Computer modeling of ductile iron solidification for

thin wall casting

W. Kapturkiewicz, A.A. Burbelko

AGH University of Science and Technology

Krakow, Poland

e-mail: kaptur@agh.edu.pl

Abstract — A solidification model for ductile iron, using a generic

growth law for austenite and eutectic grains (without calculation

of carbon diffusion through the austenite shell) and including

Weibull formula for nodule count has been presented. From this

model, the following can be determined: cooling curves, kinetics

of austenite and eutectic nucleation, austenite and eutectic

growth velocity, volume fraction, distribution of Si and P in both

austenite and eutectic grains. In the paper only some of these

data were presented. The model has been well developed for thin-

walled castings, where the undercooling below the equilibrium

temperature is very high and the solidification process is very

fast. The correctness of the mathematical model has been expe-

rimentally verified in the range of the most significant factors,

which include cooling curves, the value of maximum undercool-

ing, and the graphite nodule count interrelated with the casting

cross-section. Literature offers practically no data on so con-

fronted process model and simulation program.

Keywords: modeling, ductile iron, solidification process

I. INTRODUCTION

Nodular graphite cast iron, also known as ductile cast iron, ductile iron, nodular iron, spheroidal graphite iron (SGI), has major applications in critical engineering parts due to its ex-cellent properties and castablility. The mechanical, physical and utilization properties of this cast iron depend on the num-ber of the graphite grains and on the individual matrix constitu-ents.

Most of the computer modeling programs described in lite-rature are devoted to eutectic transformation [1-6] under the pre-assumed stationary conditions of carbon diffusion in auste-nite. In [5] a physical model of solidification of the nodular graphite cast iron which quantitatively accounts for the forma-tion of non-eutectic austenite during cooling and solidification of hypereutectic as well as hypoeutectic cast iron has been presented. In investigation [6], process modeling techniques have been applied to describe the multiple phase changes oc-curring during solidification and subsequent cooling of near-eutectic nodular graphite cast iron, based on the internal state variable approach. The diffusion model of graphitization in nodular cast iron casting has been presented in [7].

According to [8] at the eutectic temperature, austenite den-drites and graphite spheroids nucleate independently in the liquid. This mechanism has been confirmed by both [9, 10] for hypo-eutectic and eutectic, as well as hyper-eutectic SGI.

Concluding from that, it has been introduced by [11] for-mulae uninodular models assume a basic unit of solidification formed by a graphite nodule and austenite shell covering the nodule, and multinodular ones assume that each unit of solidi-fication is formed by a grain of dendritic austenite containing several graphite spheres. Celentano et al. [11] has presented model, in which velocity of nucleation depends on current liquid fraction.

Recently, a tendency for production of thin-walled castings has been observed [12-14]. In this technology, the process of solidification is very far from equilibrium conditions, the soli-dification is fast and the undercooling is very high. Hence, there are some problems with simulation of a typical diffusion model.

The aim of this study was to develop a model of SGI solidi-fication, using knowledge available so far, confronted with an experiment in respect of both the cooling curves as well as the grain distribution and graphite nodule count in real casting.

II. MODEL OF PROCESS

The model combines a macro model (heat transfer in cast-ing) with micro model (nucleation and growth of grains). Heat transfer in casting depends on the cooling conditions created by foundry mould.

According to the analysis of reference literature presented above, it has been assumed that, irrespective of the fact that molten metal may have the chemical composition correspond-ing to a eutectic one (eutectic saturation coefficient SC ≈ 1.0), it is possible that austenite dendrites and graphite spheroids will nucleate independently in the liquid. The mechanism of the diffusion growth of nodular graphite has been disregarded, assuming that the leading factor in the process of the grain growth is the kinetic undercooling at an austenite-liquid phase boundary, including the growth of both eutectic grains and austenite dendrites (for which an approximate spherical growth with amendment in Kolmogorov equation [15] has been

adapted). It has also been assumed that the eutectic nodule count is equivalent to graphite nodule count.

A. Heat transfer

The macro temperature field in casting-mold system is:

v

s

c

qTa

T 2

where T, τ – temperature and time, a – thermal diffusivity, qs – heat generation rate of phase transformations, cv – volumetric specific heat.

