Solitliłkation of i\ldah a111l..\lln.d KrL e pn i ę cte: M e talt t S tor>ńw. IR 1'1. ISSN H.WX - 'J.\Xfl NUMElUCAL ANALYSIS OF FOI{CU> CONVECTION INFLI IJ<:N('E ON Tli E COUHSE OF SOLI I>IFICATION PROCESS jan Szajnar Fot/1/ilrv 111.\lifllft'. Silnilfil /'edtnic lll l i ni\'1'1'\ ll\', (.'!i ll'ic. l'olrllul ABSTRACT In this paper the mathcmatical model and numerical computations cor1eerning the pure metal (Al 99.7) solidification in rotational magnctic field are prcscnted. This ticld causes the convectional movemcnts of molten metal (forced convection) and in sub-arca corrcsponding to liquid stale t he heat transfer conditions change in essential way. This process has unquestionable influence on the course of solidification and temperature field in casting domain. The aim of researeh was to eonstruci the computer program simulating heat transfer in considered area at the same time the convectional movements of molten metal should be taken into account. The problem was solved on the basis of eontroi volume method (in enthalpy convention). In this paper the macroscepie approach is prcscnted it means trat boundary-initial problem described by Fourier's equations and adequate conditions i s analyzed. In the fina! part of the paper the results of computations are shown. The problems concerning the crystallization of metal (macroscopic-microscopic scale) will be a subject of future research works. ANALIZA NUMERYCZNA WPLYWU WYMUSZONEJ KONWEKCJI W CIEKłYM METALU NA PRZEBIEG PROCESU KRZF.PNIĘCIA STRESZCZENIE W pracy przedstawiono model matematyczny i obliczenia numeryczne dotycz4ce krzepnięcia czystego metalu (Al 99. 7) w obecności wirującego pola magnetycznego. Wirujące pole magnetyczne powoduje ruchy konwekcyjne fazy cieklej w krzepnącym odlewie (konwekcja wymuszona). Tak więc w podobszarze zajmowanym przez ciekły metal zmieniają si ~ istotnie warunki przepływu ciepła co ma niewątpliwy wpływ na kinetykę krzepnięcia i pole temperatury w krzepnącym odlewie. Celem pracy była konstrukcja programu komputerowego symulującego przepływ ciepła w odlewie przy uwzględnieniu wpływu przemieszczania się cieklego metalu w rozważanym obszarze. Zaclanie rozwiązano wykorzystując metodę bilansów elementarnych (control volume methocl) w ujęciu entalpowym. Wyniki oblicze11 zostały przedstawione w kork owej C7c~ci artykułu. W pracy prezentowane jest ujęcie makroskopowe bazujące na równaniach energii 117. upełnionych odpowiednimi warunkami brzegowo - początkowymi. Problemy dotyojcc krv,talizacji mctal11 w polu magnetycznym będą przedmiotem dalszych badań.
170 MATI-IEMATIC:\1. l >I.'>C'Rli"IION A non -stla!lly tempc r;~tu re lido in the domain of solidified and cooling metal is described by a system of tw11 d,fferential cquat1ons conceming a liquid and solid parts of casting, namely at".(x, t) ] c (T )p (T) +w gradt.(x,t) = [ "' "' "' '" at (l) = div[a",(t".)gradt.,(x, t) j, m=l, 2 where c,., p,., t..m are the thermophysical parameters of liquid and solid state, wis a velocity field in considered object, T, X, r are the temperature, spatial co-ordinates and time. The expression in b racket s on t he left s ide o f eqn (l) i s called a substantial derivative. The velocity field i s generated by action o f rotational magnetic field and w=o for XE D 2 (t), w;ćo for XED 1 (t). In this paper it is assumed that intensive stirring of molten metal caused by magnetic field action takes place in the all sub-area of liquid zone except of boundary layer close to interface r (comp. Fig. l). Because it is rather difficult to determine in convincing way the thickness of this layer therefore the some values of this quantity in numerical s ~ mulation has been taken into account. flow o f me l t D 0 (t) f* (t) l l l l / mol d Fig. l. The ronsidert"-"j suh-areas
l i l The adequate differential equations and bottndary - initial conditions have been transformed to so-called enthalpy convention. The physical enthalpy of metal related to <111 unit of volume is defined as T H (X. t) = f c ( 11) p (li) d li + 11 (T) L o (2) where T< T' ll(t) = l o' l ' T< T' T~ T' (3) L is a latent heat related to an unit of volume, T' - solidification point (Fig. 2). [J /m 3] L Fig. 2. Entalphy distribulion The energy equations expressed in cnthalpy convention are of t he form at the same time a,.=>...,! p.., c,.
