Numerical modelling of the solidification of ductile iron

Journal of Crystal Growth 191 (1998) 261 267 Numerical modelling of the solidification of ductile iron J. Liu*, R. Elliott Manchester Materials Science Centre, University of Manchester, Grosvenor Street, Manchester M1 7HS, UK Received 12 November 1997 Abstract Numerical calculations are presented describing the solidification of a ductile iron based on the Stefanescu macroscopic heat transfer-microscopic solidification kinetic model but using a different kinetic model than that used by Stefanescu. The results show that the kinetic model used influences the recalescence behaviour predicted by the modelling. Cooling curves calculated with the present model show reasonable agreement with experimentally measured cooling curves for four different cooling rates. 1998 Published by Elsevier Science B.V. All rights reserved. 1. Introduction The solidification of ductile iron is a coupled phase transformation and mass transfer problem. The relationship between the phase transformation and heat evolution is central to the problem because latent heat affects the temperature field which, in turn, influences the microstructural evolution. Temperature recovery [1], specific heat and enthalpy methods [2] have been used to model the release of latent heat. However, these methods use equilibrium thermodynamics to describe a nonequilibrium process. A significant improvement is to use the latent heat method [3 9] which uses solidification kinetics to express the latent heat release term and to model the solidification process. The work of Stefanescu and co-workers is promin- * Corresponding author. ent in this approach. They developed a solidification model which combines a macroscopic heat transfer model with a macroscopic solidification kinetic model and treat the latent heat released as a heat source term in the governing heat transfer equation. The macroscopic heat transfer equation is then solved by finite difference or finite element methods. The starting point of the microscopic kinetic growth model for the eutectic structure is the assumption of instantaneous nucleation and, thereafter, a constant nucleus or grain density. If the density, N, and the growth rate are known, the fraction of the solid during solidification, f can be calculated from the Johnson Mehl equation [10] f "1!exp (! NπR), (1) where R is the eutectic cell radius. The grain impingement factor used in deriving Eq. (1) is (l!f ). 0022-0248/98/$19.00 1998 Published by Elsevier Science B.V. All rights reserved. PII S0022-0248(97)00860-9

262 J. Liu, R. Elliott / Journal of Crystal Growth 191 (1998) 261 267 Fig. 1. (a) Schematic phase diagram and (b) carbon concentration versus eutectic cell radius diagram. The latent heat release term can be obtained from the rate of formation of solid phases, f /t, which is given by f t "(1!f )4πNR R t, (2) where R /t is the rate of growth of the austenite shell, Fig. 1. The eutectic grain or cell consists of a graphite nodule surrounded by the austenite shell, Fig. 1. It is generally accepted that most of the growth of this eutectic cell is controlled by the diffusion of C through the austenite shell that separates the graphite nodule and the liquid. The growth rate of the austenite shell under steady-state conditions is obtained from the equation dr dt " D R C!C R (R!R ) C!C, (3) where D is the diffusion coefficient of C in austenite; R the radius of the graphite nodule; R the radius of the austenite shell; C the concentration of C in the austenite shell at the G/A interface; C the concentration of C in the austenite at the L/A interface; C the concentration of C in the liquid at the A/L interface. These concentrations can be determined from the phase diagram shown in Fig. 1. It should be noted that the C distributions in the austenite shell in this figure should be a convex curve and not a concave one, as has been depicted in some previous treatments. Eq. (3) is based on the assumption that graphite grows by a steady state diffusion of C from the austenite and uses a mass balance of C content at the L/A expressed by the equation d dt 4 3 πr ρ (C!C) "4πR D dc. (4) dr This equation assumes that the rate of C entering the austenite shell from the liquid at the L/A interface equals the rate of C diffusion away from the interface into the austenite. Graphite nodule growth was modelled using the equation ρ π((r )!(R ))C "ρ π((r )!(R ))C, (5) where ρ and ρ are the densities of graphite and austenite, respectively; C is the concentration of C in graphite. This equation assumes that the rate of increase of C in graphite equals the rate of increase of C in the austenite shell or the C content needed for graphite growth equals the C needed for austenite shell growth. Using Eqs. (3) and (5) the radius of graphite nodule and that of the austenite shell and the growth rate of austenite can be calculated numerically at each time step and the latent heat released calculated through the equation qr " f t, (6) where is the latent heat of a volume element. This solidification model for graphite nodule and austenite shell growth is an improvement on earlier models. However, out of necessity, it still makes simplifying assumptions. For example, the mass balance equation used, Eq. (5), may be questioned. The amount of C entering the austenite from the liquid is not necessarily equal to the amount of carbon entering graphite from the austenite. Graphite and austenite growth are influenced by three processess:

1. C enters the austenite from the liquid as the shell grows; 2. C diffuses from the L/A interface towards the G/A interface; 3. C deposits onto the growing graphite nodule. J. Liu, R. Elliott / Journal of Crystal Growth 191 (1998) 261 267 263 At the L/A interface the composition of austenite and liquid are determined and maintained by the partition coefficient and the driving force is the thermodynamic free energy difference between the liquid and austenite phases. C diffusion in the austenite shell from the L/A to the G/A interface is governed by Fick s law with a driving force proportional to the concentration gradient in the radial direction. Finally, the driving force for C deposition on the nodule is the difference in the thermodynamic free energy of graphite and austenite. Hence, the C entering the austenite from the liquid is not necessarily equal to the C entering the graphite from the austenite phase as assumed in Eq. (5). The complete analysis should consider the balances at the L/A and G/A interfaces for the whole microvolume element and the effect of interface movement. This is a difficult problem to analyse. In the present paper we describe an alternative model for graphite growth that considers the C mass balance at the G/A interface rather than the L/A interface. Numerical calculations are presented and the resulting cooling curves compared with those calculated using Stefanescu s model and experimentally measured curves. 2. Mathematical model The solidification of the ductile iron keel block in Fig. 2 was modelled. The following assumptions were made: 1. The keel block was considered as a two-dimensional shape as the central area is long and thin. 2. The physical constants of ductile iron and mould are given in Table 1 and are assumed to be independent of temperature. 3. Fluid flow in the liquid is neglected. 4. The volume change during solidification, shrinkage and porosity are neglected; the ductile iron is Fig. 2. Keel block showing dimensions in mm on the right-hand side and thermocouple positions, 1 4, on the left-hand side. Table 1 Constants used in numerical calculations Constant Unit Value Refs. Constants for cost iron ρ g/cm 7.0 [8] ρ g/cm 7.0 [8] ρ g/cm 7.40 [13] ρ g/cm 2.11 [13] K cal/( C cm s) 0.08 [8] K cal/( C cm s) 0.08 [8] C cal/(g C) 0.202 [8] C cal/(g C) 0.202 [8] cal/g 55.0 [5] D cm/s 410 * Constants for sand ρ g/cm 1.5 [8] K cal/( Ccms) 1.610 [8] C cal/(g C) 0.27 [8] H cal/(cm s C) 4.510 H cal/(cm s C) 1.6510 [8] Note: For D literature shows a range of 910 110. 410 was used for the calculations. considered to be a continuity throughout the solidification process. 5. The instantaneous nucleation and, thereafter, a constant nucleus density is assumed for each volume element; the nucleus density depends on the local cooling rate.

