f solidification for aluminum

Section I: Basic and Applied Research

Numerical Simulation of Solidification for Aluminum-Base Multicomponent Alloy

Kenichi Ohsasa

(Submitted 29 November 2000)

Solidification simulation of an aluminum-base multicomponent alloy was carried out by a methodcombining thermodynamic analysis using Thermo-Calc and heat-transfer calculation. An Al-9.5% Si-3% Cu-1% Mg-0.8% Fe (all mass%) aluminum-base multicomponent alloy was usedfor the simulation. The effect of latent heat on the heat-transfer calculation was considered byusing an enthalpy method. The temperature-enthalpy curves for both an equilibrium state andnonequilibrium state with assumptions of no diffusion in the solid were calculated by usingThermo-Calc. A small casting with a cylindrical shape was used for the heat-transfer simulation.The vertical cross section of the casting was divided into rectangular grids, and the enthalpychange of each grid was numerically calculated. The calculated enthalpies in the grids wereconverted for each time-step into temperatures by using the temperature-enthalpy curve. Acasting experiment was carried out under the same conditions as those of the simulation, andthe calculated cooling curves obtained under the nonequilibrium condition agreed with the experi-mental ones.

geous for multicomponent alloys because the features of an1. Introductionalloy, such as specific heat, liquidus, and solidus tempera-tures, change in fraction solid during solidification, and evo-Numerical simulation of the solidification process of alution of latent heat are all totally involved in the temperature-

casting based on heat-transfer analysis is commonly carried enthalpy relationship of the alloy.out prior to the real casting operation. Prediction of the tem- In this study, thermodynamic analysis using ThermoCalc[9]

perature field in the solidifying casting enables foundry engi- was applied to calculate the temperature-enthalpy relation-neers to design an appropriate shape of casting and optimum ship of an aluminum-base multicomponent alloy. Solidifica-pouring procedure so as to avoid the formation of casting tion simulation of a small casting based on the enthalpydefects such as shrinkage cavities. In order to simulate the method was carried out using the calculated temperature-temperature field in a solidifying alloy casting, alloy proper- enthalpy curve. The validity of the calculated temperature-ties such as liquidus and solidus temperatures, solidification enthalpy curve was examined by comparing calculated cool-path, and latent heat are required in addition to thermal prop- ing curves in the casting with experimental results.erties such as thermal conductivity, density, and specific heat.In binary alloy systems, liquidus and solidus temperaturescan be obtained from the published phase diagram; however, 2. Methodmost commercial alloys are multicomponent systems, anddetermination of accurate alloy properties is difficult. Further-more, there are few published data on the solidification path 2.1 Analysisand latent heat of commercial multicomponent alloys.

Multicomponent Alloy. An aluminum-base multi-Several techniques are used to incorporate the latent heatcomponent alloy that is used for the castings of engine partsevolution into the heat-transfer calculation of a solidifyingin the car industry was used for the simulation. Nominalalloy: (a) temperature recovery method,[1–3] (b) equivalentcomposition of the alloy is shown in Table 1. The alloyspecific heat method,[3–5] and (c) enthalpy method.[3,6–8]

contains many elements, and silicon, copper, and magnesiumAmong these techniques, the enthalpy method is advanta-are added as alloying elements, whereas other elements aremixed as impurities. Among these impurity elements, ironhas a relatively higher concentration of 0.8 mass% and mayKenichi Ohsasa, Division of Molecular Chemistry, Graduate Schoolhave a non-negligible effect. Therefore, an alloy having theof Engineering, Hokkaido University, N13, W8, Kita-ku, Sapporo, Hok-

kaido 060-8628, Japan. Contact e-mail: [email protected]. composition shown in Table 2 was used in the analysis.

Table 1 Nominal composition of the multicomponent alloy (mass%)

Si Cu Mg Fe Zn Mn Ni Ti Pb Cr Al

8.5–10.5 2.0–4.0 0.6–1.5 ,0.80 ,0.50 ,0.50 ,0.50 ,0.20 ,0.10 ,0.10 Bal

Journal of Phase Equilibria Vol. 22 No. 4 2001498

Basic and Applied Research: Section I

Table 3 Thermal properties used in the simulation

Property Value

hI (W/(m2 ? K) 6000[3]

hS (W/(m2 ? K) 42[3]

« 0.80lMold (W/(m ? K)) 75.3-0.073T-0.000021T 2[10]

lAlloy (W/(m ? K)) 193.0[10]

rMold (mg/m3) 7.8[10]

rAlloy (mg/m3) 2.7[10]

s (W/(m2 ? K4) 5.670 3 1028

2f EAlloyTz

5 hS (TS 2 TAtom) 1 «s (T 4S 2 T 4

Atom) (at z 5 Z )

