paricles were placed on a free srface onl [9]. Tis meod did improe e comper memor sage... Free Srface Approimaion sing olme of Flid (OF) Meod olme of flid meod [1] (see Figre.) was deeloped o ake adanage of olme racking capabiliies of e MAC meod and redce is CP memor reqiremens. In order o minimie sorage space, onl one of e flow ariables (pressre, eloci, emperare) is sored in a single conrol olme. Following e same meodolog, flid olme fracion is sored in eac compaional olme. olme fracion is a sep fncion a can be eier ero or one. Free srface is locaed beween compleel filled and emp cells. Based on a fracion of flid in e pariclar cell, slopes and e crare of e free srface can be easil comped as well. To compe srface adecion in ime in D, e kinemaic eqaion for flid fracion can be sed: F F F (.1) were F is a fracion of flid in a cell, and are elociies in and direcion respeciel. olme of fracion meod as proen o be a robs and accrae in racking a free-srface flow [11]. I is a sbsaniall simplified MAC meod wio an added compaional coss.
16 Figre. olme of flid meod.5 Renolds s lbricaion approimaion In 1886 Osborne Renolds [1] deried e differenial eqaion goerning e pressre disribion in in film of lbricaion. If e cai is assmed o be in, and o of plane flow is negleced, e ree dimensional problem can be simplified o a wo dimensional one. Consider one dimensional Naier-Sokes eqaion: p (.) were,,, and p are densi, iscosi, and pressre respeciel. If we assmed a Deriaie of wi respec of ime is eqal o ero, eloci is independen of and doesn ae or componen, Pressre is drosaic in direcion, Cai is assmed o be in and aring slowl along e and direcions, Inerial forces are mc smaller an iscos forces,
17 en e Naier-Sokes eqaions, on being inegraed along e ickness () direcion, can en be redced o e well known Renolds lbricaion eqaion [1]: P P 6 1 1 1 (.) Here is e cai ickness wile i and i are e and direcion elociies of e pper and lower srfaces. For or case wi bo e walls of e cannel being saionar and no ariaion in flow in e direcion, Eq.. redces o S P (.) were S is flow condcance and P is liqid pressre. Eqaion (.) indicaes a e Naier Sokes eqaion is redced o a lbricaion eqaion nder e assmpion a ineria effecs can be negleced. Howeer in order o implemen lbricaion eqaion for nmerical analsis of e liqid meal-flow in e in caiies, e ineria effecs canno be eclded from or consideraion de e ig ickness-based Renolds (~ 6,) nmber of e process. Hence e lbricaion eqaion wi ineria effecs copled wi energ eqaion will be sed o nmericall compe liqid meal flow wi solidificaion in a narrow cannel..6 Scope and objecie In e presen work, nmerical algorim for modeling e sead as well as ransien flow of liqid meal and is sbseqen solidificaion in a in cai is deeloped. Meal flow appens in a in enilaion cannel a e end of e die cas process. Main prpose of is algorim is o deelop a nmerical abili o calclae qickl and wi reasonable degree of accrac ow far liqid meal will be able o rael in e enilaion
18 cannel before solidificaion occrs. I will enable engineers o design die enilaion cannels wio soling e fll Naier-Sokes eqaion, wic akes a long ime owing o a er large aspec raio of e cannel geomer, i.e., e cannel ickness is in millimeers wile e poenial leng of e liqid meal flow is in ens of cenimeers. Objecies of is esis proposal are: 1. Deelop a -D eqaion se for e flow and solidificaion of liqid meal in in caiies sing e Renolds s lbricaion approimaion. Deelop a nmerical algorim o anale sead-sae and ransien meal flow in e in cannel wi solidificaion. alidae or nmerical simlaion agains eperimenal daa To aciee ese objecies, we will presen e resls in e following wo pars: (a) a proof-of-concep sd in caper wi a simpler, sead flow of meal in a cannel wile e skin of solid meal growing on e cannel walls, (b) a more adanced, ransien flow in e cannel in caper wi a moing meal-air inerface wile e solid-meal skin grows on e walls.
19 Caper : A fas nmerical simlaion for modeling simlaneos meal flow and solidificaion in in caiies sing e lbricaion approimaion Absrac: A nmerical algorim for modelling sead flow of liqid meal accompanied b solidificaion in a in cai is presened. Te problem is closel relaed o a die cas process and in pariclar o e meal flow penomenon obsered in in enilaion cannels. sing e fac a e rae of meal flow in e cannel is mc iger an e rae of solidificaion, a nmerical algorim is deeloped b reaing e meal flow as sead in a gien ime-sep wile reaing e ea ransfer in e ickness direcion as ransien. Te flow in e in cai is reaed as wo dimensional afer inegraing e momenm and conini eqaions oer e ickness of e cannel, wile e ea ransfer is modelled as a one-dimensional penomenon in e ickness direcion. Te presence of a moing solid-liqid inerface inrodces non-lineari in e resling se of eqaions, and wic are soled ieraiel. Te locaion and sape of e solid-liqid inerface are fond as a par of e solion. Te saggered grid arrangemen is sed o discreie e flow goerning eqaions and e resling se of parial differenial eqaions is soled sing e SIMPLE algorim. Te ickness direcion ea-ransfer problem accompanied b pase cange is soled sing a conrol olme formlaion. Te resls are compared wi e predicions of e commercial sofware FLOWD wic soles e fll ree-dimensional se of flow and ea ransfer eqaions
accompanied wi solidificaion. Te Renolds s lbricaion eqaions accompanied b e rog-e-ickness ea loss and solidificaion model can be sccessfll implemened o anale flow and solidificaion of liqid meals in in cannel dring e die cas process. Te resls were obained wi significan saings in CP ime..1 Inrodcion Global compeiion for manfacring speriori as enered a new sage. As economiss prediced for qie some ime, ere is no a single conr or a region wic can claim absole world dominance in manfacring capabiliies. Widespread se of nmerical analsis sofware and free ecange of informaion allow engineers arond e world o design, anale, and bring o manfacre new prodcs in record imes. Die cas indsr is no an ecepion. Flow, ermal, and disorion analses are e inegral par of deeloping die cas process parameers as well as die-cas die design. B de o an increase in complei of par design, i akes longer o go rog e complee nmerical analses ccle; in man cases, i akes seeral ieraions o aciee e desired resls. Wi e deelopmen of faser compers as well as more efficien and accrae nmerical approimaions, engineers can eamine more design opions and aciee beer resls in a mc sorer ime. Howeer, in spie of e laes adances in nmerical simlaions, deailed eaminaions of e flow and solidificaion inside in cannels remain callenging.
1 Liqid flow and solidificaion in cannels is a comple penomenon wic gained mc aenion of researcers in e pas few decades. Complei of e flidflow psics and solidificaion, as well as canges in e flow regime along e leng of e cannel, creae qie a few callenges in e deelopmen of nmerical algorims o predic e locaion and sape of e liqid-solid inerface as well as eloci and emperare disribions in e cannel. Deailed descripions of e flid flow, ea ransfer and solidificaion in e sraig cannels was condced b Epsein and Cng [1]. Te nmerical analsis of flid flow and solidificaion in cannels reqires e solion of e D Naier Sokes eqaions. Te in caiies wi ig leng-oickness aspec raios reqire qie a large nmber of compaional cells in order o aciee accrac and conergence. Man nmerical algorims were deeloped o anale flow and solidificaion beween wo parallel plaes. In order o simplif e D problem, i is redced ino a D one, were e original goerning eqaions are conered from e Caresian coordinae ssem ino e crilinear coordinaes. Te nmerical model deeloped b B. Weigand e. al [15] sccessfll soled e wo-dimensional Naier-Sokes eqaion copled wi e ea ransfer eqaion. Te nmerical analsis of ea ransfer dring solidificaion in a cannel, in mos cases, reqires e conersion of e ea condcion eqaion from e Caresian coordinaes ino e crilinear coordinaes as well [16]. Tog seeral nmerical meods ae been proposed o model solidificaion of maerials in e recen pas, no all of em are sefl for modelling e flow and solidificaion in in cannels. For eample, a generalied finie difference meod was sown o be an efficien ecniqe o model e solidificaion of meals in [17]. Howeer,
e model was deeloped for saionar liqid meals and reqires a nmerical procedre for idenificaion and generaion of nodes rogo e compaional domain; sc a meod will be difficl o implemen in in cannels as i will reqire o generae large nmber of compaional nodes. Similarl, a celllar aomaon model, sed for microscopic modelling of ea ransfer and copled wi e finie olme meod for macroscopic modelling of solidificaion process, was inrodced b Yao e al. [18]. Te model allows for accrae predicion of e solidificaion parameers in bo macro and micro scales. Howeer, e model is implemened for saionar flids onl and reqires finie olme descriiaion in e direcion of solidificaion. A nmerical algorim for modelling wo-pase flow was proposed in [19] were forcing erms are added o e Naier-Sokes eqaions o accon for e properies ariaion beween e wo flids. Te meod doesn accon for e ineria drien flows as well as does no predic solidificaion, and reqires compaional descriiaion of e domain in e direcion of e inerface beween wo flids dring adecion. In e las few ears, seeral finiedifference and finie-olme based meods ae been deeloped and implemened o model solidificaion of meals dring casing [-1]. Tese models concenrae on deeloping accrae and efficien algorims o predic emperare disribion dring e solidificaion process. Sc models reqire a large nmber of compaional cells o be generaed in e direcion of solidificaion-fron moion and do no accon for e effecs of e ineria drien flow on emperare disribion in e liqid pase. Anoer approac is o coner e original D goerning eqaions ino e dep-aeraged eqaions; is approac is widel sed in e sallow-dep flid-flow models [-]. Howeer, ese algorims sill reqire e solion of e rblence
models. Seeral rblence models ae been deeloped o be sed wi e depaeraged goerning eqaions [-6]. Alog e aboe-menioned approaces simplif e goerning eqaions, e inclsion of rblence and oer deails in nmerical models reqires sbsanial amon of CP ime o aciee conerged solion. Moreoer, conersion of e goerning eqaions ino crilinear coordinaes in [-6] creaes added compleiies in e deelopmen of e nmerical algorim. In is paper, we propose a simpler, less-compaionall epensie approac were e ree-dimensional problem of flow and solidificaion in a in cannel is redced o a wo-dimensional one based on e Hele-Saw approimaion [7]. Tis approac is based on e Renolds lbricaion eor. Fndamenal assmpion of e lbricaion eqaion is a in in, slowl-aring caiies wi e flow a relaiel small Renolds s nmbers, e ineria forces are mc smaller an e iscos forces and can be negleced. In sc a siaion, e ree-dimensional Naier-Sokes eqaion can be redced o a Renolds s lbricaion eqaion and sed o analse flows in in caiies [1]. In spie of is limiaions, e Renolds s lbricaion formlaion remains e fondaion of e nmerical analsis in in caiies. Owing o e ig-speed nare of e die cas process [8], ineria effecs in e meal flow canno be negleced. Some aemps were made o inclde e inflence of ineria in e lbricaion eqaion. For eample, alidi of inegraion of e goerning eqaion oer a cai ickness afer assming a parabolic disribion of e eloci was eperimenall confirmed [7]. Similarl, e ineria effecs in in-cannel flows
were inclded in e lbricaion eqaion and alidi of e alered lbricaion eqaion for a wide range of Renolds nmbers was esablised [-1]. In is paper, a nmerical solion of flow in a in cai sing e lbricaion approimaion along wi a conrol-olme based solidificaion model will be presened. Te saggered grid arrangemen is sed o discreie e goerning eqaions. Ten an ieraie SIMPLE algorim is sed o sole e discreied eqaions for momenm in e cenre-line D plane wiin e cannel, wile anoer ieraie sceme is sed o model e o-of-plane solidificaion. Before presening a deailed flow model, seeral dimensionless parameers ae o be eamined in order o idenif e driing forces conrolling e flow in in cannels dring e die-cas operaion. Relaie imporance of e inerial forces compared o e srface ension forces can be eamined sing e Weber nmber: l We (.1) were, l,, and are densi of liqid meal, aerage eloci, eig of e cannel, and srface ension, respeciel. sing e die-cas alminim properies lised in Table.1 and assming e aerage flid eloci in e enilaion cannel o be 1.6 ms -1 wile aking e cannel gap o be.5 m, e corresponding Weber nmber will be 65*1.6 *.5 We.9.86 Tis ale indicaes a e iniial ineria forces, before e meal flow in e cai is affeced b solidificaion, are almos wo imes iger an e srface ension forces.
