2/4 Ahivs o Foundy, Ya 22, olum 2, 4 Ahiwum Odlwniwa, ok 22, oznik 2, N 4 PAN Kaowi PL IN 642-538 MICO/MACO MODEL OF OLIDIFICAION UING HE CONOL OLUME MEHOD B. MOCHNACKI, M. CIEIELKI 2 hnial Univsiy o Czsohowa Insiu o Mahmais & Compu in ul. Dabowskigo 73, 42-2 Czsohowa, Poland UMMAY In h pap h onol volum mhod is applid o numial modlling o hmal posss poding in h solidiying asing. h modl blongs o h goup o h 2 nd gnaion ons (h mio/mao appoah). h domain onsidd is ovd by h msh o onol volums. h onol volums ospond o h inal shap o gains, whil hi nal poins o h posiion o ysal nuli. h numial modl o ysallizaion poss bass on h onol volum mhod (CM). In h inal pa o h pap h xampl o ompuaions is shown. Ky wods: modlling o solidiiaion, mio/mao appoah, onol volum mhod. INODUCION h onol volum mhod (CM) onsius h iv ool o numial ompuaions o h ha ans posss. h domain analyzd is dividd ino n volums. h CM algoihm allows o ind h ansin mpau ild a h s o nods osponding o h nal poins o onol volums. h nodal mpaus an b ound on h basis o ngy balans o sussiv volums. In his pap h onol volums ospond o h inal shaps o gains (in oh wods, o h pimay suu o h asing). In od o assu h onss and xanss o h algoihm poposd w gna h onol volums in h shap o h hissn polygons [] (h sos po. d hab. inż., moh@main.pz.zs.pl 2 mg inż., maiusz@k2.pz.zs.pl
62 joining h nighbouing nods a ppndiula o h suas limiing h onol volums - s: Fig.). L us onsid h onol volum wih nal nod X. I is assumd h ha h hmal apaiis and apaiis o innal ha sous a onnad in h nods psning lmns, whil hmal sisans a onnad in h sos joining h nods. h ngy balan o h onol volum an b win in h om [2,3] H Q () wh H is a hang o onol volum nhalpy duing im inval, Q - h ha ondud a h im om h adjoining nods o h nod X, - a man apaiy o innal ha sous in h onol volum. 2 n Fig.. Conol volum ys.. Objęość konolna h hang o nhalpy o h lmn duing h im is ual o H ( ) (2) wh is h volumi spii ha,, dnos wo sussiv im lvls. I w assum ha h ha luxs lowing o h lmn a popoional o h mpau dins a h momn, hn w shall obain a solving sysm o h yp xplii shm. o Q A (3) wh is h hmal sisan bwn poins X and X, A sua limiing h domain in diion. I w dno as and h mpoay gain adiuss a
63 momn and by h h disan bwn h nods X, X hn ( ) L h λ λ (4) wh λl and λ a h hmal onduiviis o liuid and solid phass. h oh diniion o hmal sisan should b inodud o h bounday volums [3]. L us wi h balan uaion in xplii shm ( ) A (5) o Φ (6) wh ΦA /. Dnoing W Φ (7) w hav n W (8) a h sam im n W W (9) In od o assu h sabiliy o abov xplii shm h oiin W mus b posiiv.
64 2. HE OUCE FUNCION MODELLING h apaiy o innal ha sous suls om h omula L () wh L is h volumi lan ha, is h solid sa aion a h nighbouhood o onsidd poin om asing domain. o, h apaiy o innal ha sous uals L () h valu o o h onol volum w din as ollows π 3 2 2 ( ) (2) L us dno as h undooling blow h solidiiaion poin a h nod X : * (3) Aoding o [4] () 2 d u µ (4) d wh µ is h gowh oiin. In numial alizaion w inodud h appoxima om o uaion (4), namly and * ( ) µ (4) (5) Nx h loal apaiy o sou union has bn dmind.
65 3. HE EXAMPLE OF COMPUAION h mao onol volum om domain o aluminium asing has bn onsidd. On h ona sua bwn asing and mould h subsiu ha ans oiin α[w/m 2 K] has bn assumd. Along h ohs suas limiing h volum h no-lux ondiion has bn akn ino aoun. h inpu daa onning h maial paams an b ound in [2]. In Figu 2 h xampl o ad loal pimay suu is shown, whil in Figu 3 h ooling uvs a sld poins a makd. On an noi ha h alsn is good visibl i is a ypial au o h 2 nd gnaion modls soluion. h nx sah will onn h onsuion o h modls onning h whol asing domain (h s o big onol volums wih innal suu). a) b) Fig. 2. Exampl o loal pimay suu a) a 5s, b) inish suu ys. 2. Pzykład lokalnj sukuy piwonj a) po 5s b) sukua końowa 68 66 67 [ C] 66 * 66 * 659 [ C] 658 65 64 5 5 2 [s] a) b) Fig. 3. a) h ooling uvs a 3 poins b) zoom ys. 3. a) Kzyw sygnięia w 3 punkah b) powiększni 657 656 B C 655,6,8 2, 2,2 2,4 2,6 [s] A
66 h pap has bn sponsod by KBN (Gan No 7 8B 8 8). EFEENCE [] J. Okisz, Moda óżni skońzonyh, w: M.Klib (d.), Mody kompuow w mhani iała sałgo, PWN, Waszawa, (995). [2] B. Mohnaki, J.. uhy, Numial mhods in ompuaions o oundy posss, PFA, Caow, (995). [3]. zopa, J. idlki, Modlling o solidiiaion using h onol volum mhod, olidiiaion o Mals and Alloys, 2, 44, 349-354 (2). [4] E. Faś, W. Kapukiwiz, H.F. Lopz, Mao and mio modlling o h solidiiaion kinis o asing, AF ansaions, 92-48, 583-59 (993). [5] W. Longa, Kzpnięi odlwów, Śląsk, Kaowi, (985). EZCZENIE MIKO/MAKO MODEL KZEPNIĘCIA Z WYKOZYANIEM MEODY BILANÓW W pay pzdsawiono algoym wykozysująy modę bilansów lmnanyh do modlowania posów iplnyh w kzpnąym odlwi. Modl nalży do zw. modli II gnaji (podjśi miko/mako). ozważany obsza pokyo siaką objęośi konolnyh, kóyh kszał odpowiada końowmu kszałowi zian, naomias punky naln yh objęośi odpowiadają położniu zaodków kysalizaji. Algoym numyzny bazuj na pwnj odmiani mody bilansów. W końowj zęśi pay zamiszzono pzykład oblizń numyznyh. nzowała Po. Ewa Majhzak