(Eq 6)

Fig. 1 Shape of the casting and finite difference grid used forwhere hI is the heat-transfer coefficient at the casting-moldnumerical simulationinterface (W/(m2 ? s)); TAl and TM are the temperatures ofthe alloy and mold at the interface, respectively; rI is the

Table 2 Chemical composition of the inner radius of the mold (m); ZC is the position of the bottommulticomponent alloy used for the simulation of the mold (m); hS is the heat-transfer coefficient at the(mass%) surface of the mold (W/(m2 ? s)); TOM is the temperature

of the outer surface of the mold (K); TAtom is the ambientSi Cu Mg Fe Altemperature (K); « is the heat emissivity of the mold; s is

9.5 3.0 1.0 0.8 Bal the Stefan-Boltzmann constant (W/(m2 K4)); R is the outerradius of the mold; TS is the temperature of the top surfaceof the casting; and Z is the position of the top surface (m).

Heat-Transfer Calculation Based on the Enthalpy The initial condition is T 5 900 K at t 5 0.Method. A casting with a cylindrical shape of 35 mm in In order to solve Eq 1 to 6 numerically by using a finitediameter and 40 mm in height was used for the simulation. difference method, the longitudinal cross section of the cast-Figure 1 shows the shape of the casting and the mold used ing and mold was divided into rectangular grids with 0.25for the analysis. The thickness of the sidewall and the bottom mm on the r and z coordinates. Since the shape of the castingof the mold is 5 mm. The basic heat conduction equation in is axial-symmetric, a half of the cross section was divided,enthalpy form is expressed as follows: as shown in Fig. 1. The finite difference form of Eq 1 is

expressed as follows:Ht

5lr F2T

r 2 11r

Tr

12Tz2G (Eq 1)

H t1Dt 5 H t 1l ? DtrDx 2 FT t

i1l, j 1 2T ti, j 1 T t

i2l, j

where H is enthalpy (J/mol), t is time (s), l is thermal conduc-tivity (W/(m?s)), r is density (Kg/m3), T is temperature (K),

11R 1T t

i1l, j 2 T ti2l, j

2 2G 1l ? DtrD2

z(T t

i, j11 1 2T ti, jr is the radial-coordinate (m), and z is the vertical-coordi-

nate (m).The mathematical expressions of the boundary condi- 1 T t

i, j21) (Eq 7)tions are

The values of the thermal properties used in the calculationare given in Table 3.T

r5 0 (at r 5 0) (Eq 2)

Figure 2 shows a schematic illustration of the temperature-enthalpy curve of an alloy. The change in enthalpy of gridsat each time-step was numerically calculated, and the new

2lAlloyTr

5 2lMoldTr

5 hI (TAl 2 TM) (at r 5 rI) (Eq 3)enthalpy value H t1Dt was converted into temperature by usingthe temperature-enthalpy relationship.

Temperature-Enthalpy Curves. Temperature-enthalpy2f EAlloy

Tz

5 2lMoldTz

5 hI (TAl 2 TM) (at z 5 ZC) (Eq 4) curves of the multicomponent alloy for an equilibrium condi-tion and nonequilibrium condition were calculated by usingThermo-Calc. In the present paper, the term “nonequilibrium

2f EMoldTr

5 hS (TOM 2 TAtom) 1 «s (T 4OM 2 T 4

Atom) condition” is used for the Scheil model.[11] The Sheil modelis based on the following assumptions: (1) no diffusion insolid, (2) complete mixing in liquid, and (3) local equilibrium(at r 5 R) (Eq 5)

Journal of Phase Equilibria Vol. 22 No. 4 2001 499


Fig. 2 Schematic illustration of the temperature-enthalpy curve ofan alloy

Fig. 3 Temperature-enthalpy curve of the aluminum-base multi-at the solid-liquid interface. On the other hand, the word term component alloy under the equilibrium condition“equilibrium condition” is used for the following conditions:(1) complete diffusion in solid, (2) complete mixing in liquid,