5 Te raio beween e srface ension and iscos forces as o be sdied ne. Te Capillar nmber, wic represens e raio of e iscos forces compared o e capillar forces, can be epressed as Ca (.) Table.1 Properies of A8 alminim [] 65[kg m - ] 9 [J m -1 s -1 K -1 ] Hea condcion coefficien (k s, k l ) Specific ea (C) 98 [J kg -1 K -1 ] Liqid meal iscosi 1 - [kg m -1 s -1 ] Solidificaion poin 58 [ C] Laen ea (L f ).971 5 [J kg -1 ] Meal densi sing e ales lised in Table.1, e Capillar nmber can be esimaed o be Ca 1 *1.6.86 1.86e Tis resl indicaes a e srface ension forces are wo orders-of-magnide larger an e iscos forces. Ealaion of Eqs.(.1) and (.) lead o e conclsion a inerial forces dominae e meal flow in in cannels dring e die-cas operaion.. Deelopmen of sead-sae solion of flow in in cai Meal flow and solidificaion in a in cannel is a sbjec of is sd. Molen meal is fed from e lef of e cannel (see Figre.1) in posiie direcion. Flow is indced de a pressre difference beween e lef side (inle) and rig side (ole) of e cannel. In e presen sd, i is assmed a a sead flow of meal as been
6 esablised before e onse of solidificaion a e walls. (Sc an assmpion is jsified since e filling of sc cannels appen wiin a second.) Afer a sfficien amon of ea as been eraced from e meal, a solid-liqid inerface formed ne o e cannel walls grows and mees a e cenre of e cannel. Flid flow is assmed o feed e solidificaion fron wile e ea is being eraced. Figre.1. Sraig cannel wi a recanglar cross-secion: e liqid meal eners from e lef-mos secion in e - plane, flows along e direcion, and en eis from e oer end. Bo e meal and cannel are a spereaed emperare iniiall. Te cannel walls are sddenl cooled o a emperare below e solidificaion emperare. Solidificaion frons will be forming near e walls of e cannel, propagaing inside e molen meal. Dring solidificaion, e meal is moing nder a pressre-drien flow wi a prescribed inle eloci. Now we presen a comparison of e pical speed wi wic e meal solidifies erss e speed wi wic e meal passes rog e cannel sc a comparison will elp s o ignore solidificaion dring e filling of e cannel. A pical solidificaion rae S inside e orional cannel can be fond [5] sing properies of alminim, sown in Table.1 as
7 S T T m w ks (.) L f H o were, T m, Tw, L, H, f k s 65 1 S 9 5. 65*.97e *.1 7e 5 and - emperare of e meal, emperare of e wall, laen m s ea of fsion, aerage eig of e cannel, ea ransfer coefficien, and densi of e alminm, respeciel. Meanwile e pical rae for meal flow in e cannel * wio e solid-liqid inerface being presen is 1 ms -1. Te caracerisic ales clearl sow a e rae of solidificaion is mc smaller en e rae of meal flow in e cannel. In fac, as e solid-liqid inerface conerges a e cenre of e cannel, e rae of meal flow increases, and e raio of e solidificaion rae o e flow rae frer redces and goes almos o ero. Based on ese conclsions, we are jsified in deeloping or nmerical algorim for ransien solidificaion in e cannel accompanied b liqid-o-solid ea condcion, wile reaing e meal flow o be qasi-sead. Te proposed nmerical algorim is deeloped based on e assmpions, a a ime greaer an ero, e liqid meal is enering e cannel wi is emperare aboe e meling poin. De o eir low ermal resisance, e cannel walls are assmed o remain a a consan emperare below e meling poin o indce solidificaion. Since e ariaion in e solid-laer ickness wi posiion along e cannel leng is small, qasi sable-sae can be assmed for e ea condcion in e solid. Te liqid-meal emperare is aken o be a consan, wile e meal eloci a e cannel enrance is * Te pical eloci corresponds o e end of e die cas process, afer e main cai is filled and liqid meal is in e enilaion cannel.
8 considered o be fll deeloped and sead. All psical properies for bo e liqid and solid pases are considered consans. Problem is applicable o e meal die-cas process inoling flow in in caiies. A brief descripion of e process is gien below. Tin-wall casings, flow in enilaion cannels, ec., are some eamples were e proposed algorim can be ilied. Te proposed nmerical solions can be sed o redce e nmber of design ieraions emploing e fll D simlaion algorim. One of e was o redce e compaional ime is o redce a -D problem o a -D one. In e presen case, e die-cas mold cai is in and ence e flow in e erical direcion is negleced. Assmpion of negligible inerial forces allows one o redce e Naier-Sokes eqaions o e Renolds lbricaion approimaion. B sc an approimaion is alid onl for small Renolds nmbers. A major caracerisic of e die casing process is meal flow nder ig pressre and eloci, and since e Renolds nmbers can aciee qie ig ales, e ineria effecs canno be negleced. Hence a modified lbricaion approimaion afer inclding e inerial effecs is emploed o predic e ickness-aeraged in-plane flow in e die-cas mold. Te presened algorim considers e ickness-aeraged -D sead-sae flow in a die-cas mold in e in-plane direcions and 1-D along-e-ickness ransien earansfer. A se of nonlinear parial differenial eqaions is deeloped o sole for flow wic are en discreied sing e finie difference sceme afer emploing e SIMPLE algorim. A direc conrol-olme based formlaion is proposed o model ea ransfer and solidificaion along e ickness direcion. Te wo ses of eqaions are soled in an ieraie manner sing Malab and e obained resls are alidaed b
9 comparing wi e ones acieed sing e commercial sofware FLOWD, were e conrol olme meod is sed o discreie e goerning eqaions wile e enalp meod is sed o esimae emperare disribion in e cai.. Goerning eqaions We are considering a ree-dimensional flow in a sraig cannel wi recanglar cross-secion as sown in Figre.1. Te flow is considered o be incompressible, iscos, and Newonian. De o e fac a e rae of meal flow in e cannel is mc iger an e rae of solidificaion, a sead-sae flow a a gien imesep wi ransien ea condcion from liqid ino solid is assmed. Since e inerial effecs caraceried b ig Renolds s nmber are dominan in e flow, so e graiaional forces are negleced. Te goerning eqaions are epressed in e Caresian coordinae ssem wi coordinae in e direcion of flow (along e cai leng), in e direcion normal o e flow (along e cai wid), and in e direcion ranserse o e - plane (along e cai eig);,, and w are e corresponding elociies. Te goerning eqaions sed are e conini, momenm, and energ eqaions in e liqid and solid pases wi momenm and energ bondar condiions specified a e cannel walls, inle, and ole as well as a e solid-liqid inerface. Locaion and sape of e solid-liqid inerface is fond as a par of e solion of e FLOWD [] is a general prpose commercial CFD sofware wic soles ree-dimensional flid-flow and solidificaion problems sing e finie differen approimaion. FLOWD ilies e olme-of-flid ecniqe and e FAOR meod o rack free srfaces as well as solid-liqid inerfaces. Te wo eqaion k-e model is sed o resole e rblen properies of e flow. Te aeraged Naier Sokes eqaions copled wi e energ eqaion allow e sofware o aciee an accrae solion for rblen meal flow ndergoing solidificaion.
presened algorim. Te sead-sae conseraion eqaions goerning e ranspor of mass, momenm and energ are epressed as follows. Liqid region Conini eqaion: w (.) Momenm balance eqaions: p w (.5a) p w (.5b) p w w w w w w w (.5c) Energ balance eqaion: In differenial form, e energ balance eqaion is epressed as q T C T C (.6) were, q and T C,,, are specific ea, emperare of e meal, ime, and ea fl, respeciel. Solion of e goerning eqaions (.) o (.6) presens seeral problems. To begin wi, e conecie erms in e lef and side of Eq. (.5) are non-linear. All eqaions are copled becase eloci componens are presen in eac eqaion. On comparing e
1 rae of solidificaion (see Eq..) o e rae of flow, one can define a solidificaion parameer as S s (.7) Based on e flow and solidificaion caracerisic of e presened problem and e caracerisic -direcion eloci of =1m/s, e solidificaion parameer in or case will be.518 s 1..518 Meal flowing in in cannels is affeced b solidificaion, wic resls in e presence of a growing solid-liqid inerface ne o e cooler cannel wall. As e ime progresses, e solid-liqid inerfaces from e wo walls conerge a e cenre of e cannel; in oer words, a progressie redcion of e effecie cannel gap. Wen, e conseraion of mass of e flowing meal indicaes a s. In is limi wen s, e effec of solidificaion in erms of momenm ransfer on e meal flow is negligible, and ence e eloci field can be ncopled from e emperare field [1]. In oer words, we can se e ero eloci a e inerface o model e meal flow wile e energ eqaion is sed o esimae e cannel gap. In order o frer simplif e goerning eqaions, we condced an order-ofmagnide analsis o deermine e imporance of eac erm on e flow caracerisics. nder is, e dimensionless ariables were defined as
o o m w m o o H s s p L L H p H Lw W w w T T T T H H L L,,,,,,,, (.8) were, ildes designae dimensionless qaniies, s and W L, _, _,,, are leng of e cannel, dimensionless emperare, reference ime, reference eloci in and direcion, reference eloci in direcion L H _ and locaion of e solid-liqid inerface, respeciel. Owing o a small aspec raio of e cai eig o is leng and wid, leng and wid of e cai are considered on e same order of magnide and will be denoed b L along bo and. For noaional conenience, e ildes are dropped from non-dimensional ariables. On non-dimensionaliing e conini eqaion, Eq. (.), we ge _ w H W L L o (.9) Mlipling all erms of Eq. (.9) b _ L will resl in _ w L H W o (.1) We ae o define e caracerisic eloci in direcion. In order o insre a all e erms of Eq. (.1) are on e same order of magnide, e caracerisic eloci in direcion is defined as:
L H W L H W o o _ 1 (.11) Eq. (.11) indicaes a e caracerisic eloci in direcion is mc smaller an ose in and direcions, i.e., W becase H o <<L. Afer absorbing is conclsion, e reslan non-dimensional conini eqaion, Eq.(.1), redces o w (.1) Sbsiion of dimensionless ariables in e momenm balance eqaions, Eq. (.5), leads o p L H w L H o o _ (.1a) p L H w L H o o _ (.1b) p w w L H w L H w w w w L H o o o _ (.1c) De o e small cai aspec raio, i.e. L H o <<1, all erms on e order L H o or iger can be negleced. Ten e in-plane momenm balance eqaions resl in p w L H o _ (.16a) p w L H o _ (.16b) wile e momenm eqaion in e direcion ranserse o e flow redces o
p (.17) Eq. (.17) indicaes a e flid pressre is niform in e direcion regardless of e ineria effecs in e flow and ence, e pressre is p = f (,, ) regardless of e ig- Re caracer of e flow. Preios work on ig-speed flow in in cannels [9] as assmed a parabolic disribion of flow elociies. We also will assme a parabolic disribion of eloci along e and direcions for frer analsis: ) )(, ( (.a) ) )(, ( (.b) On being inegraed oer e ickness of e cannel, e conini eqaion, Eq. (.1), becomes (.1) Noe a based on e no-peneraion bondar condiion on op and boom, and e small cai sie in e direcion, w eloci ariaion is negligible and is se o ero (i.e., w ). On non-dimensionaliing Eq.(.16), e in-plane wo-dimensional momenm eqaions are epressed as P P (.)
