Krzepnięcie Metali i Stopów, 18 PL ISSN

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1 Slidificati f Metais ad Alłys Krzepięcie Metali i Stpów, 18 PL ISSN NUMERICAL SIMULATION OF SOLIDIFICATION PROCESS IN DOMAIN OF CONTINUOUS CASTING WITH COMPLEX SHAPE Adam Kapusta ad Adrzej Wawrzyek Silesiali Tecflllica/ Uiversily, GliH ice, Plr111d ABSTRACT I the paper a algrithm f temperature field cmputati i dmai f ctiuus castig f "dg' s bae" shape is preseted. The algrithm cstitutes a cmpsiti f three methds: alteratig phase truc ati methd (APTM}, aalytical-umerical methd called R- fuctis e ad fiite differece e ( FDM}. I fial part f t he paper umer i cal examples illustratig discussed algrithm is shw. NUMERYCZNA SYMULACJA CIĄGŁEGO KSZTAŁCIE ODLEWANIA WLEWKA O ZŁOŻONYM STRESZCZENIE W pracy przedstawi algrytm wyzaczaia pla temperatury we wlewku typu "psia kść" pdczas dlewaia ciągłeg. Algrytm te bazuje a trzech metdach: metdzie przemieej fazy (APTM}, aalitycz-umeryczej metdzie wykrzystującej pjęcie R-fukcji raz metdzie różic skńczych. Rzwiąza przykład umeryczy ilustrujący prezetway algrytm. INTRODUCTION The cmplex gemetrical shape f csidered bject ca be i simple way take it accut wig t itrducti t umerical simulati s-called R-fuctis. The details f this apprach are preseted i [4), [8) whereas the prblem f RFM applicatis i the rage f thermal thery f fudry have bee discussed i [2), [5). I particular the umerical mdel simulatig the slidificati ad clig prcesses i the

2 vlume f cper ctiuus castig has bee aalyzed i [2). I this paper the geeralizati f RFM algrithm fr the case f mre cmplex gemetrical cditi will be shw. Applicati f geeralized APTM allws t liearize a liear slidificati prblem by the crrecti f cal culated ethalpy field fr successive levels cf time (see [l), [3), [5)). It is pssible t esicter the bjects fr which the ethalpy related t a uit f vlume is described as fllws (see Fig.l.) r m-l H(T) =f c (t) p (t) dt +E p:xl(r-r' i l (l) TJ (u) = { ' l ' us:o, u> O (2) where : c, p - are the thermphysical parameters; K, - the latet heat fr " i" phase chage; m - umber f subdmais i csidered bject; ~ - iterface fr trasiti i ~ i+l (see Fig.l. ). H i=3 ~ / i=l Figura l Ethalpy distributi T

3 89 The cmputati f apprximate ethalpy field at mmet t P' ' (the ethalpy distributi fr tp is kw) is realized i m stages. Fr every stage 1 the whle castig area i s frmally reduced t e f the phases by "additi" t distiguished pi:r;1t f a certa i quati ty f ethalpy ( clder phases) r "substracti" the ethalpy fr warmer subdmais. After this rebuildig f pseudiitial cditi e btais tbe bmge us budary prblem wbich ca be slved i simple way. At tbe same time tbe ed f every stage f cmputati depeds tbe substracti r additi previusly mdified values f etbalpy. Tbe fial sluti cstitutes tbe ew pseudiitial cditi fr tbe ext step f slidificati mdellig. Tbe defiiti f R-fuctis bas bee itrduced by Rva cbev (see [6) 1 [7)) ad als applied by tber autbers (4], [5). Tbese appracb allws t take it accut, i exact way, the sbape f aalyzed gemetrical bje et ad the budary e ditis give tbe uter surfaces f csidered dmai. GOVERNING EQUATIONS Let us csider tbe area Q = Q (x, y). Tbe etbalpy field i tbis dmai is described by liear Furier 1 s equati. O the parts f its budary tbe cditis f 1st r 3rd kid are give, it meas that fr k c parts the ethalpy distributi is kw ad fr rest f parts tbe beat flux ctiuity is as sumed. Tbe budary cditi f tbe 2d kid i s treated as particular case f 3rd e. Tbe fucti fulfillig all budary cditi (s - called ge eral structure f sluti) is f tbe frm wbere: k L gl = i l fi U) i - l L f 1 w? g2 = i k+l L -1 U> i E i :: t i=k+l L g3 = i=k+l - 1 U>j L w? L g, = i=k+l L j-1 i k+l - 1 U> t - l hl U>j - 1 U>; (4)

4 <)() ad w = w(x,y), w, = wi(x,yl - aalytical descripti f whle area Q ad its part; I the Eq. (3), the fucti which ctais the certai ukw parameters a 11 time.at the same The cstats f 1 ad h 1,.i = k -t l,..., are described by the Egs (8) - (10). The aalyzed dmai ca be preseted as a result f peratis fiite umber Of subdmaifls (f r example Q = U 0 1 ). Substitutig the perati U ad by Rvachev's peratrs (see [6), [7) l, e btais the a alytical descripti f Q. DETERMINATION OF TEMPERATURE FIELD IN THE OBJECT a ~ b :_ Figure 2 Crss secti f ctiuus castig I t he p a per the lateral secti f ctiuus castig f "dg's be", preseted i Fig. 2, will be csidered. Take it accut t h e ge - metrical s ymmetry, t he quter f the secti shuld b e aalyzed. This e ca be treated as a cmpsiti f the fllwig subdmais