172 Along the surface r (t) the Stefan boundary condition is accepted J - a 1 n gradh 1 (X, t) = - a 2 n gradh 2 (X, t)+ Lv" l Al = Az + L (5) The. sense of A 1 and A 2 symbols is shown in Fig. 2, v. is a solidification rate in normai to interface r direction. Along the contact surface between casting and mold the 1st kind condition resulting from Schwarz' solution is assumed (6) where He is the enthalpy corresponding to contact temperature found on the basis of Schwarz' solution [1]. In conventionally assumed limits of considered object the adiabatic conditions in the form XEr.: n gradh(x, t) = O (7) have been taken into account. Initial condition describing enthalpy field at the moment t=o is of the form t= O: H(X, O) = HP. (8) The physical experiments show that molten metal stirring caused by rotational magnetic field is very strong and the temperature of liquid core is practically homogenous. In numerical reali zation this effect can be achieved by the acceptation for this sub-area the thermophysical parameter called effective thermal conductivity [2]. This one is a multiple of real thermal conductivity A 1 and in practice the values of range ~=7>.. 1 are assumed. The introducing of ~ in mathematical description allows to neglect the substantial derivative in eqns (l) and (4) however it is necessary to consider three sub-domains, namely solid state D 2 (t), boundary layer D 1 (t) and shifting melt area D 0 (t). Finally one obtains the system of equations ah (X, t) "" = div[a grad H (X, t)j, m =0, l, 2 at "" "' where index O identilies the convectional movements sub-area. Along the surface between D 0 (t) and D 1 (t) the conditions of temperature and heat flux continuity should be assumed. CONTROI. VOLUME METHOD In this papcr the object shown in Fig. l has been considered. lt was assumed that solidificat.ion process procecds in normai to mold surface direction and the problem was treated as a one d i mensionaj - at the same time in adequate energy bajances the considered form o f liquid - solid in terface h as been taken in to account (moditication o f so-called shape functions in adequate difference equations). Control volume method depends on the (9)
11\ discrt'tintion of consrdern l an a 011, Jn ncnr ary, nltllrlcs <t ihi lll.tkillg up of encr)!y halancc.> for distinguishcd vollltnc ~. J'ltc SILpol.\1 rcsulrin g from l ill it' dr s,,, rization is 1 nnsidcted. For elementary volume.1 V,, one obtai11\ thc fpllov.ing c qu a t~o tl\ (lo) where T(ll) is an inver.\c fun ction to fun CIIOil // -= /1(7') (comp. Fig. 2), 1 1, t 1 " - the poinls being the heginning and end of time interval tj./,.,e" distinguished dircction of heat flow, R, -!herma! resistances in dircction' e. <t>, - sltape functions which are the ratio of a surface area!'j.s, Iimiting element tj. V,, in.. e" direction to volume tj. V., [3, 4], m - the number of considered directions. Above bajances written for all eontroi volumc\ eonstilule a lincar sysrem of equations from which on the basis of initial and next pscudo-initial condition (cnthalpy field at the moment t f) one can directly fin d t he cnthalpy field for 1 1 1 lcvcl o f t i me. Presented algorithm belongs to the group of explicit schemcs it means that main matrix of the system has only one non zero diagonal This schcme must fulfili the stability condition [5] Iimiting acceptable interval of time!'j.t. As it was mcntioncd in chapter l the algorithm of heat transfer computations in considcrcd ohjcct has bccn used to numerical simulation of several geomctrical variants (thickness of hnundary layer o). THE RESULTS OF NUMERICAL SlMULATIONS The solidification process in area of aluminium casting has hccn considered. The following thermophysical parameters for Dm(r) have hcen acccptcd: c 0 =C1, 'A 0 =7>...1, p 0 =p 1, c 1 =1050 [J/kgK], >...,=100 [W/mK], p 1 =2700 [k g/m 3 ], c 2 = 1290, >... 