5 Afer inegraing Eq.(.) across e cai ickness from o, e momenm eqaions become 6 p (.a) 6 p (.b) were, H Re modified Renolds nmber, as a cai aspec raio L and Re as a Renolds nmber wi e H For a deailed deriaion of e eqaion se Eq.(.), see Appendi A (Appendi A sows e deriaion of e eqaions for e ransien-flow case. Deriaion of e sead-sae flow eqaions, Eq. (.), is e same, ecep for e absence of e ransien erm).. Energ eqaion A general balance eqaion for energ is deeloped for e arbirar conrol olme sown on Figre.. On inegraing e conseraie form of Eq. (.6) oer e conrol olme afer neglecing ariaions in flid properies and eloci, we ge C CT CT q d (.) On rewriing e las wo erms of Eq. (.) as srface inegrals, e energ balance eqaion oer e fied conrol olme canges o Deriaion of e Eq.. is for wo dimensional ariaion of e cai ickness. Te second erm of e eqaions as denominaor of 6 insead of 1 in [1].
6 CT nda q nda C CTd ˆ ˆ (.5) CS CS Te conrol olme can be reaed as an open ssem a ecanges ea wi is srrondings and were mass can flow in and o, ence Eq. (.5) represens e energ balance a can be described as: Rae of ea accmlaion in conrol olme = ne rae of ea ranspor ino conrol olme (b flid flow) - ne rae of ea ransferred o of conrol olme o srronding rog condcion. Noe a de o ig Pecle nmbers (5) inoled in is problem, e energ ransfer beween e flid meal and e cannel wall, or beween e flid and solidified meal, is drien b conecion; e ea ransfer rog e liqid meal is aken o be prel conecie as well. Te condcion erms are ignored. Figre.. A pical conrol olme, defined arond e nodes of e mid-leel - plane, is sed o model e -direcion ea loss and sbseqen solidificaion, in e in cai. Bondar and iniial condiions
7 A e iniial ime =, e same niform emperare, T = 65 o C, is applied o liqid meal ling wiin e compaional domain. A e enrance ( = ), e flid emperare is se as T= 65 C. A e solid-liqid inerface, T s = T l = T mel. Te wall emperares a = and = are se o 1 o C. Te flow is drien b a niform eloci imposed a = locaion. Te oflow bondar condiion specified a e end of e cai, =L, is. Te no-slip bondar condiions are applied a e walls. Addiionall, a no-peneraion bondar condiion, in e form of e eloci gradien in e direcion normal o e wall being eqal o ero (i.e., ), is applied. An addiional condiion is needed o epress e eloci of e moing solid-liqid inerface as a fncion of ea ransfer in bo e solid and liqid pases. Tis is called e Sefan condiion and can be epressed [] as ds d T s T l L f ks s kl s (.6) were, k is ea ransfer coefficien, sbscrips s and l are designae solid and liqid meal, respeciel. In order o esablis e alidi of Eqs. (.1), (.) and (.5) a form e goerning eqaions for e presened problem, e were soled nmericall and e resls were compared wi e solion of e incompressible Naier Sokes eqaions fll-copled wi e ree-dimensional energ eqaion dring solidificaion a was soled sing e commercial sofware FLOWD.. Solion procedre Te ssem of dimensionless eqaions, Eqs.(.1)-(.), and Eq.(.5) gies a
8 complee maemaical formlaion of e presened problem of liqid-meal flow and solidificaion in a in cannel. Te solion inoles deerminaion of eloci and emperare disribion in e liqid pase, as well as e emperare disribion in e solid pase, of e in cannel. Te goerning eqaions in a liqid pase are copled rog e inerface (Sefan) condiion, Eq.(.6). Te solion of e Sefan condiion gies e locaion of e solid-liqid inerface as a fncion of ime and posiion along e leng of e cannel. Te problem is soled in a sraig cannel of recanglar cross-secion sown in Figre.1. A niform eloci is applied a e = locaion o drie e flow. Consan emperares are specified a = and = walls, wile e walls a = and = ma are considered adiabaic. Owing o e weak copling beween e momenm and energ eqaions, e emperare disribion wiin e compaional domain can be soled firs. Tis esablises e locaion and sape of e solid-liqid inerface, and s defines e bondaries of e liqid domain. Momenm eqaions are en soled sing e SIMPLE [7] procedre were e momenm and conini eqaions are soled in a copled manner. Te momenm eqaion, Eq (.), ses e gessed pressre field and soles for e preliminar elociies and. Ten e modified conini eqaion, Eq (.1), is sed o calclae e correced ale of e pressre field: P a (.7) In e sal incompressible form, e mass conseraion or e conini eqaion does no ae an pressre erm. An arificial compressibili erm a as o be added o e
9 modified conini eqaion, Eq.(.7), o allow for a solion of P. Te sal ales are.1 > a > 1. i is aken o be.8 [8] in e presen algorim. Solion procedre is described in Figre., and can be broken down as follows: 1. Gess pressre ales in e firs ime sep.. Sole e momenm eqaion o esimae e preliminar ales of e and eloci componens.. se e modified conini eqaion, Eq. (.7), o correc e pressre ales.. Correc elociies sing e new pressre ale and conine ieraing nil e conini Eq. (.1) is saisfied. A is poin of analsis, eig of e cannel in e momenm eqaions sill remains nknown. In e absence of solidificaion, is eqal o e cannel eig, and e momenm and conini eqaions alone will allow s o esimae e eloci and pressre disribions in e cannel. In order o close e ssem of goerning eqaions, e energ balance eqaion as o be soled o find emperare disribion in e cannel. Based on e resl of e energ eqaion and sing Sefan condiion, Eq. (.6), locaion of e solid-liqid inerface can be esablised for eer pariclar ime sep. On knowing e locaion of e solid-liqid inerface, ale can be pdaed and sed in e momenm eqaion for e ne ime sep. I is assmed a ere is a perfec conac beween e solidified meal and walls of e cannel. Walls of e cannel are assmed o ae ig ermal mass and condcii, and erefore, eir emperares remain consan dring e calclaion procedre.
Sar Make iniial gess of pressre field Sole e discreied form of e momenm eqaion, Eq. (.), o obain e iniial and elociies Sbsie e iniial elociies ino pressre correcion Eq. (.7) Sole pressre correcion Eq. (.7) o obain correcion ale se pressre correcion ale o obain correced elociies. se correced elociies o saisf e conini eqaion.1 No Tes for conergence Yes se final eloci ales o sole for energ balance, Eq. (.5) Calclae locaion of solid liqid inerface sing Eq. (.6) pdae local (Flid region eig) ales (Appendi C) Moe o e ne ime sep Figre. Flow car for e solion algorim
1 Noe a e momenm eqaions were soled in dimensionless form wile e energ eqaion was soled in dimensional form..5 Resls Goerning eqaions were soled as indicaed in secion. (Solion procedre). Te maerial properies sed in e resls presened in is secion are sown in Table.1. Te proposed algorim is erified for flow and solidificaion in a sraig cannel of a recanglar cross-secion (Figre.1). A e iniial ime sep iself, e flow is considered fll deeloped. Flow is drien b a niform aial eloci imposed a e enrance of e cai a =. A e ime =, meal emperare is considered o be 6 o C and a niform emperare of 1 o C is applied o e op and boom of e cai ( direcion). A e inflow bondar, e meal emperare is se o a consan 6 o C. Analses were rn for 1s. eloci, emperare disribion, and locaion of solid liqid inerface were ploed a ree locaions. eloci =1m/s was applied a = locaion. Proposed algorim was erified agains resls obain sing e commercial sofware FLOWD wic simlaed a fll-copled ree-dimensional flow analsis wi solidificaion. Cannel (Figre.1) dimensions are 11.1 (mm) in e,, and direcions, respeciel. Grid independence was insred b comparing -D resls of e analsis wi grid densiies 11,,,, 55, 66 sown in Figre.. Since e difference beween 55 and 66 resls are less an.1%, e analses were condced originall wi 55 grid. In order o redce anglari in e Te mes densiies are for soling e -aeraged eloci fields along and direcions
inerface-locaion plos, e mes densi along e ickness -direcion was laer aken o be 15 grid poins. Figre.. Grid independence sd condced a =.5 plane. [Te cai wid in direcion was non-dimensionalied as /L, Eq(.8), afer sing e leng of e cai as L=.1m. Te eloci was rendered dimensionless as /, Eq(.8), afer emploing e caracerisic eloci ale of =1 m/s.] Te goerning eqaions, Eqs. (.1) and (.), were soled sing e algorim described in e las secion. Conergence of e solion was jdged b e maimm cange in eac ariable ales dring eac ieraion. Te solion was considered conerged wen canges in a dimensionless ariables ale was less an 1-8. To erif analses obained sing e presened algorim, ree-dimensional flow and solidificaion solions from e commercial CFD code FLOWD were obained sing e same bondar and iniial condiions. Presened resls inclde flid eloci, emperare disribion, as well as locaion of e solid-liqid inerface. Tree conrol poins along e direcion a dimensionless locaions =., =.5, and =.9 were cosen for e plos of -aeraged elociies based on e solidificaion paerned obsered in e cai.
Resls presened in Figres.5a,.5b, and.5c sow eloci ariaion along e cai leng were e elociies prediced b or program are compared wi e elociies prediced b FLOWD. We obsere a a fairl close flow-predicion is made b or simlaion based on e lbricaion approimaion. We also obsere a e - direcion eloci increases wi. a. b. c. Figre.5. eloci desribion; a) eloci disribion a =., b) eloci disribion a =.5, c) eloci disribion a =.9 [Te cai wid in direcion was non-dimensionalied as /L, Eq(.8), afer sing e leng of e cai as L=.1m. Te eloci was rendered dimensionless as /, Eq(.8), afer emploing e caracerisic eloci ale of =1m/s.] Temperare disribion sown in Figres.6a,.6b, and.6c are ploed a e same locaions as sed for Figre.5. Temperare disribion, as i falls below liqids emperare or meling poin of Alminim (Table.1), sggess e presence of solid-
liqid inerface some disance awa from e cai wall. Moing solid-liqid inerface redces cai eig, and as a resl, cases an increase in e mel eloci (Figre.5) de o conseraion of mass. a. b. c. Figre.6. Temperare disribion along e cai ickness a ime=1s: a) Temperare disribion a =.; b) Temperare disribion a =.5; c) Temperare disribion a =.9. Coordinae in e direcion was non-dimensionalied as / H o, Eq(.8), wile sing H o =.1m as e cai ickness. Eolion of e solid-liqid inerface along e cannel leng is sown sing Figres.7a and.7b. We noe a some discrepanc eiss beween e lbricaion approimaion solion and e FlowD solion in e beginning. Howeer, we aciee a beer conergence of resls as e ime increases. Te difference in e resls ma be aribed o e rblen nare of e flow emploed in FLOWD simlaion: as e cannel eig decreases, rblence is less prealen in e flow, and e resls prediced b e presened algorim are closer o e FLOWD solion.
5 a. b. Figre.7. Eolion of e solid liqid inerface wi ime for =1: a) Te inerface locaion a =.5s, b) Te inerface locaion a =1s. [Te cai leng in direcion was non-dimensionalied as /L, Eq(.8), afer sing e leng of e cai as L=.1m. Coordinae in e direcion was non-dimensionalied as / H o, Eq(.8), wile sing H o =.1m as e cai ickness.] A e specified locaions, differences in e posiion of e solid liqid inerface are significan enog o case isible eloci differences in Figre.5. As e aboe gien discssion indicaes, eloci canges cased b e redcion in cai eig corresponds nicel wi e canges in eloci esimaed b FLOWD. All resls are wiin 1% of e solion obained b rnning ree- dimensional analses iliing e commercial sofware FLOWD. A significan compaional adanage is acieed rog a dramaic redcion in CP ime. Owing o e simplificaion of e goerning eqaions sing e lbricaion approimaion, e CP ime for e proposed algorim was obsered o be s. In conras, e CP ime for e corresponding ree-dimensional analsis wi FLOWD sofware was 1 min. Tis 6 fold redcion in CP ime clearl demonsraes a e proposed algorim based on redced psics is qie fas wio a significan sacrifice in e accrac.
