Numerical modeling of coupled heat transfer and phase transformation for solidification of the gray cast iron

Computational Materials Science 68 (2013) 160–165

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Numerical modeling of coupled heat transfer and phase transformation for solidification of the gray cast iron Masoud Jabbari a,⇑, Azin Hosseinzadeh b a b

Department of Mechanical Engineering, Technical University of Denmark, Nils Koppels All, 2800 Kgs. Lyngby, Denmark Department of Chemical Engineering, University of Tehran, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 11 September 2012 Received in revised form 22 October 2012 Accepted 30 October 2012 Available online 23 November 2012 Keywords: Numerical model Austenite Graphite Cementite Cooling rate Hardness

a b s t r a c t In the present study the numerical model in 2D is used to study the solidification bahavior of the gray cast iron. The conventional heat transfer is coupled with the proposed micro-model to predict the amount of different phases, i.e. total austenite (c) phase, graphite (G) and cementite (C), in gray cast iron based on the cooling rate (R). The results of phase amount are evaluated to find the proper correlation in respect to cooling rate. The semi-empirical formulas are proposed to find a good correlation between mechanical property (hardness Brinell) and different phase amounts and the cooling rate. Results show that the hardness of the gray cast iron decreases as the amount of graphite phase increases. It also increases by increased amount of the cementite and the cooling rate. These formulas were developed to correlate the phase fractions to hardness. Results are compared whit experimental data and show good agreement. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Cast iron (CI) remains the most important casting materials with over 70% of the total world tonnage [1]. Based on the shape of graphite, cast iron can be divided into the lamellar (flake) and spheroidal (nodular) graphite iron. In recent years, the numerical simulation and computer aided modeling of casting solidification are developed. Compared with the traditional experimental based design of casting [2], the numerical simulation has a great quantity of potential in increasing the productivity of metal casting industry by shortening time. The microstructure of CI is characterized by graphite lamellas, which disperse into the ferrous matrix. Foundry can influence the nucleation and growth of graphite flakes, and the size and type of graphite flakes enhance the desired properties. The amount of graphite and the size, morphology, and distribution of graphite lamellas are critical in determining the mechanical behavior of CI [3]. There is an extensive and increasing effort on the numerical simulation of different processes, including solidification, heating–cooling, and a variety of casting processes [4]. To determine the condition and optimum values, the simulation of solidification process is done by computer software or database programming.

⇑ Corresponding author. Tel.: +45 45254734; fax: +45 45930190. E-mail address: [email protected] (M. Jabbari). 0927-0256/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2012.10.034

The program output provides the details on determination of the high stress point, porosity distribution and microsegregation, cooling curves, flow rate, module mapping, and solidification process. Prediction of microstructure evolution in solidification is a key factor in controlling the solidification microstructures, properties, and quality of final casting products [5–7]. Many papers have been published on the micro-modeling of cast iron solidification [8,9]. However, most of these were in the case of ductile iron. Since the solidification of gray cast iron is affected by the different thermal condition and the graphite is in different shapes, modeling of microstructure evolution in this case is somehow complicated [9]. Moreover, the mechanical property (hardness) of the cast products is of great importance to predict. However, the hardness of the cast parts are difficult to predict, since different parameters have to be taken to account [10]. In the present study the heat transfer equations was coupled with the micro-model for the microstructure in the mid-section of a cast part (2D) to find the volume fraction of different phases. The equations were discretized in the Finite Difference (FD) fashion and programed in visual FORTRAN90. In the approached program, the Cp (heat capacity) effective method was used to calculate the latent heat. The volume fraction of total phase (the primary c and secondary c in the form of pearlite), graphite, and cementite were all calculated. Finally, this information was connected with the mechanical properties to develop a new equations between the microstructure and the mechanical properties.

