ANALYSIS OF SEGREGATION PROCESS USING THE BROKEN LINE MODEL. NUMERICAL REALIZATION

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1 31/10 Archives of Foundry, Year 2003, Volume 3, 10 Archiwum O dlewnictwa, Rok 2003, Rocznik 3, Nr 10 PAN Katowice PL ISSN ANALYSIS OF SEGREGATION PROCESS USING THE BROKEN LINE MODEL. NUMERICAL REALIATION J. S. SUCHY 1, B. MOCHNACKI 2, M. PRAŻMOWSKI 3 1. AGH, Cracow, 2, 3 Technical University of Opole SUMMARY In the paper [1] the basic idea of simplified model of macro-segregation process called 'a broken line model' is presented. Here, the details concerning the algorithm realizing the numerical simulation of the process are discussed. The examples of computations concern both the constant solidification rate and the time-dependent one. Key words: macrosegregation, numerical modelling, broken line model 1. INTRODUCTION In the thermal theory of foundry processes one can find the solutions concerning the kinetics of casting solidification obtained under the assumption that the temporary temperature field in domain considered is described by a certain class of functions (for instance the parabolas method proposed by Wiejnik [2]). The authors of this paper propose the similar approach to the macro-segregation process modelling. The temporary concentration field can be quite exactly approximated by the broken line. The first sector of line correspond to boundary layer, while the second one to the region in which the concentration field is homogeneous (Figure 1). The number of information concerning the physical aspects of the process (solidification rate, thickness of the boundary layer etc.) assures the univocal determination of the parameters of assumed function. The start point of the algorithm consists in the computations of the direction of sector corresponding to the boundary layer, next the mass balance of alloy component allows to determine the others parameters of the broken line. 1 prof. dr hab. inż. 2 prof. dr hab. inż. 3 dr inż.

2 THE BROKEN LINE MODEL We consider the solidification problem for which the temporary position of interface and the function determining its dislocation are known. The mass balance for the neighborhood of moving boundary leads to following condition [3, 4],, z2 x t z1 x t d D2 D1 k z1 t n n dt x x 1, (1) where z 1 (x, t ), z 2 (x, t ) are alloy component concentrations in the liquid and solid subdomains, D is a diffusion coefficient, /n is a normal derivative, x = ξ corresponds to the solid-liquid interface, k is a partition coefficient. If the mass transfer in the solid body is neglected (D 2 = 0) and the 1D problem is considered then z1 x, t d D1 k z1 t n dt x 1, (2) Fig. 1. The broken line model Rys. 1. Metoda linii łamanej On the basis of the last formula we determine the slope of the first section of the broken line for x = ξ (ξ is a multiple of assumed step Δx = h). Next on the basis of balance for time t corresponding to x = ξ, namely z2 xdx z11 xdx z12 L Lz0 (3) 0 0 where z 11 is a linear function approximating the concentration field in domain of

3 237 boundary layer, z 12 is a constant value (see: horizontal sector in Figure 1) we can calculate the alloy component concentration for x = ξ. In this way the set of parameters determining the course of broken line is known. 3. THE EXAMPLES OF COMPUTATIONS We consider the plate (2L = [m]) made from Al-Si alloy (z 0 = 0.05). The constant solidification rate dξ/dt = [m/s] has been assumed. The partition coefficient: k = 0.2, the diffusion coefficient: D 1 = [m 2 /s], the thickness of boundary layer δ = 0.5, 1, 1.5 [mm], correspondingly [5]. In Figure 2 the changes of the concentration in the liquid state for x=ξ(t ), while in Figure 3 the changes of concentration for x = ξ(t ) + δ are shown. =1,5 mm =1,0 mm =0,5 mm t [mm] Fig. 2. Concentration for x = ξ ( t ) Rys. 2. Stężenie na froncie krzepnięcia The solutions shown in Figures 2 and 3 show that the assumption concerning the thickness of boundary layer does not cause the essential differences in the calculated courses of boundary and internal concentrations.

4 238 =1,5 mm =1,0 mm =0,5 mm t [mm] Fig. 3. Concentration for ξ ( t ) + δ Rys. 3. Stężenie dla ξ(t) + δ 0,11 t=40 t=30 t=12,5 t=4 0,0 1,5 3,0 4,5 6,0 7,5 9,0 x[mm] Fig. 4. Concentration field (δ = 1.5 [mm]) Rys. 4. Pole stężeń (δ = 1.5 [mm])

5 239 0,11 t=40 t=30 t=12,5 t=4 0,0 1,5 3,0 4,5 6,0 7,5 9,0 x[mm] Fig. 5. Concentration field (δ = 0.5 [mm]) Rys. 5. Pole stężeń (δ = 0.5 [mm]) 5 t=40 s 0 t=360 s t=900 s 5 t=1200 s 0 0,0 1,5 3,0 4,5 6,0 7,5 9,0 x[mm] Fig. 6. Concentration field for v (t) Rys. 6. Pole stężeń dla v (t) In Figures 4 and 5 the concentration profiles for different times are shown. The thicknesses of the boundary layer are equal to 1.5 and 0.5 [mm], correspondingly. The solution presented in Figure 6 has been found for the variable solidification

6 rate. The well known equation ξ = kt has been taken into account (k = ) and the solidification rate resulted from the differentiation of the formula discussed. From the numerical point of view a such problem is not more complicated than the problems discussed previously. REFERENCES [1] J. S. Suchy, B.Mochnacki, Analysis of segregation process using the broken line model. Theoretical base, Archives of Foundry, 10, [2] A. I. Wiejnik, Tieoria zatverdevanija otlivki, Maszgiz., Moskva, [3] E. Majchrzak, B.Mochnacki, J.S.Suchy, Numerical model of macro-segregation during directional crystallization process, Journal of Materials Processing Technology, 78, 1998, [4] B. Mochnacki, E.Majchrzak, J.S.Suchy, Boundary element model of coupled heat and mass transfer in solidifying casting, International Journal of Cast Metals Research, Vol. 12, No 4, 1999, [5] J. S. Suchy, Segregation of alloy components during the directional solidification, Silesian Technical University Publishers, Mechanics, Gliwice, Prac finansowana ze środków KBN grant nr 4T 08B STRESCENIE OPIS PROCESU SEGREGACJI A POMOCĄ MODELU LINII ŁAMANEJ. REALIACJA NUMERYCNA W pracy [1] omówiono istotę uproszczonego modelu makrosegregacji nazwaną przez autorów metodą linii łamanej. W niniejszym artykule przedstawione zostaną niektóre szczegóły dotyczące algorytmu realizującego obliczenia numeryczne. Przykłady obliczeń dotyczą zarówno zadań, dla których założono stałą szybkość krzepnięcia, jak i zadań, dla których szybkość krzepnięcia zmienia się w czasie. Recenzowała Prof. Ewa Majchrzak