6.6 Conclsion Resls of e presened analses indicae a Renolds lbricaion approac can be sccessfll implemened o inesigae e flow and solidificaion of e molen meal in in caiies dring e die cas process. Te proposed.5d algorim allows one o esimae e ickness-aeraged liqid-meal eloci in e plane of e cai sing e finie difference meod; en a finie-olme based algorim allows one o esimae emperare disribion along e ickness direcion as well as locaion of e solid liqid inerface. Te nmerical simlaion based on e algorim is erified b comparing is predicions wi e solion of e ree-dimensional Naier-Sokes eqaion fll copled wi ree dimensional energ eqaion as prediced b e commercial sofware FLOWD. Resls indicae a e proposed simlaion is fairl accrae in predicing e aeraged eloci fields, emperares along e ickness, and gap icknesses inside e cai. Considering small error and significan saings in compaional ime, e proposed algorim can be sed o redce ime on e iniial sages of process deelopmen of e die-cas process. I will epedie flow analsis of e die casing process b sing e presened algorim in cases were e ig aspec raio of e in cai reqires large nmber of e compaional cells o aciee e conerged solion. I can be especiall sefl in analing flid flow and solidificaion in enilaion cannels of e die-cas die.
7 Caper : A Fas Simlaion of Transien Meal Flow and Solidificaion in a Narrow Cannel Absrac: A fas nmerical algorim for modelling e ransien flow and solidificaion of liqid meal in a narrow gap is presened. Te problem is closel relaed o e die-cas process, and in pariclar o meal flow in in enilaion cannels. Afer inegraing oer e cannel ickness and emploing e lbricaion approimaion, e Naier-Sokes eqaions are redced o -D eqaions for modelling e in-plane flow. Te flow model is soled along wi a ea balance eqaion afer inclding e effecs of solidificaion in a conrol olme. Te flow ariables and emperare disribion are soled in ree sages. In sep one, commercial sofware FLOWD is ilied o sole -D Naier-Sokes eqaions copled wi e ea balance eqaion for flow and solidificaion in e main cai. In sep wo, e flow and ea ransfer ariables from e main model are ransferred as e enrance bondar condiion for e proposed nmerical simlaion. And finall in sep ree, e meal flow and solidificaion in a in cannel is modelled sing e -D eqaions copled wi e 1-D ea balance eqaion. Since e solid-liqid inerface inrodces non-lineari in e flow, e -D flow eqaions are soled ieraiel wile a saggered grid arrangemen as reqired b e SIMPLE algorim is sed for discreiaion. Laer, e proposed simlaion is applied o predic pars prodced b e ig pressre die cas process (HPDC). Te model is alidaed b comparing is resls wi ose obained from e commercial flow-and-solidificaion sofware FlowD as well as wi e eperimenall measred secondar dendrie arm spacing (SDAS)..1. Model deelopmen In pical applicaions relaed o a ig pressre die-cas process, meal as o flow rog a combinaion of in (1- mm) and ick secions (-8mm). Commercial sofware
8 allows one o se differen mes densiies based on e scale of e compaional domain. As can be seen in Figre.1, e enilaion (or e gas eacaion) cannel is an order of magnide smaller an e res of e casing. Sbsanial redcion in e cell sie is reqired in order o aciee a conergen solion in a in cannel porion of e casing. Owing o e fac a flow in e cai of e die-cas die is igl rblen, e mes as o be sfficienl refined in order o resole e flow accrael. Figre.1 A scemaic sowing e ick and in secion of a pical die-cas par Frer increase in mes densi o accommodae flow wiin in enilaion cannels is sall impracical, and as a common pracice, e flow analses are condced in e main cai onl. Te algorim presened in is paper will allow one o eend e flow and solidificaion analsis ino in cannels also. Te compaional domain is diided ino wo regions (see Figre.). In e main cai, e flow and solidificaion analsis is condced sing commercial sofware, wic soles e reedimensional Naier-Socks eqaion copled wi energ eqaion. In e in-cannel region, e lbricaion eqaion, copled wi ea balance eqaion deeloped for a conrol olme defined arond a node of e -D in-plane mes, is soled in a finie ime ineral ieraiel. Owing o e fac a bo models are soled in differen dimensional
9 and ime scales, a procedre was deeloped o ransfer e flow-ariable ales from e main cai o e in-cannel ia e inerface (or c-off ) plane. Figre.. A scemaic sowing e main cai and e in cannel of a die-cas die conneced rog a c-off plane.. Main cai model Flow ariables and emperare disribion in e main cai are soled sing e general-prpose commercial CFD sofware FLOWD. Te sofware proides solions o ree-dimensional flid and ea flow and solidificaion problems sing e finie difference approimaion. Te Naier Sokes eqaions copled wi e energ eqaion aciee an accrae represenaion of e rblen flid and ea flow accompanied wi solidificaion. Te wo-eqaion k-ε model is sed o resole e properies of rblen flow. A conrol olme meod is sed o discreie e goerning eqaions and e enalp meod is sed o ealae emperare disribion in e die cai. Te OF (olme of flid) meod is sed o accrael rack free srface eolion in e compaional domain.
5. Tin cannel model Owing o e small aspec raio (eig o leng raio) of e cannel, Ho 1 (.1) L c and e problem of modelling meal flow in e enilaion cannel can be redced o e solion of e in-plane lbricaion eqaion combined wi e solion of e o-ofplane ea ransfer and solidificaion model (see caper as well as []). To ake ino consideraion e ig-eloci flow obsered in e die-cas process, e ineria-effec erms were added o e final lbricaion eqaions according o e recommendaions of [, ]. In order o model die-cas die filling wi liqid meal, an era ransien erm as o be incorporaed ino e momenm eqaion, Eq. (.5), for e sead-flow case discssed in secion. Te goerning eqaions in differenial form are sown below. Conini eqaion: (.) Momenm balance eqaions: p (.) Energ balance eqaion: T C C T q (.) An order-of-magnide analsis was condced in order o simplif e goerning eqaions and o deermine e imporance of eac erm dring ig-speed flows in in
51 cannels. Onl a smmar will be presened ere e deails of is redcion in e goerning eqaions for in-cannel flows can be fond in caper. and Appendi A. For is redcion (caper. and Appendi A), e following dimensionless ariables were sed: p L L H p W Lw w H L L,,,,,,, (.5) Te earlier-gien goerning eqaion can be re-epressed in dimensionless form, afer dropping e inconenien ilde signs, as: Conini eqaion w (.6) Momenm eqaion p w L c _ (.7a) p w L c _ (.7b) Te momenm eqaion in e direcion ranserse o flow redces o p (.7c)
5 impling a flid pressre is niform in e direcion regardless of e ineria effecs in e flow. Te dimensionless grops sed in is deelopmen are: Renolds nmber Lc l Re (.8) Modified Renolds nmber [9] Re (.9) wi Lc o Eqs. (.7a) and (.7b) can now be epressed as p w (.1a) p w (.1b) Once again, based on e recommendaions of [], we assme a parabolic eloci disribion along e and direcions: ) )(, ( (.11a) ) )(, ( (.11b)
5 Afer inegraing oer e ickness of e cannel, e conini eqaion, Eq. (.6), becomes (.1) Noe a based on e no-peneraion bondar condiion w and small sie of e cai in e direcion, w eloci ariaion is negligible, and is se o ero. Afer inegraing across e ickness of e cai from o, e momenm, Eq. (.1), becomes p 6 1 (.1a) p 6 1 (.1b) Noe a e onl difference beween Eqs. (.) and Eqs.(.1) is e presence of e firs (ransien) erms. Energ eqaion Te general ermal energ balance eqaion (Eqs. (.), (.), and (.5)) are described in Secion. and are sed in e ransien model as well. Bondar and iniial condiions Te in-cannel geomer sed for model alidaion is sown in Figre.. A e iniial ime =, e same niform emperare, T = 6 o C, is applied o e liqid meal ling wiin e compaional domain. A e enrance ( = ), e flid emperare is se as T= 6 C. A e solid-liqid inerface, T s = T l = T mel. Te wall emperares a = and = are se o 6 o C. Te flow is drien b a niform eloci imposed a =
5 locaion. Te oflow bondar condiion specified a e end of e cai, =L, (afer e liqid as gone beond e ei) is. Te no-slip bondar condiions are applied a e lef and rig walls in e direcion. Addiionall, a smmer bondar condiion, in e form of e eloci gradien in e direcion normal o e wall being eqal o ero (i.e., ), is applied. Figre. Te in-cannel geomer sed for model alidaion: e lef- and rig-side segmens are 1 mm and.5 mm ick, respeciel. (Te oer dimensions are gien in Figre.11 and Table.1) Noe a Eqs. (.1), (.1) and (.5) form e goerning eqaions for e in cannel problem.. Copling of e main cai and in cannel flows Main cai and in cannel are diided b c-off plane (see Figre.). In order o ensre a smoo ransiion of elociies and pressre from e main cai ino e in cannel region, e pressre, eloci and emperare ales ae o mac a
55 e c-off plane. Te meal eloci in e main cai is maced o e meal eloci in a in cannel b e fac a flow olme spplied from e main cai is eqal o e olmeric flow in e cannel: H (.1) Te emperare and pressre disribions a e ei of e main cai are inegraed oer e ickness of e cai H, sc a ese aeraged ales are applied as a emperare and pressre bondar condiions for e in cannel secion a e c-off plane. Te ime-sep sie as o be addressed as well. Owing o e difference in e leng scales of e wo regions, e ime scales of e main cai and in cannel regions are relaed as main cannel (.15) Te parameer λ ad o be cosen sc a e solion in e in cannel was kep sable. Based on e difference in e main cai and in cannel leng-scales, e scale parameer was cosen o be λ = 1..5. Discreiaion of goerning eqaions Te compaional domain as o be sbdiided on smaller conrolled olmes were e aerage flow ariables can be soled for sing e discreied form of balance laws..5.1 Main cai Owing o e fac a FLOWD [9], a commercial CFD sofware, was sed for comping flow and emperare qaniies in e main cai, discreiaion of goerning
56 eqaion for e main cai is no gien in is paper. Aomaic srcred grid generaion fncion of e sofware was sed o discreie e compaional domain. Te fracional area/olme meod, FAOR [11], is sed for modelling e comple geomer of or main cai..5. Tin cannel To aoid nrealisic beaior of e momenm eqaions, a saggered grid arrangemen is sed o discreie e goerning eqaions, Eqn..1 (a) and (b), in e in cannel sing e SIMPLE algorim. Te principal of e -D saggered grid arrangemen (Figre.) is a e scalar qaniies sc as pressre, emperare, and eig of e cannel are sored a e cenre of e conrol olme. Howeer, sorage of eloci componens are saggered wi respec o e faces: elociies are sored a e wes and eas faces, wile elociies are sored a e nor and so faces. Te discreiaion of e goerning eqaions in in cai is described in deail in Appendi B; onl eqaions in e final form are gien below in secion.6.. Figre.. Te saggered grid arrangemen (based on SIMPLE algorim) was emploed for soling e in-plane flow ariables.