5 = ( (x, y) : x~ O) 0 2 ={(x,y): y~o) 0 3 ={(x,y): b - y> O} 0 4 ={(x,y): y-(a-c)/b (x-c) > ~} (5) ad the aalytical frm f budary descripti is as fllws w he re w(x,y) =w 1 (x,y) A 0 w 2 (x,y) A 0 w 3 (x,y) A 0 w 4 (x,y) (6) W2 =Y (1)3 = b -y () 4 = y- (a-e) /b (x-e) (7) The eaidered castig was made frm carb steel. Alg the surfaces a 1 i a. the!1mgeus 2 d kid budary cditis have bee assumed, at the same time fr 0 3 i. the 3 rd kid es. Heat trasfer cefficiets fr clig zes are as bellw: Ci Ci Ci 1300 [W/m 2 deg] 950 [W/m 2 deg] 550 [W/m 2 deg] (primary clig ze) 1 (secdary clig ze - sectr A); (secdary clig ze - sectr B). The legth f ctiuus castig muld: l u = 0.8 [m]. whereas fr secdary clig zes 5 [m]. The pullig rates: V 1 = O. 015 [m/s) ; V 2 O. 020 [m/s); v, O. 025 [m/s]. have bee csidered. The assumed iput date crrespds t typical techlgical cditis fr radial ctiuus castig plats (radius f

6 92 c urvature R- 1 0 [m]). The 3 rd kid cditis writte i ethalpy cveti a c crdig t the phase which i eaidered iterval f time shuld be take it accut is f the frm - slidified part f castig a H = ~ T - _!!H= f a A w A 3 3 -h H (8) - mushy ze (9) - mlte metal (10) wher e A2 i A4 (see Fig.1.): T,' A 2 =f c (t) p (t) dt + p 1 K 1 ; T; A,= f c(t) p (t) dt + p 1 K 1 +p 2~ (11) ad f.= f 3, h.= h 3 The results f umerical simulati are shw i Figs.3-5. ct. C 411SO: tl Q rwul d IIII sgctr A f ~Qcd~ry IIII cali zg ~Q C t r B f ~ e c d~ r y cli y ZO lił Figure 3 Rage f mlte metal f r pullig rate v = [m/s]

7 (M) cet. c.=.s;tig r>uld cctr A f s;c d.. r~ c cal i z Q ~gct r 8 f s cd~ry cli zg, Figure 4 Rage f mlte metal fr pullig rate v = 0.02 [m/s] [M) cet. cas:ti r>uld s:ec tr A f r;c}cddr lj clig zq s e ctr 8 f &Gr ~ daru r c1 l i g- zp Figure 5 Rage f mlte metal fr pullig rate v = [m / s] The r esults f cmputatis have bee c mpa r ed w:i t h umerica.l es quted i [9] ad the y are ccurret.

8 94 REFERF.:NCES [l] Kapusta A., Mchacki B. The Aalysis f Heat Trasfer Prcesses i the Cylidrical Radial Ctiuus Castig Vlume. Bull. f the Pl. Ac. f Sc., Tech. Sc., Vl. 36, N. 5-6, 1988 [2] Kapusta A., Wawrzyek A. Applicati f Rvachev's Fucti i Numerical Simulati f Mvig Budary Prblems. Slidificati f Metals ad Allys, Vl.16, pp.ls0-158, 1994 [3] Mchacki B., Majchrzak E., Kapusta A. Heat Trasfer Prcesses i Slidifyig Ig t. Free ad Mvig Budary Prblems, Berli ad New Yrk, 1991 Numerical Mdel f ad Clig Steel Vl. l, de Gruyer [4] Wawrzyek A. Applicati f R -Fucti Metbd fr Cmputati f the Temperature Field i the Vlume f a C tiuus Castig Muld. Bull. f the Pl. Ac. f Sc., Tech. Sc., Vl. 39, N. 4, 1991 [5] Maj chrzak E., Wawrzyek A. Utilizati f R-Fuctis Methd i Numerical Madellig ąf Slidificati?rcess. Vl. 1: Cducti, Radiati ad Phase Chage, Cmputatial Mechaice Publicatis, Elsevier Applied Sciece, Ld ad New Yrk, 1992 (6] Rvachev L.V., Slesarek A.P. Algebra f Lgic ad Itegral Trasfrmatis i Budary Value Prblems. Naukva Dumka, Kiev, 1976 [ 7J. Rvachev L.V., Apprximati Thery Dumka, Kiev, 1979 Rvachev V.A. N-Classical Methds f i the Budary Value Prblems. Naukva (8] Grzymkwski R., Wawrzyek A. N-Classical Apprach t the Sl u t i f the I i tial PrbJem i Arbitrary Dmai. Prc. f the 1st Burpea Cferece Numerical Methds i 'Egieerig, Brussels, Septerber 1992