264 J. Liu, R. Elliott / Journal of Crystal Growth 191 (1998) 261 267 The macroscopic heat transfer equation for twodimensional Cartesian coordinates is K ¹ ¹ #K x y #qr "ρc ¹ t, (7) where ¹ is the temperature, t the time, K the thermal conductivity, ρ the density, C the specific heat and qr the latent heat release term. The rate of formation of solid phases, f /t, is calculated from Eq. (2) and the latent heat release rate is calculated from Eq. (6) as in the Stefanescu model. However, it is now based on a C balance at the G/A interface. Eq. (5) is replaced by d dt 4 3 πr ρ (C!C) "4πR D dc. (8) dr In turn, the graphite nodule growth rate at r"r is given by dr " D R ρ C!C dt R (R!R ) ρ C!C. (9) The austenite shell and graphite radii can be calculated numerically through Eqs. (3) and (9), then the growth rate of the austenite is calculated from Eq. (3). The values of R and dr /dt, calculated from the above are extended values which do not consider the grain impingement. The true rate of evolution of solid fraction can be calculated from Eq. (2) using the treatment of Johnson and Mehl [10] and Avirami [11]. The latent heat release term, q, is calculated from Eq. (6). In the solution of the macroscopic heat transfer equation, Eq. (7), the initial temperature of the liquid is assumed to be uniform and is taken as the experimentally recorded highest temperature during filling. The initial mould temperature is assumed to be room temperature, ¹. The boundary condition between mould and air at boundaries OC, AB and BC in Fig. 2 is K ¹ x "H (¹!¹ ), K ¹ y "H (¹!¹ ). (10) At the boundary OA, the condition is ¹ "0. (11) x The boundary condition between the mould and casting is K ¹ x "H (¹!¹ ), K ¹ y "H (¹!¹ ). (12) Eq. (7) with boundary conditions (10) (12) was solved by the finite difference method using the alternating direction implicit scheme. This scheme has the advantage of stability and the calculations are rapid since tridiagonal equations are obtained and a direct method is used to solve the discretizated algebraic equations. The heat source term is zero when the temperature is above the solidification temperature or the node is totally solidified. For intermediate temperatures the nodule count was calculated using the experimentally determined relationship between nodule count and local cooling rate and the latent heat release term was calculated as described above. Nonuniform meshes were used for the mould and casting to increase the accuracy. Total meshes were 10290 for mould and casting and a time step of 0.5 s was used. Further refinement of meshes and decrease of time step did not result in a significant change in the solution fields. Therefore, the calculations were considered to be independent of mesh size and time step. 3. Experimental results and discussion The composition of ductile iron used was 3.55% C, 2.51% Si, 0.556% Mn, 0.016% P, 0.012% S, 0.042% Mg, 0.311% Cu, 0.152% Mo. Keel blocks were cast with type K thermocouples located in four positions marked in Fig. 2. Cooling curves were recorded for each position using a multichannel data logger connected to a computer. Samples were taken from close to the four thermocouple positions and polished for microstructural observation. Image analysis was used to determine the nodule count; only graphite particles with diameters greater than 8 μm were counted.

J. Liu, R. Elliott / Journal of Crystal Growth 191 (1998) 261 267 265 Fig. 3. The relationship between nodule count and cooling rate. Stefanescu s relationship is presented by open circles and the present relationship by open squares. The cooling rates in the present study were slower than those in Stefanescu s study. The open circles in Fig. 3 are measurements reported by Stefanescu and the open squares are measurements made in this study. The best fit relationship used by Stefanescu was N"!65.150#409.07 log dt d¹. (13) Eq. (13) predicts a negative nodule count for lower cooling rates in the present study. Consequently, it cannot be considered as a universal relationship and used in the present analysis. The best fit relationship for the present iron over the cooling rate range studied is N"74.1#101.1 log dt d¹. (14) Numerical calculations started from the highest pouring temperature and terminated when all the nodes reached or were lower than 1060 C. Fig. 4 shows the calculated and experimental cooling curves and those calculated using the Stefanescu model for the four locations. Both numerically calculated curves show recalescence. This is the effect of the micro diffusion controlled kinetic growth model. The only difference in the two calculations is the graphite growth equation, other equations and system parameters were the same. Consequently, the curves for liquid and solid cooling and the total solidification time are the same. The total recalescence, that is, the temperature difference between the highest and lowest temperatures is the same. However, the predicted recalescence behaviour is different. The cooling curve is predicted to recalesce earlier and more rapidly with the Stenfanescu model although both models assumed instantaneous nucleation at the same temperature. This results from the prediction of a larger graphite radius and a higher growth rate for austenite at the early stages of solidification with the Stefanescu model. The difference in predicted solidification temperature means that the present model suggests a greater susceptibility to carbide formation. It is interesting to note that some intercellular carbide was observed in the solidified microstructure. Fig. 4 also shows that there is a reasonable agreement between the experimental cooling curves and those calculated using the present model. However, the shape of the cooling curve is determined by the heat extraction from the casting as well as the solidification kinetics. The accuracy of the different parameters in Table 1 and the use of a constant value for heat exchange coefficient at the casting/mould interface may not represent the real situation. The use of more accurate values for the parameters, of a heat exchange coefficient that varies with position and time taking into account the influence of mould filling and the running and gating system should lead to more accurate calculated cooling curves. Fig. 5 shows how the nodule count influences the calculated cooling curve. The same values were used for the different parameters in calculating the curves except for the nodule count relationship. For the light lines the relationship used was N"2 74.1#101.1 log d¹ dt. (15)