57.6. Solion procedre Solion procedre in e form of a flow car is described in Figre.5. Iniial condiions in e main cai inclde meal emperare, cold camber die-cas macine plnger-eloci profile, iniial emperare of e die-cas die..6.1 Solion procedre in main cai Before calclaions of e meal flow in e main cai can begin, die emperare disribion as o be esablised. sing a FLOWD feare called "ermal die ccling", e die is filled wi molen meal insananeosl and eld for e draion of e dwell ime (5s). Dring is ime, e condcion eqaion o predic emperare in e die-cas die is soled in order o accon for ea ransfer from liqid meal o e die seel. Ten e liqid meal is remoed from e die insananeosl, and eernal cooling is e applied o e die in order o correc e die seel emperare. Tis process is repeaed nil e die emperare aains a qasi-sable sae. Te final calclaed die emperare was compared o e emperare of e die-cas die sing a ermal imaging camera. Process was repeaed wi adjsmen in ime and efficienc of e eernal cooling, nil error of emperare disribion was less an fie percen. Wen e ermal qasi-sable sae is acieed, flow analsis in e main cai can begin. Te final die emperare is sed as a bondar condiion for frer flow and solidificaion analsis in e main cai. Meal in e cai is en drien b e plnger of e die cas macine. (Figre.6 sows e scemaics of a pical die-cas die wi so clinder and pison assembl.)
58 Sar Sole for eloci, pressre, and emperare disribions in e main cai Transfer e calclaed ariables from e main cai ino e in cannel Find locaion of air-meal inerface, based on eloci disribion, Eq.(.15). Find eig of liqid domain from emperare disribion and sing Sefan condiion, Eq(.6) Sole for elociies and emperare in e in cannel sing Eqs. (.1) and (.5) Find new locaions of solid-liqid (Appendi C) and meal-air inerfaces Eq.(.15) Is >? No Yes End Figre.5 Te proposed solion algorim for soling flow and emperare ariables in e in cai
59 Figre.6 A scemaic of a die-cas die wi so sleee and plnger: 1) So sleee, ) Plnger, ) Saionar alf of e die-cas die, ) Ejecor alf of e die-cas die, 5) Mold cai, 6) enilaion cannel. Te plnger eloci profile (Figre.7) is ransferred ino FLOWD from e so monioring ssem isitrak 5. Calclaions in e main cai conine nil e liqid meal as reaced e c-off plane, wic is posiioned on e border beween e main cai and in cannel. A is poin, all flow and emperare ariable ales are ransferred from e main cai ino e in cannel sing e procedre olined in Secion. Tracking of free srface beween liqid meal and air Fracional olme of flid (OF) [11] meod is sed o rack e meal-air free srface inerface. Meod is based on e assmpion a fncion F, wic represens fracion of flid in e cell, can assme ales beween and 1: ale 1 is assigned o e cell fll of 5 Te so monioring ssem is for racking e plnger eloci dring e injecion process.
6 flid, wile is assigned o an emp cell (see Figre.). Based on is assignmen, eer cell wi fncion F ales beween and 1 conain a free srface. Te imedepended fncion F is goerned b e eqaion F F F (.16) wic saes a e ale of F moing wi e flid remains ncanged. Te OF meod is based on e assmpion a e local rae of cange of F in a cell is eqal o e amon of F fling rog e faces of e cell. Te oal amon of flid olme crossing e face of e cell per ni cross-secional area is ol ; is is mliplied b e cell face-area o esimae e amon of flid passing rog in a ni of ime. Figre.7. Tpical plnger eloci profile (IPS = inces per second) posiion is e disance plnger raels dring e die cas process..6. Solion procedre in in cannel Te problem is soled in a sraig cannel wi a recanglar cross-secion (Figre.). Compaion begins wi an esimaion of emperare disribion in e compaional domain. I esablises e locaion and sape of e solid-liqid inerface in e region
61 raersed b e liqid meal. Te solid-liqid inerface defines e bondar of e liqid domain. Ten e momenm eqaions can be soled o define e disribion of ickness-aeraged elociies in e compaional domain. Tis eloci disribion is en sed o define e posiion of e free srface in e ne ime sep. Te assmpions adaped for e deelopmen of e proposed algorim are: 1. Flid is incompressible and Newonian.. Flow is laminar.. Hea condcion in e direcion of meal flow in e cannel is negligible.. Hea resisance of e cannel walls are negligible Solion of e ickness-aeraged conini and momenm eqaions Discreied forms of Eqns. (.1), (.1), and (.5) can be epressed as follows. In e following eqaions, Eqs. (.17) (.19), indices in capial leers indicae e primar grid were indices in small leers indicae e secondar grid (see Appendi B for more deails). Conini eqaion: (.17) i 1, J i, J I, j 1 I, j Momenm eqaionin e direcion: n1 n 1 I, J I, J 6 n n I J I J 1,, i 1, J n I J I J 1,, i 1, J I, J 6 Pn Pn n I 1, J I, J * I, J n n I, J i 1, j 1 n i 1, j 1 n i 1, j n i 1, j (.18a)
6 In e direcion: n1 n 1 I, J I, J 6 n I J, 1 I, j 1 I, J 6 Pn Pn I, J 1 I, J * I, J n n I J I J, 1, I, j 1 n n I, J I, J i 1, j 1 n i 1, j 1 n i, j 1 n i, j 1 (.18b) Energ eqaion Hea balance eqaion, Eq. (.5), discreied oer e conrol olme in a finie ime-ineral resls in C pi 1, J C pi, j Ti 1, J TI, j C p C pi, j Ti, j I, J TI, J C pi, j 1 qw TI, j 1 (.19) Te momenm eqaions are soled sing e SIMPLE procedre [7] wic is sed o cople e momenm and conini eqaions. Te momenm eqaions, Eqs. (.18a) and (.18b), se e pressre field ransferred from e main cai and sole for preliminar elociies and. In is incompressible form, e mass conseraion eqaion doesn' ae a pressre erm. Howeer, an arificial compressibili erm, a, as o be added o e modified conini eqaion, Eq.(.7), o allow for e solion of pressre P. sal ales of e arificial compressibili erm are.1 > a > 1.; i is aken o be 1. [9] in e presened algorim. Te modified conini eqaion, deried from Eq. (.7), is sed o calclae e correced ales of e pressre field:
6 n1 n n1 n1 n1 n1 P I, J PI, J i1, J i, J I, j1 I, j a (.) Solion of e conini and momenm eqaions procedre can be broken down ino following seps: 1. se e pressre ransferred from e main cai in e firs ime-sep.. Sole e momenm eqaion o esimae e preliminar ales of and.. se e modified conini eqaion, Eq. (.), o correc pressre ales.. Correc e and elociies sing e new pressre ales and conine e ieraions nil e conini eqaion, Eq.(.17), is saisfied. A is poin of e analsis, e eig of e in cannel,, in e momenm eqaions sill remains nknown. In e absence of solidificaion, e momenm and conini eqaions alone will allow one o esimae e eloci and pressre disribions in e in cannel beind e fron in a gien ime sep. In order o close e ssem energ balance, Eq. (.19) as o be soled; solion of is energ balance eqaion allows one o esimae e emperare disribion in or calclaion domain. Based on e resl of e energ eqaion and rog e se of e Sefan condiion, Eq. (.6), e locaion of e solid-liqid inerface can be esablised for eer ime-sep (see Appendi C for deails). Te locaion of e solid-liqid inerfaces allows one o pdae e ales and o se em in e momenm eqaion for e ne ime-sep. I is assmed, in is algorim, a ere is perfec conac beween e solidified meal and walls of e cannel. Wall emperare of e cannel remains consan dring e calclaion procedre is means a e walls ae ig ermal mass and condcii.
6.7 Smmar of Nmerical Meods Te presened nmerical algorim allows one o esimae e flow and solidificaion ariables dring e die-cas operaion in a mold conaining a ick main cai and a in enilaion cannel. Flow and ea ransfer in e main cai is esimaed sing e commercial sofware Flow-D. Te c-off plane separaing e main cai from e in cannel is sed o ransfer flow and emperare ariables from e main cai o e in cannel as bondar and iniial condiions in e proposed algorim. Te flow elociies and ime seps are ransferred according o e lengscale difference beween e wo flow regions. elociies and pressre disribion in a in cannel are esimaed sing e lbricaion approimaion afer aking ino accon ineria effecs and solidificaion in e meal. Te -D flow field is redced o a ickness-aeraged -D flow field wic is o be soled in e cenral plane of e in cannel. Te liqid meal-air free srface is racked sing e OF meod. Te cannel gap is esimaed afer incorporaing solidificaion rog 1-D o-of-plane ea ransfer in e cannel. Owing o e simplificaion of e algorim based on e lbricaion approimaion, e proposed nmerical approimaion is epeced o offer significan amon of saings in compaional ime..8. alidaion of e Proposed Nmerical Algorim process. Casing sown in Figre.8 is prodced b a cold camber ig pressre die-cas
65 Figre.8 A picre (a fll so ) of a par made sing e die-cas process. Te oerflows are creaed wen e meal fron, afer filling e main cai, fills p e macined oerflow pockes in e die-cas mold. enilaion cannel is las o fill-p. Te plnger eloci profile is ransferred ino FLOWD from e so monioring ssem isitrak and is sown in Figre.7. Te molen-meal (Alminim A8) emperare was se a 65 C, and e iniial die-emperare was se a 15 C. Trog e insananeos fill-ccle process of FLOWD 6, e die emperare was made o reac a qasi-sable sae. Ten a prescribed plnger eloci was applied o fill e cai of e die-cas die. Resls of e flow analsis are sown in Figre.9. Final emperare disribion in e cai of e die- cas die filled wi liqid meal is illsraed in Figre.1. 6 Twen fie insananeos fill ccles were applied.
66 Figre.9 Flow analsis resls sing FLOWD of e meal flow and solidificaion in e main cai. (Te eloci is in m/s.) Figre.1 Temperare disribion in e considered cai of e die-cas die, filled wi liqid meal a e end of e fill process. (Te emperare is in C.) eloci, emperare, and pressre disribion were eraced from e main cai a e locaion of e c-off plane. All e flow and meal emperare qaniies
67 were ransferred ino e proposed algorim as e iniial condiion for e sbseqen nmerical simlaion in e in cannel. In order o erif e eloci disribion prediced b e proposed algorim, e were compared wi e ones obained from e analsis condced b e commercial sofware FLOWD. A separae reedimensional model of e in cannel was impored ino FLOWD. Te eloci and emperare disribions, conered ino in-cannel scale, were applied as a bondar and iniial condiions. Te emperare and eloci a e inle of e cannel were considered consan. Bo FLOWD and or proposed nmerical algorim were se o rn nil e solid-liqid inerfaces conerged a e middle of e cannel. In order o esablis e alidi of goerning eqaions [Eqs. (.1), (.1) and (.5)] for modelling e flow and solidificaion of liqid meal in a in cannel, e reslan simlaion was applied o a enilaion cannel of recanglar cross-secion (Figre.11). Figre.11 A scemaic of e considered enilaion cannel: e sepped profile is necessar o solidif and conain e oerflowing meal. Te dimensions a, b and c are lised in Table.1.
68 Table.1 Tin cannel dimensions a(1) a() b c(1) c() Dimensions (mm) 1.5 5 5 Te proposed goerning eqaions were soled nmericall and e resls were compared wi e solion of e incompressible Naier Sokes eqaions fllcopled wi e ree-dimensional energ eqaion dring solidificaion a was soled sing e commercial sofware FLOWD. Finall, a parameric sd was condced o define e relaion beween meal progress in e enilaion cannel and ree parameers: iniial eloci, emperare of e cannel walls, and e ickness of e cannel. Deeloped cre can be sed as a gide in designing enilaion cannel of e die-cas dies. Properies of e commercial Alminim allo, A8, were sed for or nmerical calclaions. Resls of e nmerical predicion of e flow and solidificaion of e molen meal flowing in a in cannel b or code were compared wi e resls obained from e commercial sofware FLOWD as well as e acal casing lengs. Te prediced rae of solidificaion was frer alidaed b comparing e calclaed (based on e resls obained sing e proposed nmerical algorim) and measred secondar-dendrie-arm spacing..9 alidaing eloci disribion in e cannel Meal flow in e cai, as conrolled b e solid-liqid inerface locaion in e cannel, was sdied. Mes densi was se a 5 X 15 cells along e and direcions, respeciel. enilaion cannel in FLOWD was mesed wi 1X1X5
69 compaional cells in,, and direcion, respeciel. Decision on mes densiies in bo algorims were based on or mes independence analsis, and were carried o nil differences in e obained resls did no ar more an.1%. eloci disribions in a cannel was eraced a fied locaions (see Figre.1) a a gien ime from e nmerical resls gien b e proposed algorim and e resls obained from e commercial sofware FLOWD; sc comparisons are sown in Figre.1. Figre.1 A scemaic sowing a pical cross-secion of e sepped enilaion cannel. Secions I and II are sed for e comparison of predicions b e proposed algorim and FLOWD. Figre.1 Comparison of eloci disribions in e enilaion cannel: (Lef a.5s) secion I and (Rig a.5s) secion II of Figre.1. Noe a e plo gies elociies aeraged along e ickness direcion.