2 =250, p 2 =2380, L=29000(J-p 2 [J/m 3 ], T'=660 C, Tr=700, Tc=650. The function descrihing the physical enthalpy [J/m 3 ] is of the form T< 660 T> 660 ( 11 ) whereas an inverse function T(H) H H< 18.71 I 10 8 2.835 10 6 660, H - 62.7768 lo 7 3.07 10 6 H E < 18.711 10 8, 26.541 10 8 > H > 26.541 10 8 (12) The step of mesh in normai direction to mol d smface i s h = 0. 12[ mm 1 at t he same t i me t he shape of eontroi volume corresponding to r has hcen takcn in to account. In Fig. 3 the kinetics of so1idification for o,-hx>, o2=0.7.'i and 03=0.25 [mml is shown. One can notice that the solidification processin immovahle volume o f molten metal <h -+oo)
174 proceeds slowly in comparison with solidification rate for the case o< oo and this rate increases with the rate acceleration of molten metal it means with reduction of boundary layer thickness. The very interesting is a fact that in initial stage of duration of the process (comp. Fig. 3) the phenomena proceeds conversely. It results from very intensive heat abstraction (superheating enthalpy) from liquid area to inteńace r and then the crystallization rate is smaller than in typical solidification conditions. After the essential reduction of temperature in D 0 (t) the process proceeds in expected way. ~[mm] 0.0 +----t------ł----1----+------ł----ł----' 0.0 0.5 1.0 1.5 20 2.5 30 t [s] Fig. 3. Kinetics o f solidification l: o,... oo, 2:.5 2 =0. 75mm, 3:.5 3 =0.25mm Very distinct are the qualitative and quantitative differences of temperature fields in considered area (comp. Figures 4, 5, 6). As one can anticipate the temperature in D 0 (t) quickly tends to homogenous field and its value only slightly surpasses the solidification point. Presented above model comprises the numerous simplifications. They as a ruje could be discarded but only by essential extension of algorithm and adequate computer program. It should be pointed that from the other hand the experimental verification in fuli confirmed the effectiveness of proposed numerical model.
17 ~ Fig. 4. Temperature field (ó -+oo) e11 ee7 &ej / 1/ l l ~ 1--- e--? V V l-----' ~ - s 1.5s 3s if; V i 655 o 2 3 6 ~[mm] Fig. 5. Temperature field (ó=0.75 mm)
176 l s l.ss 38 855-1-o::.. -+---+---+3---+.---+5--~6 ~[mm) Fig. 6. Temperature field (ó=0.25 mm) ACKNOWLEDGEMENT This research work has been supported by KBN (Grant No 3 0862 91 Ol). The author wish to thank for prof. B. Mochnacki for the help in the range of this paper preparing. REFERENCES [l] W.Longa. Solidification of Castings., Śląsk, Katowice 1985 [2] R.Grzymkowski, B.Mochnacki, The Analysis of Solidification Process in Continuous Steel Casting Domain, Solidification of Metais and Alloys, Ossolineum, 2, (1980), pp. 69-125 [3] J.Szargut, Thermal Computations in lndustry Furnaces, Śląsk, Katowice 1977 [ 4] B. Mochnacki, J.S.Suchy, Modeliing and Numerical Simulation of Casting Solidification, PWN Warszawa 1993 [5] A.A.Samarski, Teoria raznostnych schem, Nauka, Moskwa 1983