B. Volume fraction

In order to calculate the true volume fraction of solid (fS), one must include the effect of grain impingement. This value can be described by Kolmogorov equation [15]:

efS 1

t t

t

tdduts0

3

3

4

where - so-called "extended" volume of all solid grains, t' -

nucleation time, (t') - rate of the grain nucleation, u( ) – linear velocity of the growth (inner integral – grain radius), s –shape coefficient, e.g. s=1 for globular grains and s=0.3 for dendrites.

C. Nucleation

It is well known that liquid cast iron contains undissolved particles of various sizes. Hence, upon alloy undercooling beyond a critical value, these particles exceed the minimum sizes needed for stable growth. Hence, growing nuclei are con-tinually developed until the time when the metal attains its maximum level of undercooling. Afterward, with the progress of recalescence, no new nuclei form because all the particles larger than the critical size (which corresponds to maximum undercooling) were already exhausted. Activation of smaller particle substrates as active nuclei will require undercooling, which will have to exceed the maximum value. To compute the density of the formed austenite nuclei the following relation-ship has been adapted [16]:

2TN

where Ψγ – nucleation coefficient of austenite grains, ΔTγ – undercooling with the reference to equilibrium austenite tem-perature.

For number of graphite nodule count the Fraś [17] equation has been used:

ese TbNN exp

where Ns – overall nucleation site density in the melt, b – nu-cleation coefficient, ΔTe – undercooling with the reference to equilibrium eutectic temperature.

D. Growth of grains

For the austenite linear growth velocity the classic law [19] is used:

5.2Tu

where μγ – austenite growth coefficient.

Rate of growth for eutectic grains:

2

eee Tu

where e – eutectic growth coefficient.

E. Equilibrium temperature and segregation

The equilibrium austenite liquidus line T and eutectic tem-perature Te can be represented by linear functions of carbon, silicon and phosphorus concentration in liquid cast iron [20]:

LLL PSiCT 5.025.01131636

LLe PSiT 88.1425.51154

where CL, SiL, PL – weight percent of C, Si and P in liquid.

The solute concentration in the solidifying phases is strongly influenced by the magnitude of the diffusion coeffi-cients. Hence, for solute of relatively high diffusivity (e.g. carbon in austenite), the solute concentration in the liquid phase can be approximated by the mass balance. Alternatively, the Scheil equation can be used in dealing with low diffusivity solutes, such as in the case of silicon or phosphorus in aus-tenite.

A set of the above equations, after transformation to a diffe-rential form, was solved by the finite difference method, ap-plying an iteration procedure (secant method). The simulation program operating in Delphi environment was prepared in 1D and 3D systems.

The verification of the developed model was confronted with the results of an experiment which in more detail was described in [21].

The parameters adapted in modeling are given below and in Table 1 and 2. The first five parameters in Tab. 2 concern the nucleation and growth, the next ones – thermal conductivity, specific heat and density for casting and mould material.

F. Parameters of experiment and for modeling

TABLE I. CHEMICAL COMPOSITION OF CAST IRON, WT% [21]

Melt No. C Si P SC Std. devia-

tion of SC

1 3.62 2.68 0.020 1.0496 0.002

2 3.73 2.57 0.013 1.0705 0.006

3 3.62 2.65 0.014 1.0463 0.004

Mean Val. 3.66 2.63 0.016 1.055

TABLE II. PARAMETERS FOR MODELING

Property Value Units

Ψγ 5x106 cm-3 K-2

μγ 5x10-5 cm s-1 K-2

Ns 5x107 cm-3

b 50 K

μe 1x10-7 cm s-1 K-2

λc 0.37 W cm-1 K-1

cc 0.753 J cm-3 K-1

ρc 7.3 g cm-3

Lγ 1952.4 J cm-3

Le 2028.8 J cm-3

λm 0.0103 W cm-1 K-1

cm 1.09 J cm-3 K-1

ρm 1.73 g cm-3

III. DISCUSSION OF RESULTS

Some attention deserves the fact that studies in both va-riants, i.e. simulation and experiment, were carried out on the Fe-C-Si-P alloy of practically eutectic composition (SC=1.05) which, considering an equilibrium course of the solidification process, allows us to expect total absence of the austenite in structure. Yet, modeling carried out for a non-equilibrium system has revealed an important share of austenite (disclosed in further drawings). Another proof is the result of modeling excluding the possibility of austenite nucleation – Fig. 1, where the plotted cooling curve is basically different from both the experimental curve (dots) and simulation curve (solid line). The said drawing, displaying the results obtained on a 10 mm thick cast plate, also shows the characteristic point A, which re-flects the thermal effect caused by the growing dendrites of austenite. Modeling for different casting wall thicknesses indicates large variations in the cooling curves (Fig. 2), especially as regards the thin-walled castings.