2.2 Experimentand (3) equilibrium in the entire solid and liquid regions.The procedure for calculating the change in the fraction A steel mold with a cylindrical shape of 35 mm in diame-

of each phase under the nonequilibrium condition by using ter, 40 mm in height, and 5 mm in thickness was used forThermo-Calc is as follows. Temperature is lowered by a the casting experiment. The aluminum-base multicomponentsmall degree of DT from the starting temperature. Change alloy, the composition of which is shown in Table 1, wasin the fraction solid, Dfs, and changes in the concentration melted in an electric furnace and poured into the mold. Theof each element in the liquid, DCL(i), are calculated by consid- pouring temperature was 900 K, and the casting was carriedering that those changes occur in an equilibrium manner. out in air. Temperatures at a half of the height of the castingBased on the assumption of no solid diffusion, the solidified were measured by alumel-chromel thermocouples. In orderpart is no longer concerned with the solidification process to increase the accuracy of the temperature measurement inand is abandoned, and a new liquid composition is adopted the casting, thermocouples of 0.3 mm f and 1 mm in beadas the initial composition. This procedure is repeated until diameter were directly set in the mold cavity without anythe sum of fS reaches unity. From a mathematical point of protective tubes.view, if the temperature interval T approaches 0, the errorarising from temperature discretization decreases. However,from the point of view of numerical calculation, a smaller 3. Results and Discussiontemperature interval increases the number of numerical calcu-lations and results in an increase in “round-off error.” Hence,the estimation of an appropriate temperature interval is 3.1 Simulation under the Equilibrium Conditionimportant. Based on the results of preliminary calculation,[12]

a temperature interval of 1 K was used for the present Figure 3 shows the calculated temperature-enthalpy curveScheil simulation. of the aluminum-base multicomponent alloy under the equi-

Enthalpy of the multicomponent alloy at each temperature librium condition. Several changes in the slope of the curvewas calculated by proportioning the enthalpies of phases in the solid-liquid coexisting region (mushy zone) correspondbased on the fraction of each phase calculated by the proce- to the formation of new phases from liquid.dure described above. Figure 4 shows simulated cooling curves obtained by using

the temperature-enthalpy curve under the equilibrium condi-tion shown in Fig. 3. Solidification starts at 862 K, andH(T ) 5 o

n

iHi (T ) ? fsi (T ) (Eq 8)

several changes, corresponding to the crystallization of newphases, appear in the cooling curve, and the end of solidifica-tion can be clearly seen from the abrupt increase in coolingwhere n is the number of phases existing at T (K), Hi (T ) is

the enthalpy of phase i at T (K), and fsi is the fraction solid rate at 794 K. It is also seen that the temperature gradientin the radial direction in the casting is quite small becauseof phase i at T (K). The SGTE solution database was used

in the Scheil simulation and the calculation of the enthalpy the thermal conductivity of an aluminum-base alloy is rela-tively high.of each phase.



Fig. 4 Simulated cooling curves in the casting under the equilib- Fig. 6 Solidification sequence of the multicomponent alloy calcu-rium condition. The vertical position of the cooling curves is the lated by Scheil simulationhalf height of the casting, and the radial positions are (a) center(center of the casting), (b) middle (middle position between the

at about 850 K, and the cooling curves are similar to thecenter and the surface of the casting), and (c) surface (at the surfaceof the casting) calculated ones in the early stage of solidification. However,

solidification was completed at about 775 K with a tempera-ture arrest due to an invariant reaction. The occurrence ofthe invariant reaction at the end of the solidification differsfrom the calculated cooling curves shown in Fig. 4. Theoccurrence of the invariant reaction is ascribed to limiteddiffusion in solid. An equilibrium solidification path is neverachieved under conventional casting conditions with a finitecooling rate.

3.3 Simulation under the Nonequilibrium Condition

Figure 6 shows the solidification sequence of the alloycalculated by Scheil simulation with Thermo-Calc, i.e., underthe condition of no diffusion in solid. The sequence of thecrystallization of phases during solidification is summarizedas follows:

• 862 K primary: Al-rich solid solution;• 837 K secondary: Si;• 835 K third: Al13Fe4;• 818 K fourth: Mg2Si; and• 777 K fifth: Al2Cu.

Fig. 5 Experimentally measured cooling curves in the casting. Thevertical position of the cooling curves is the half-height of the

Solidification of the alloy starts at 862 K, followed by crystal-casting, and the radial positions are (a) center (center of the casting)lization of Si, Al13Fe14, and Mg2Si, and is completed at 777and (b) surface (at the surface of the casting)K by the invariant eutectic reaction, L → Al 1 Si 1 Al13Fe14

1 Mg2Si 1 Al2Cu. The predicted sequence of the crystalliza-3.2 Experimental Resultstion described above was confirmed by the results of aquenching experiment.[12]Figure 5 shows experimentally measured cooling curves

at the locations of the center and surface of the casting. It The temperature-enthalpy curve of the alloy with a non-equilibrium state was calculated from Eq 8 and is shown incan be seen from the cooling curves that solidification started



The adopted thermal properties, especially the heat-transfercoefficient at the casting/mold interface, may not be valid toapply the simulation. However, this is another subject to beinvestigated and is not the main subject of the present study.The changes in slope corresponding to crystallization of eachphase and the temperature arrest due to the invariant reactionare reproduced in the simulated cooling curves. It can beconcluded that the application of thermodynamic analysisis effective for solidification simulation of aluminum-basemulticomponent alloys.