7 Solion obained from FLOWD akes ino accon e rblen caracer of e flow were i emplos fll-copled ree-dimensional momenm and energ eqaions. Trblen qaniies were resoled sing e wo-eqaion k model. As obsered in Figre.1, e ickness-aeraged eloci disribion, obained from e lbricaion approimaion based simlaion, do no differ b more an 5% from e fll D flow and solidificaion simlaion. Howeer, de o e rblence caracer of e flow, e reslan eloci disribion obained sing FLOWD eibi plg-pe flow profile. Assmpions of laminar flow and parabolic eloci profile in e proposed algorim eplain e differences in eloci profiles obained from FLOWD and e algorim. Te in-plane eloci profile approimaed b e lbricaion formlaion becomes cred as flow progresses along e cannel. On e oer and, e eloci profile prediced b FLOWD appears plg-like, wic is pical of rblen flows..1 alidaing solidificaion analsis Solidificaion analsis is an imporan par of e nmerical approimaion of e die-cas process. I defines e bondar of compaional domain dring liqid-meal flow simlaion, as well as mecanical properies of e die-cas par [5]. Iniial meal emperare was eraced from e flow and solidificaion analsis condced in e main cai. Temperare of e meal was kep consan a e inle dring nmerical simlaion of flow and solidificaion in e enilaion cannel. Temperare of e cannel walls sed in e simlaion is based on e measred emperare of e die seel of e die-cas die. Twen consecie measremens sing an infrared deice ( ermal gn ) as sown a aerage emperare of e cannel walls is 6 C. De o e qick
71 filling of e cannel, wall emperare was considered consan. In order o erif e accrac of emperare disribion obained sing e proposed algorim, ales of meal emperare were eraced a wo specified locaions (Secions 1 and ) sown in Figre.1. Along-e-ickness emperares presened in Figre.1 sow a er good correlaion beween resls from e proposed algorim and resls from e commercial sofware FLOWD. Figre.1 Comparison of emperare disribions in e enilaion cannel: (Lef a.5s) secion I and (Rig a.5s) secion II of Figre.1..11 alidaing locaions of liqid meal-air and solid-liqid inerfaces As was described aboe, e proposed algorim allows one o predic e locaion of e solid-liqid inerface as well as e locaion of e free srface as e eole dring compaional ccle. Locaion of e solid-liqid inerface is fond sing e Sefan condiion, wile e free-srface posiion is defined b e OF meod. eloci and emperare disribions are calclaed in e liqid region.
7 a. b. c. d. e. Figre.15 Locaions of e free srface: Proposed nmerical algorim (op), FLOWD (boom) a a).1s, b).18s, c).9s, d).5s, e).6s [In all iews, e orional ais along direcion is along e cannel leng, wile e erical ais is e wid of e cannel (m) in direcion] Locaions of e free srface a arios imes are sown in Figre.15. As i can be seen, ere is good correlaion in e locaion of e free srface as prediced b or
7 algorim and FLOWD. Howeer, an obseraion can be made on sapes of e free srfaces prediced b e wo simlaions. Te proposed algorim predics a smoo, second order cre as a free srface, compared wi e almos fla free-srface prediced b FLOWD. Acal casing of e meal solidified in e enilaion cannel sows a bo algorims prediced correc leng of meal flow in e cannel (noe a nmericall esimaed flow-lengs were fond o be wiin 5% of e flow-lengs seen in an acal casing), b FLOWD prediced correc sape of e solidified meal 7 (see Figre.16). An eplanaion of e difference in e free-srface sapes comes from e assmpion of laminar flow in e proposed algorim; FLOWD ilies e woeqaion k-ε model for modelling e rblen meal flow, wic conribes o is abili o predic correc free-srface sape. 7 Te secions of casings (Figre.16) ae confirmed a e wa sape of e solidified meal as prediced b FlowD is qie accrae.
7 a. b. Figre.16 Eperimenall obsered solidified meal in e enilaion cannel; a) Measred leng of meal flow in e enilaion cannel afer solidificaion sops i; b) Enlarged image of e solidified meal in e cannel
75 Locaions of e solid-liqid inerface as prediced b e wo simlaions, a arios imes, are sown in Figre.17. I can be obsered a as e meal flows in e cai, ea ges eraced b e cannel walls eld a consan emperare, below e emperare of solidificaion. Meal solidificaion on walls occrs a some disance from e cannel inle. As meal conine o flow ino e cai and reaces e inner secion of e cannel, e meal sars solidifing a e cross-oer poin also. I can be eplained b e fac a since e eloci in e consriced par of e cannel is iger, i increases e ea ransfer beween walls of e cannel and e liqid meal. Bo algorims predic correc locaions of e areas were consricions de o meal solidificaion on walls are formed inside e cannel, and e area were e op and boom solid-liqid inerfaces conerge in e middle of e cannel and sop e meal flow. Tis locaion is also confirmed b e acal casing (Figre.16), were e darker area in e beginning of e inner area of e cannel 8 indicaes were e solidificaion occrs firs. 8 Te dark color of e meal is de o e miing of brn die-release agen wi liqid meal a e conracion of cannel.
76 a. (Figre.17)
77 b. (Figre.17)
78 c. Figre.17. Locaion of e solid-liqid inerface prediced b simlaions: e proposed algorim (op), commercial sofware FLOWD (boom) a a).6s, b).55s, c).6s. Te ellow (lig) color represens e liqid meal wile e green (dark) color signifies solidified meal..1 erificaion of cooling rae sing measred secondar dendrie arm spacing. Ofen erificaion of e cooling rae of e molen alminim is condced b comparing e calclaed and measred Secondar Dendrie Arm Spacing (SDAS) []. Tree samples were prepared o measre SDAS. Preparaion procedre inclded: a) cing e
79 sample, b) moning e sample in a fire sing an epo, c) grinding and polising wi progressiel redced gri sie nil a planer srface is acieed, d) cemical ecing sing a reagen. Keller s reagen was sed o prepare sample sown in Figre.18; e prpose of is sep was o iglig e microsrcre of e specimen. Figre.18 Cross secion of e casing sed o measre SDAS (X magnificaion) Tree separae samples were prepared sing procedre described aboe. Measred SDAS were in e range.17.19 m. In order o erif rae of solidificaion of e presened algorim and commercial sofware FLOWD, solidificaion cre was ploed (see Figre.19). Based on e calclaed rae of solidificaion, SDAS can be esimaed sing e formlae
8 T 5.5 G 1 M (.1) were, G is a emperare gradien, solidificaion rae, and T is a emperare range beween e sar and end of secondar dendrie grow. Te consan M is esimaed as M D lnc / c e (.) m1 c k d c e were D,, c e, c, m, and kd are diffsii, Gibbs Tompson coefficien, eecic concenraion, sole concenraion, liqids slope, and disribion coefficien respecil. sing e ales from Table., e consan M can be calclaed as M 5e 9 5.56e 8 ln.1 /.9.51.161.1.9 1.1e 15 Terefore, e calclaed SDAS is 9 588.7 5.51.1e 18 15 5. m Resls of is calclaion sow a bo FLOWD and proposed algorim predic correc rae of solidificaion of e liqid meal. Table. ales of ariables in Eq. (.1)-(.) [] D(m /s) (K m) c e c k m(k/%) G(K/s) (K/s) 5e-9 5.56e-8.1.9.161.5 15
81 Figre.19 Temperare isor a e cenerline of secion I in Figre.1; e cooling rae, ν, is obained from e slope of e cre..1 Significan Improemen in Compaional Speed Te mos beneficial adanage of e proposed nmerical algorim is e sbsanial redcion in CP ime. Proposed algorim reqired onl 8 mines o aciee a conerged solion compared o 5 mines reqired b e commercial sofware FLOWD for e case sown in Figres.15 and.17. Tis remarkable redcion in compaional ime is an imporan feare of e proposed algorim e lile error in is predicions are a small price o pa for a man-fold improemen in e compaional speed.
8.1 Frer alidaion rog a Parameric Sd Frer erificaion of e proposed model was condced b canging iniial cai ickness, cannel-wall emperare, and liqid-meal eloci a e inle. For e parameric sd, e cannel was considered o be of a consan eig. Te liqid-meal flow-leng in all ree cases was compared wi e resls obained sing FLOWD. Comparisons of ese nmerical resls are sown in Figres....1.1 Effec of canges in enilaion-cannel ickness Firs parameric sd was condced b canging e ickness of e enilaion cannel and keeping e wall emperare and liqid-meal iniial eloci consan. Cannel ickness was canged from.5 mm o 1 mm, wi iniial meal eloci se a 1 m/s and wall emperare kep consan a 6 C. Flow analses were rn ree imes wi a.5 mm increase in e enilaion cannel ickness. Resls of e sd are sown in Figre. were flow leng (e leng e liqid meal moes in e enilaion cannel before coming o a sop) is ploed as fncion of cannel ickness. We see a remarkable conergence beween predicions of or algorim and FLOWD. Te monoonicall-increasing resl indicaes an increase of e liqid-meal flow leng can be epeced b increasing e cannel ickness. Dobling e cannel ickness resls leads o a similar increase in e flow leng.
8 Figre. Meal flow-leng s. cai ickness resl of e parameric sd..1. Effec of canges in wall emperare of e enilaion cannel Ne parameric sd was condced b canging e emperare of e enilaioncannel walls. Te cannel ickness was se a 1 mm and e walls emperare was iniiall se a 6 C wi e iniial liqid-meal eloci of 1 m/s. For consecie simlaions were condced wi emperare of e wall increased b C eac ime. Te relaion beween e wall emperare and e liqid-meal flow-leng is sown in Figre.1. Once again, a remarkable mac is obained beween e predicions of or proposed algorim and FLOWD. Resls indicae a e wall emperare of e cannel as onl a slig effec on e flow leng of e liqid-meal.
8 Figre.1 Meal flow-leng s. wall emperare resl of e parameric sd..1. Effec of canges in e iniial meal eloci ariaion of e iniial liqid-meal eloci and is effec on e flow leng in e enilaion cannel was also sdied. Te liqid-meal eloci was iniiall se a.5 m/s, wi e enilaion cannel ickness a 1 mm and e cannel-wall emperare a 6 C. For consecie simlaions were rn wile increasing e inle meal-eloci b.5 m/s eac ime. Te reslan correlaion beween e liqid-meal iniial eloci and flow leng is sown in Figre.. Resls indicae a flow leng of e liqid meal increases wi an increase in e meal eloci a e cannel enrance. Howeer, afer e meal speed reaces 1.5 m/s, e increase in e flow leng is mc less an seen beween e elociies of.75 1.5 m/s. I can be eplained b e fac a conecie ea-ransfer coefficien beween liqid meal and cannel wall is an increasing fncion
85 of e meal eloci 9 : i increases wi an increase in e meal eloci, and conseqenl, e liqid meal solidifies mc faser in e cannel. Also noe a e accrac of proposed algorim deerioraes wi an increase in e inle speed. I can be eplained b e fac a as e Renolds nmber increases wi an increase in e inle speed of e liqid meal, rblence in meal flow become more prononced and e lbricaion approimaion emploed in or algorim is rendered increasingl less accrae. Figre. Meal flow-leng s. meal eloci a e enrance of e enilaion cannel resl of e parameric sd. 9 Increase in e liqid-meal eloci increases e Renolds nmber, wic in rn increases e Nssel nmber. Since Nssel nmber is proporional o e conecie ea ransfer coefficien, i increases ea ransfer coefficien as well [8].