266 J. Liu, R. Elliott / Journal of Crystal Growth 191 (1998) 261 267 Fig. 4. Comparison of the cooling curves predicted by the two models and the experimentally measured cooling curves for the four thermocouple positions: ( ) present model calculations; ( ) Stefanescu model calculations; (...) experimentally measured cooling curve; () first temperature recorded experimentally. It can be seen that the model predicts that increasing the nodule count results in a lower undercooling on the cooling curve. This is in agreement with the predictions of heterogeneous nucleation theory and experiment. It also predicts that the melt is less susceptible to carbide formation. Fig. 6 shows how the solid fraction evolves with solidification time. It can be seen that the change in solid fraction is not constant. It is lower at the beginning and in the later stages of solidification. This is because at the beginning the radius of the eutectic cell is small, so the increase of solid fraction is small. In the last stages, the solid fraction is high, so the factor (1!f ) is small which results in a small solid fraction increase. The calculation of the growth rate of graphite and austenite and solid fraction evolution can be used to calculate the degree of solute segregation during solidification. This analysis is presented elsewhere [12].

J. Liu, R. Elliott / Journal of Crystal Growth 191 (1998) 261 267 267 1. the choice of solidification kinetic model influences the recalescence section of the cooling curve, 2. the present model predicts a later and slower recalescence during solidification, 3. the cooling curve calculated with the present model shows reasonable agreement with experimental cooling curves, 4. increasing ductile count is predicted to result in a lower undercooling during solidification. Acknowledgements Fig. 5. Cooling curves and solid fraction curves for the four thermocouple positions calculated with the present model. The authors would like to thank Prof. F.R. Sale for providing laboratory facilities and EPSRC for financial support. References Fig. 6. Cooling curves calculated with present model for four thermocouple positions and two nodule counts. 4. Conclusions Numerical calculations for the solidification of a ductile iron using a modified diffusion controlled solidification kinetic model show that [1] N.R. Eyers, I. Hartree, Phil. Trans. Royal Soc. Series A 240 (1946) 1. [2] M. Rappaz, D.M. Stefanescu, Metals Handbook, vol. 15, 9th ed., ASM International, Metals Park, OH, 1988, p. 833. [3] K.C. Su, I. Ohnaka, I. Yamauchi, T. Fukusako, in: H. Fredriksson, M. Hillert (Eds.), Physical Metallurgy of Cast Iron, North-Holland, New York, 1984, p. 168. [4] F. Edward, in: H. Fredriksson, M. Hillert (Eds.), Physical Metallurgy of Cast Iron, North-Holland, NY, 1984, p. 191. [5] J. Liu, F. Weinberg, Physical Metallurgy of Cast Iron IV, Material Research Society, 1990, p. 477. [6] D.M. Stefanescu, C.S. Kanetkar, Computer Simulation of Microstructural Evolution, in: D.J. Sroloritz (Ed.), TMS- AIME, 1985, p. 171. [7] D.M. Stefanescu, G. Upadhya, D. Bandyopadhyay, Metall. Mater. Trans. 21 A (1990) 997. [8] S. Chang, D. Shangguan, D.M. Stefanescu, AFS Trans. 99 (1991) 531. [9] S. Chang, D. Shangguan, D.M. Stefanescu, Metall. Mater. Trans. 23 A (1992) 1333. [10] W.A. Johnson, R.F. Mehl, Trans. AIME 135 (1939) 416. [11] M. Avirami, J. Chem. Phys. 7 (1939) 1103. [12] J. Liu, R. Elliott, Proc. 4th Decennial Int. Conf. on Solidification Processing, Sheffield, July 1997, p. 502. [13] P. Magnin, R. Trivedi, Acta Metall. Mater. 39 (1991) 453.