3.4 Limited Diffusion in Solid

Under real solidification conditions of industrial castingswith a finite cooling rate, diffusion in solid is limited. Thetemperature-enthalpy relationship under limited diffusionconditions should be located between those of the extremeconditions of equilibrium and the Scheil model. In order tosimulate the solidification process under limited diffusionconditions, the fraction solid of the primary Al-rich solidsolution should be modified as follows:

Fig. 7 Temperature-enthalpy curve of the aluminum-base multi-component alloy under the nonequilibrium condition

f 8s 5 f ns 1

a1 1 a

( f es 2 f n

s) (Eq 9)

Here, f es is the fraction solid under the equilibrium

condition, f ns is the fraction solid under the nonequilibrium

condition (Scheil model), and a is a parameter expressingthe degree of diffusion in the solid[13] and is defined as a 5(DS ? uf)/l2, where DS is the diffusion coefficient of an elementin the solid (m2/s), uf is local solidification time (s), and l isa half of the secondary arm spacing (m). In Eq 9, f 8sapproaches f n

s when a becomes smaller, and f 8s approachesf e

s when a becomes larger. This simple method does not havea strict theoretical background but may be useful as an engi-neering approximation.

4. Conclusions

A method was proposed for simulating the solidificationprocess of an aluminum-base multicomponent alloy usingthe apparent temperature-enthalpy curve under the conditionof no diffusion in solid. Thermo-Calc was used to calculatethe enthalpy of the alloy, and the temperature-enthalpy curvewas obtained by proportioning the enthalpies of phases basedFig. 8 Simulated cooling curves in the casting under the nonequi-on the fraction of each phase calculated by Scheil simulation.librium condition. The vertical position of the cooling curves is the

half-height of the casting, and the radial positions are (a) center Solidification simulation of a small casting based on the(center of the casting), (b) middle (middle position between the enthalpy method was carried out, and the calculated coolingcenter and the surface of the casting), and (c) surface (at the surface curves in the casting agreed with the experimental ones.of the casting)

ReferencesFig. 7. Enthalpy of the alloy discontinuously decreases at

1. E.A. Mizikar: Trans. AIME, 1967, vol. 239, p. 1747.777 K due to the invariant eutectic reaction. 2. I. Ohnaka and T. Fukusako: Trans. Iron Steel Inst. Jpn., 1977,Figure 8 shows simulated cooling curves obtained from the vol. 17, p. 410.

temperature-enthalpy curve shown in Fig. 7. The simulated 3. I. Ohnaka: Introduction of Computer Analysis for Heat Transfercooling curves are not exactly the same as the measured and Solidification, Maruzen, Tokyo, 1985, p. 167.ones, because accuracy of the solidification simulation also 4. R.A. Roberts, R.P. Date, C.R. Loper, and D.R. Poirier: Trans.

AFS, 1968, vol. 76, p. 573.depends on the thermal properties used in the simulation.



5. C.R. Loper, R.W. Heine, and R.A. Roberts: Trans. AFS, 1969, 10. S. Nagasaki: Data Book of Metals, Japan Institute of Metals,Maruzen, Tokyo, 1993, p. 1.vol. 76, p. 373.

6. W. Koppe and S. Engler: Giesserei, 1962, vol. 49, p. 265. 11. E. Scheil: Z. Metallkd., 1942, vol. 34, p. 70.12. K. Ohsasa, M. Shoji, and T. Narita: J. Jpn. Foundry Eng. Soc.,7. W. Heine: AFS Trans., 1965, vol. 73, p. 34.

8. R.A. Johns: AFS Trans., 1980, vol. 88, p. 77. 2000, vol. 72, p. 525.13. H.D. Brody and M.C. Flemings: Trans. TMS-AIME, 1966, vol.9. B. Sundman, B. Jannson, and J.-O. Andersson: CALPHAD,

1985, vol. 9, p. 261. 236, p. 615


f solidification for aluminum

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