86.15 Smmar and Conclsions A nmerical algorim based on e lbricaion approimaion is deeloped o sd ransien flow and solidificaion in a in enilaion cannel of e die-cas die. Procedre consiss of diiding casing ino wo regions. Firs solion is obained in e main cai were some commercial CFD sofware can be sed o esimae flow and emperare disribions. Ten e resls a c-off plane beween e main cai and enilaion cannel are ransferred ino proposed algorim as iniial condiion a e enrance of e enilaion cannel for frer analsis. Te flow goerning eqaions are inegraed along e cannel ickness sing e lbricaion approimaion o ield goerning eqaions for.5-d cannel flow. Te ea balance eqaion oer e conrol olme ielded a emperare eqaion afer incorporaing e Sefan s condiion for meal solidificaion. Te finie difference meod based on SIMPLE algorim wi saggered Caresian grid arrangemen is sed for nmerical approimaion of e discreied goerning eqaions. olme of flid meod is ilied o define posiion of e free srface. Resls of e presened algorim are alidaed b comparing em wi predicions of e commercial CFD sofware FLOWD for a casing prodced b e cold-camber ig-pressre die-cas process. Te algorim resls are in good agreemen (wiin 5%) wi e predicions of e commercial CFD code as well as obseraions in acal casings. A good agreemen is also acieed beween e direcl measred SDAS (secondar dendrie arms spacing) from an acal casing and e eoreicall esimaed SDAS sing or algorim and FLOWD resls.
87 A parameric sd of on liqid meal flow and solidificaion in a enilaion cannel sing bo e algorim and FLOWD resls sows a e meal flow-leng increases wi e increase in e enilaion-cannel ickness, wall emperare, and iniial enrance eloci. Dobling e cannel ickness leads o a similar increase in e flow leng. Canges in e cannel ickness inflence e liqid-meal flow-leng e mos, and e cannel wall emperare inflences e flow leng e leas. Tese resls can be sed dring e iniial sages of e die-cas die design o esimae e leng of e enilaion cannel. Alog proposed algorim did no predic e correc sape of e free srface of e liqid meal flow in e enilaion cannel, and e sape of e solid-liqid inerface, i correcl prediced flow leng of e meal in e cannel, and e locaion were solidificaion occrs along e cannel leng. Despie being sligl inaccrae, e mos significan adanage of e proposed nmerical algorim is a sbsanial redcion in CP ime e proposed algorim reqired merel 8 mines o aciee a conerged solion compared o e 5 mines reqired b e commercial sofware FLOWD.
88 Caper 5: Some Concerns and Fre Researc Direcions In e las wo capers, e algorim deeloped sing e lbricaion approimaion and e simpler laminar flow is sown o be a sefl ool for predicing flow leng, gap wid, and emperare disribion dring e flow of e liqid meal in in cannels. I was fond o be especiall sefl for predicing e flow and solidificaion caracerisics in e enilaion cannels of e die-cas die. Howeer, e presened algorim reealed some of is limiaions as well. For eample, e algorim was deeloped and erified for simple recanglar cannels. Fre deelopmen of e algorim sold inclde deriaion of e goerning eqaions for flow in more comple saped cannels. To anale e flow and solidificaion caracerisics of liqid meal flowing arond corners and in wa saped cannels, i ma be necessar o inclde e cenripeal-pe inerial forces as well as o se e crilinear coordinaes. Modificaions of some basic assmpions ma also be necessar. For eample, assming a slg-flow eloci profile raer an a parabolic profile ma be closer o e real rblen flow obsered in e cannel. Tese pes of canges sold be a par of e direcions aken for fre code deelopmen. Presened algorim was deeloped sing a consan ermo-psical properies of e liqid meal. Fre deelopmens sold inclde e se of e solidificaion model a ilies e emperare-percenage of solids cre obained from e solidificaion cre and measred sing a ermocople in conjncion wi e Forier ineracion procedre [1]. I will allow one o se e presened algorim o calclae flow and solidificaion caracerisics for a arie of commercial allos For eample, e solidificaion cre for e alminm allo A8 is sown in Figre 5.1. Te cre sape
89 represens a balance beween ea los b e meal dring solidificaion and ea generaed dring pase cange. As obsered in e figre, e laen ea of fsion is no a linear fncion of ime. In binar allos, e ea is released oer a range of emperares, so solidificaion does no progress in an orderl manner. Firs par of e solidificaion cre indicaes rapid release of e laen ea, nil all sperea is los. Howeer, solidificaion doesn affec e liqid meal nil e emperare falls below e liqids line (poin 1 on e solidificaion cre). Tis poin is also called e emperare of liqids arres; is is e emperare a wic e primar dendries sar o form. Te ne imporan poin on e solidificaion cre, poin, is corresponds o e ime close o e end of e solidificaion process. In order o beer deermine e caracerisic poins of e solidificaion process, e firs-deriaie cre is oerlaid on e solidificaion cre and is sown in Figre 5.. Te firs-deriaie cre no onl elps o deermine canges in e solidificaion process, b also allows one o disingis e poin a wic e primar and secondar dendries are formed. As i can be obsered in e figre, e firs-deriaie cre increases in ale (p o e poin 1 in Figre 5.), indicaes e beginning of solidificaion. And en, a e end of e solidificaion process (poin in Figre 5.), i decreases. Te region beween poins 1 and on e firsderiaie cre indicaes free grow of e primar dendries. Te emperare range, as seen beween poins 1 and, is were dendrie arms increase in sie, b are no e ocing eac oer. Poin is called e dendrie coerenc poin and corresponds o e fracion of solids in e liqid meal wen dendrie arms come in conac wi eac oer. Afer a poin, free grow is no possible an more, and e region beween poin and poin corresponds o e penomena of dendrie ickening and formaion of e rigid
9 skeleon. Te region of e firs-deriaie cre beween poins and 5, corresponds o e end of solidificaion. Deried relaions will allow one o se a emperare-dependen laen ea ale, insead of e sal consan one. Figre 5.1 Te measre solidificaion cre for alminm A8 allo. Figre 5. Te firs-deriaie cre, obained from e slope of e solidificaion cre sown in Figre 5.1, is oerlaid on e original solidificaion cre.(y ais is liqid meal emperare, X ais is percenage of solid in meal)
91 Te mel iscosi does no pla a significan role in e flow of liqid meal in in cannels. As was sown earlier sing e Weber and Capillar nmbers, e inerial force is e dominan force dring meal flow rog e enilaion cannels. Howeer, in order o increase e accrac of predicions b e proposed algorim, e cange in e meal iscosi wi an increase in solid olme fracion sill needs o be inclded, een og mos of e cange in iscosi occrs owards e end of solidificaion [] and ence ma no affec e simlaion accrac mc. As sown in Figre 5., e primar dendries sar o form wen e meal is sill in e so sleee, i.e., a poin 1. As e meal flows rog e gaes of e die-cas die, some of e dendries break ino smaller pieces, and are carried rog e cai of e die-cas die o e oerflows and en o e enilaion cannels, s increasing e solid fracion in e mel. Te sall become ceners of ncleaion for e eqiaed dendries dring frer solidificaion. Noe a de o a small cannel ickness (.5 mm), a ig meal eloci, and a ig solidificaion rae, e colmnar dendries canno form dring solidificaion inside e in cannels and e ms one doesn deelop o inflence e meal flow. (As seen in Figre 5., e non-dendrie srcre is dominan inside e in enilaion cannel.) Figre 5. Non-dendrie srcres seen in e micrograp of a secion of a in enilaion cannel.
9 As a fre deelopmen, e ariable iscosi as a fncion of e liqid-meal emperare (and ence e solid fracion) can be sed o improe e accrac of e proposed algorim. Sc a deelopmen ma improe e predicion accrac for e analsis of meal flows in icker cannels (1- mm). Wi an increase in e accrac of emperare predicions, seeral oer imporan parameers can be prediced as well. For eample, afer implemening all e proposed canges in e code, e simlaion for predicing meal flow and solidificaion can also be sed for e sbseqen predicion of e macro and micro srcres in in cannels. Te ideas presened in is final caper, if implemened, ma improe e accrac of e proposed algorim. Howeer, eir implemenaion will inrodce addiional nonlineariies (and ence addiional ieraie loops) in e proposed (simpler) solion, and ence ma rob e meod of is adanages of faser compaional speed. Onl some fre researc will answer ese qesions conclsiel.
9 References 1. Abo die casing, Te Nor American Die Casing Associaion, arcied from e original on 1-1-15, rerieed 1-1-15.. W. F. Smi, Srcre and Properies of Engineering Allos, McGraw-Hill, 199.. Jean-Lc Caber, e al. A Hisor of Algorims: From e Pebble o e Microcip, 1999, Springer-erlag.. C. Edwards, Jr. Te Hisorical Deelopmen of e Calcls, Springer-erlag, 1997. 5. J. on Nemann and H. Goldsine, "Nmerical Inering of Marices of Hig Order" Bllein of e AMS, No. 197 6. J. D. Hoffman, Nmerical meods for engineers and scieniss, CRC Press, 1 7. S.. Paankar, Nmerical ea ransfer and flid flow, McGraw-Hill, 198. Harlow, 8. F.H. Harlow, J.E. Welc, Nmerical Calclaion of Time-Dependen iscos Incompressible Flow, 1965 Ps. Flids 8, 18. 9. B.D. Nicols, C.W. Hir, Meods for Calclaing Mlidimensional, Transien Free Srface Flows Pas Bodies, Proc. of e Firs Inernaional Conf. On Nm. Sip Hdrodnamics, Gaiersbrg, ML, Oc. - 1975. 1. R.K. Sa, M.S. Bai, Laminar conecie ea ransfer in dcs Handbook of Single Pase conecie ea ransfer, Kakac S. Sa R.K., and Ang W., Wille, New York, 1987 11. C.W. Hir, B.D. Nicols, olme of flid (OF) meod for e dnamics of free bondaries, J. Comp. Ps. 9, pp. 1-8 1981. 1. B. R. Mnson, A. P. Romaer, T. H. Okiisi, W. W. Hebsc, Fndamenals of Flid Mecanics, Wile, New York, 1.
9 1. A.Z. Seri, Tribolog fricion, lbricaion, and wear, McGraw-Hill, 1979 1. M. Epsein, F. B. Ceng, Comple freeing-meling inerfaces in flid flow, Flid Mec., 15, pp. 9-19, 198. 15. B. Weigard, H. Beer, Ice-formaion penomena for waer flow inside a cooled parallel cannel: an eperimenal and eoreical inesigaion of wa ice laers, In. J. Hea Mass Transfer, ol. 6 No. pp. 685 69, 199 16. A. Roboa, E. Moneiro, Hea ransfer in mli-block grid dring solidificaion: Performance of finie differences and finie olme meod, J. Maerial Processing Tecnolog,, pp. 51-58, 8. 17.B. Mocnacki, E. Majcrak, Nmerical modelling of casing solidificaion sing generalied finie difference meod, Maerial Science Form, 68-6, pp. 676-681, 1 18. X. Yao, M.S. Dargsc, A.K. Dale, C.J. Daidson, D.H. SJon, Inesigaion ino e effecs of ncleaion parameers on grain formaion dring solidificaion sing a celllar aomaion finie conrol olme meod, J. Maerials Researc, (9), pp. 1-5, 8 19. H. Monaeri, M. Bssmann, J. Mosagimi, Accrae implemenaion of forcing erm for wo-pase flows ino SIMPLE algorim, In. J. of Mlipase Flow, 5, pp. - 5, 1. Z. Domanski, M. Ciesielski, B. Mocnacki, Applicaion of conrol olme meod sing e oronoi essellaion in nmerical modelling of solidificaion process, AIP Conference Proceedings, pp. 17-6, 1 1.. Grodanic, Finie-difference meods for simlaing e solidificaion of casings,
95 Mariali in Tenologije, (5) pp. -7, 9. C. Yang, C. Li, Applicaion of SIMPLE algorim on non saggered grid o nmerical simlaion of e Yellow Rier flow and sedimen ranspor in Sapoo Reac, ISWREP 11 Proceedings of 11 Inernaional Smposim on Waer Resrce and Enironmenal proecion, 1, No. 58985, pp. 66 69.. M. Larmaei, T. F. Madi. Analsis of SIMPLE algorim for dep aeraged simlaions, Enironmenal dralics-proceedings of e 6 Inernaional smposim on enironmenal dralics,, pp. 967-97, 1.. A. K. Rasogi, Predicions of ea and mass ransfer in open cannels, J. of e Hdralics Diision, ASCE, 1(), pp. 97-, 1978. 5 Y. S. Cen, S.M. Kim, Compaion of rblen flow sing an eended k rblence closre model, NASA, CR 179, pp. 1-5. 6. Yako, S.A. Orsag, S. Tangam, T. B. Speiale, Deelopmen of rblence models for sear flow b a doble epansion ecniqe, Ps. Flids A, (9) pp. 151 15, 199. 7. H. S. Hele-Saw, Te flow of waer, Nare, London ol. 58, pp. -8, 1898. 8. K. enkaesan, R. Sipri, Nmerical simlaion and comparison wi waer modeling sdies of e ineria dominaed cai filling in die casing, Simlaion of Maerials Processing: Teor, Meods and Applicaions, edied b S. F. Sen and P. R. Dawson - Balkema, Roerdam, pp. 1-111, 1995.