The final number of eutectic grains depends on the value of maximum undercooling which, in turn, depends on process pa-rameters. For the same melt, the same pouring temperature, casting configuration and mould material, it depends on the casting thickness only, as shown in Fig. 3 and 4. With increas-ing casting thickness, the value of the maximum undercooling is decreasing, analogically to the number of eutectic grains (in nodular graphite cast iron associated rather with the count of graphite nodules). The values obtained by modeling (solid line) have been compared with the values measured in experiments. Besides comparison of the cooling curves, this is the most significant criterion to evaluate the correctness of a mathemati-

cal model of the process and of the developed simulation pro-gram.

1100

1120

1140

1160

1180

1200

0 10 20 30 40 50

Te

mp

era

ture

, 0C

Time, s

Plate 10 mm

Modeling without Austenite

Modeling with Austenite

Experiment

A

Figure 1. Cooling curves plotted for an interior part of the 10 mm plate.

Modelling with excluding and including the possibility of austenite nucleation.

1000

1050

1100

1150

1200

0 10 20 30 40

Tem

pera

ture

, 0C

Time, s

D = 0.6 cm

D = 0.3 cm

D = 0.15 cm

Figure 2. Cooling curves for three thickness of casting wall.

0

25

50

75

100

125

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Un

de

rco

olin

g, K

Plate Thickness, cm

Figure 3. Maximum undercooling versus casting plate thickness. Line –

modelling; points – experiment.

0.0E+00

1.0E+07

2.0E+07

3.0E+07

4.0E+07

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Num

ber o

f Eut

ectic

Gra

ins,

1/c

m3

Plate Thickness, cm

Figure 4. Number of eutectic grains (nodule count) versus casting plate

thickness. Line – modeling; points – experiment.

Some characteristic points on the cooling curves obtained from modeling were checked, comparing them with the expe-rimental values obtained by Pedersen et al. (ductile iron 3.7% C and 2.7% Si) – Fig. 5.

1080

1100

1120

1140

1160

0.1 0.3 0.5 0.7 0.9

Tem

pera

ture

, 0C

Plate Thickness, cm

Figure 5. The values of minimum temperature during solidification

compared for own modeling (solid line) and data obtained in experiments

(dotted line) according to [22].

One of the main threats in the process of making thin-walled castings from ductile iron is the problem of carbide formation. From the results of this modeling it follows (Fig. 6) that for the preset mold and metal parameters, the problem of carbide formation occurs in casting walls less than 6 mm thick - a drop in the cooling curve below the eutectic metastable temperature Tm is observed to take place.

1080

1100

1120

1140

1160

1180

0 20 40 60 80

Tem

pera

ture

, 0C

Time, s

D = 2.0 cm

D = 0.3 cm

D = 0.6 cm Tm

D = 1.2cm

Figure 6. Cooling curves for four casting thickness compared with the

eutectic metastable temperature.

IV. CONCLUSIONS

In the developed model of nodular graphite iron casting so-lidification, the correctness of the mathematical model has been experimentally verified in the range of the most significant fac-tors, which include temperature field, the value of maximum undercooling, and the graphite nodule count interrelated with the casting cross-section. Literature offers practically no data on so confronted process model and simulation program.

It has been showed, that in eutectic cast iron (SC =1) the nu-cleation and growth of austenite grains are of great importance. The cooling curves of modeling with excluding and including the possibility of austenite nucleation are quite different and the experimental curve is close to the case with austenite.

The model is well worked for thin walled ductile iron cast-ing, when the undercooling below the equilibrium temperature is very high and the solidification process is very fast.

ACKNOWLEDGEMENTS

This work was supported by NCN project No. N N508 621140.

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