96 9.. Aleksenko,.. Nakorako, B.G. Poksae, Wae formaion on a erical falling liqid film, AICE J. ol. 1, pp. 16-158, 1985.. S.C. Gpa, Te Classical Sefan Problem: basic conceps, modelling and analsis, JAI Press,. 1. J.K. Carpenner, P.H. Seen, Hea ransfer and solidificaion in planar-flow melspinning: ig weel speeds, In. J Mass Transfer,, N 9 pp. 199-7, 1997.. NADCA Prodc specificaion for die casings, Secion Table A---97. R. Mianr, K. Siddiqe, R. Kaa, Inflence of ineria and opograp in incai Psics of flids,.1 N. 5, pp. 17-1719.. C. Karcer, P.H.Seen, Hig Renolds nmber flow in a narrow gap drien b solidificaion, Psics of flids, ol. 1, nmber, pp 8-8, 1. 5. J.K. Carpener, P.H. Seen, Hea ransfer and solidificaion in planar-flow melspinning: ig weel speeds, In. J. Hea Mass Transfer ol., No 9, pp 199-7, 1997. 6. A. Reiker, K. M. Pillai A fas nmerical simlaion for modelling simlaneos meal flow and solidificaion in in caiies sing e lbricaion approimaion Nmerical Hea Transfer Par A, 6, pp. 75-1, 1. 7. K. enkaesan, R. Sipri, Nmerical simlaion and comparison wi waer modeling sdies of e ineria dominaed cai filling in die casing, Simlaion of Maerials Processing: Teor, Meods and Applicaions, edied b S. F. Sen and P. R. Dawson, Balkema, Roerdam, pp. 1-111 1995. 8. B. J. Hamrock, Fndamenals of Flid Film Lbricaion, McGraw-Hill, New York, 199. 9. FLOWD Flow Science, www.flowd.com.
97 C. Kwak, C. Kiris, Compaion of iscos Incompressible Flows, Springer, 11. 1.E. Fras, A new concep in ermal analsis of casing, Transacions of American Fondarmen s Socie, 199..11: p. 55-511..W. Kr and D.J. Fiser, Dendrie grow a e limi of sabili: ip radis and spacing, Aca Meall., 1981, ol. 9, pp. 11..J. E. Hac, Alminm: Properies and Psical Meallrg, ASM Inernaional, 198.. G.E.Toen, K. Fnaani, L. Xie, Handbook of meallrgic process design, Marcel Dekker Inc.,
98 Appendi A Renolds lbricaion eqaion afer inclding e effec of ineria Goerning eqaion of e ransien flow in non-dimensional form is gien as follows: Momenm eqaions p p (A1) were Re modified Renolds nmber. Re- Renolds nmber L - cai aspec raio L H Conini eqaion (A) Te dimensionless ariables are defined as L X, L Y, H Z, H H,,, P L L H w were _ is a reference eloci,,, and w are e elociies in,, and direcion, and H is e ickness of e cai. Since eloci w is anising a e op and boom bondaries, and is ariaion is negligible owing o a small ickness of e cai, i will
99 be se o ero. sing e conini eqaion, eqaion (A1) can be epressed in a conergen form as p p (A) On inegraing eqaion (A) oer e ickness of e cai ields d d p d d d d d p d d d (A) We will assme parabolic form of eloci disribion in and direcion: ) )(,, ( (A5) ) )(,, ( (A6) On sbsiing eloci disribion eqaions (A5) and (A6) ino eqaion (A), we ge d d p d d d d d p d d d (A7)
1 d d p d d d X (A8) d XI d p d d d (A9) sing e proper g f f g g f (A1) e erm X in eqaion (A8) can be wrien as X (A11) And en erm X1 in Eq.(A9) can be wrien as XI Since ere are no canges in and elociies in direcion i.e., and X and XI frer simplifies o X
11 XI Opening brackes in eqaions (A8) and (A9) as well as emporaril eliminaing e common direcion inegral ields: p XXX XX (A11) p XXXI XXI (A1) Opening brackes and rearranging erms in eqaions (A11) and (A1) resls in e following eqaions: (A1) ) ( XX (A1)
1 ) ( XXI (A15) XXX (A16) (A17) XXXI (A18) Sbsiion of e aboe-deried erms ino e original eqaions, Eqs. (A11) and (A1), resls in
1 p p (A19) Rearranging erms in Eq. (A19) leads o p IIIX IIX IX p III II I (A) We will bring back e direcion inegraion and inegrae along e direcion all e erms of eqaion (A) in e following secion. 6 1 d d d (A1) 1 d (A) Inegraing erm I resls in
1 5 1 5 5 5 5 5 5 5 5 d d d (A) Inegraing erm IX resls in 5 1 5 5 5 5 5 5 5 5 d d d (A) Inegraing erm II resls in 6 1 ) ( d d d (A5) Inegraing erm IIX resls in
15 6 1 ) ( d d d (A6) Inegraing erm III resls in d (A7) Inegraing erm IIIX resls in d (A8) 6 1 d d (A9) 1 (A) On sbsiing back all e aboe erms ino eqaion (A), we ge p 5 1 6 1 6 1 5 1 1 6 1 (A1) p 5 1 6 1 6 1 5 1 1 6 1 (A) On diiding is eqaion b, we ge
16 p p 6 1 6 1 1 6 1 6 1 6 1 1 6 1 (A8) A combining of similar erms resls in p p 6 1 6 1 (A9) Te second erm in e firs eqaion of (A9) can be rewrien as Since (from e conini eqaion, Eq.(.1)), e final form is Te second erm in e second eqaion on (A9) can be ransformed similarl. On sbsiing back ese ransformaions in (A9), e final form of e goerning eqaions is epressed as p p 6 1 6 1 (A)
17 Appendi B Discreiaion of momenm eqaions Te final momenm eqaions, Eqn. (.1), gien in caper are replicaed ere for conenience. p 6 6 1 (B1) p 6 6 1 (B) Figre B.1 Saggered grid arrangemen Saggered grid arrangemen is sown in Figre B.1. Tis grid arrangemen proides a sronger copling beween flid pressre and elociies, ereb improing e sabili of e solion. Te primar grid-poins, were e scalar qaniies as pressre P and
18 emperare T are sored, are denoed wi capial leers. Te secondar grid-poins, were ecor qaniies sc as elociies and are sored, are denoed wi small leers. Te discreied form of momenm eqaions on e saggered grid, sing e firs order, forward difference sceme in ime and e cenral difference sceme in space, is sown below. Momenm eqaion In e direcion: n J I P n J I P J I n n j i n j i J I n J I n J I J I J i n j i n j i n J I n J I J i n J i n J i, 1,, * 1, 1 1,,, 1,, 1, 6 1, 1 1,, 1, 1, 1, 1 1, 6 1 (B) In e direcion: n J I P n J I P J I n j i n j i J I n J I n J I J I j I n j i n j i n J I n J I j I I n j I n j, 1,, * 1, 1 1,,, 1,, 1, 6 1, 1 1,, 1, 1, 1, 1 1, 6 1 (B) De o e saggered-grid arrangemen sed for e discreiaion of or parial differenial eqaions, e ales of elociies are aailable onl on e secondar-grid poins. An inerpolaion ms be sed o deermine elociies on e primar-grid poins. Te following relaions can be sed:
19 I, J 1 1 i, J (B5) I, J i 1, J 1 i 1, J (B6) I, J 1 i, J 1 I, j (B7) I, J I, j 1 1 I, j 1 (B8) I, j 1 i 1, j 1 I 1, j 1 I, j 1 i 1, J 1 i 1, J (B9) 1 i, j 1 I, j 1 I 1, j 1 i, J 1 i, J (B1) 1 (B11) i 1, j I 1, j I, j i 1, J i 1, J 1 Figre B.. Grid lines arrangemen near e cannel walls
11 In order o eliminae e need for pressre ales a e bondaries, walls of e cannel are made o coincide wi e secondar-grid lines. Ten flid pressre is calclaed inside e cannel onl. No-slip bondar condiion can be epressed as: i, j i 1, j... i, j i 1, j... ales sc as i, J 1is reqired in e analsis and can be fond as: I, J I, J 1 I, J 1 I J 1 I, j, A e inflow bondar, elociies in e direcion of flow,, are known from e iniial condiions. elociies oside e bondar, sc as I 1, j 1, are needed for e solion, and can be fond from e erapolaion I I 1, j 1 I, j 1 1, j 1 I, j 1 I, j 1 I, j 1 I 1, j 1
111 Appendi C Esimaion of locaion of solid-liqid inerface sing e Sefan condiion, e solid-liqid inerface eloci can be calclaed as L f s n i 1 s n i k s Ts s k l Tl s (C1) Wen solid liqid inerface is formed, T s erm redces o since e inerface emperare is kep a e meling poin eerwere, wile emperare disribion in e liqid meal. T l is esimaed based on e To compe e moemen of e inerface wi ime, e 1-D eloci of e solid-liqid inerface is mliplied b e ime sep: n s n i 1 i * Ten e crren eig of e cannel is esimaed as 1 (C) n 1 n n 1 i i i (C) wi e facor of being sed o inclde e effec of inerface moemens bo from op and boom. A pical inerface moion is sown in Fig. C.1
11 Figre C.1 A scemaic sowing e grow of solid-liqid inerface de o meal solidificaion on e cannel walls.
11 CRRICLM ITAE Aleandre Reiker Place of bir Kisine SSR BS Kisine Polecnic Insie Kisine SSR Major: Mecanical Engineering PD W Milwakee, Milwakee WI Major: Engineering Professional Pblicaions: Book: Casing: An analical approac, Springer, Jl 7. Papers: 1. Mli-Sage Plnger Deceleraion Ssem, NADCA congress 8. Die-casing end-of-fill and drop forge iscomeer flow ransiens eamined wi a copled-moion nmerical model. 68 WORLD FONDRY CONGRESS. Applicaion of one dimensional nmerical simlaion o opimie process parameers of a in wall casing in ig pressre die casing process. Die Casing Engineer, Ma 9.. Simlaion of welding of alminm pars sing ig energ laser beam, Presened on 1 ASME inernaional mecanical engineering congress ancoer BC 5. Opimiaion of ig acm die cas process. Die cas Engineer, Jl 6. A fas nmerical simlaion for modeling simlaneos meal flow and solidificaion in in caiies sing e lbricaion approimaion, Jornal of Nmerical Hea Transfer 1