M. Jabbari, A. Hosseinzadeh / Computational Materials Science 68 (2013) 160–165

2. Numerical modeling 2.1. Macroscopic heat transfer Most metallic materials are produced by melting and solidification, which are generally referred to the phase transformation or boundary migration. In solidification of different alloys, three important phenomena should be considered. First, most metals and alloys shrink in the solidification process, which causes the shrinkage defects. Second, the latent heat of fusion is released at the solid/liquid interface, which affects the solidification velocity and dependent microstructures. Third, the solute is rejected or absorbed in liquid at the solid/liquid interface during the alloys solidification, which causes the micro- and macro-segregation. The differential energy balance equation, involving the liberation of latent heat of freezing, is given by:

qcp

@T @fs ¼ r  ðk M TÞ þ qHf @t @t

ð1Þ

where qHf @f@ts is the energy generation term, q is the density, k is the thermal conductivity, t is time, Hf is the latent heat solidification, and fs is the volume fraction of solidified metal. By expanding the energy generation terms in Eq. (1):

qHf

@fs @fs @T ¼ qH f  @t @T @t

ð2Þ

By rearrangement Eq. (2) in Eq. (1), the general form of the heat conduction equation can be written as below:



q cp  Hf

 @fs @T ¼ r  ðk M TÞ @t @t

ð3Þ

To solve Eq. (3), the variation of solid volume fraction should be defined as a function of temperature. The relation between solid volume fraction and temperature may be evaluated from the phase diagrams, except for the congruent solidification in pure metals, eutectic reaction, or peritectic reaction. The amount of solid volume fraction between the liquidus (TL) to the solidus (TS) in the mushy zone varies 0 to 1. In a binary alloy system, the liquidus tem from    perature T L is a function of solute concentration in liquid C L , which depends upon the solute redistribution models [11] as the following equation:

T L ¼ T L þ m  C L

TL  T TL  TS

illustrated in Fig. 1. These two temperatures are calculated by the chemical composition of melt and cooling condition. In this work, for simplicity in calculations, it is assumed that these two temperatures are related to chemical composition, as shown in the following equation:

T st ; T mst ð CÞ ¼ A þ Bð%SiÞ þ Cð%MnÞ þ Dð%PÞ

ð6Þ

where A, B, and C are constants, %Si, %Mn and %P are the concentration percentage of the silicon, manganese and phosphorus in the melt, respectively. Since there are two different solidification manners, the macromodel equation is separated into two related parts, and the rearranged equation is shown as:

qL

        @fs @fs @fs @fs þ qG LG þ qW LW ¼ qA LA @t @t A @t G @t W

ð7Þ

where the subscript A, G and W are related to austenite, gray and white solidification, respectively. With these considerations, the amount of the liquid fraction is as follow:

fl ¼ 1  fA  fG  fW

ð8Þ

In the present study, the simple mod of Oldfield’s nucleation model is used, in which the nucleation sites are already present in melt or intentionally added to melt:

N ðm2 Þ ¼ A þ B  R

ð9Þ

where N is the amount of nucleation per unit face, R the cooling rate; and A and B the nucleation constants. For gray iron, A and B are 105 and 3.36  104, respectively [12]; and for the white one, it is assumed that A and B are 0 and 105, respectively. The calculation of the volume fraction is conducted using the following equation:

  @fs ¼ 4pr2 NlrT 2 ð1  fs Þ @t

ð10Þ

where r represents the actual grain radius, N the nucleation sites, and l the growth coefficient. The simple level rule is used to calculate the different phases. However, since there are two different types of solidification, the calculations for gray iron are obtained from the stable diagram and for the white one from the metastable diagram (see Fig. 1).

ð4Þ

where TL is the liquidus temperature for the pure metal, m the slope of the liquidus line in the phase diagram, and C L is the concentration in which the liquidus temperatures is aimed to be found. Because of simplicity, the linear distribution model was used in this work to define the variation of solid fraction, shown as the following equation:

fs ¼

161

ð5Þ

2.3. Mechanical prediction The final mechanical properties of the casting parts can be predicted by modeling of the room temperature microstructure. For example, the tensile strength of gray cast iron is affected by graphite flake length, which is related to carbon equivalent, cooling rate, alloying element, and fineness of pearlite. A brief summary of mechanical prediction based on the alloying elements and the

2.2. Microstructural model Dealing with micro-modeling, the nucleation and different transformations should be considered to predict the solidified phases. One of the most important parameters in determining the characteristics of microstructure evolution is the condition of nucleation in solidification. The solidifications of commercial alloys are generally considered as the heterogeneous nucleation on the mold surface and in the bulk. Other important parameter in the solidification of gray cast iron is the cooling rate, which is used as a critical parameter in this paper. The amount of the gray or the white iron is dependent on the isotherms stable temperature (Tst) and metastable temperature (Tmst), which are schematically

Fig. 1. Schematic illustration of the stable and metastable temperature in the solidification of the cast iron.

162

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Table 1 Nominal composition of the gray cast iron used in the present study.

Table 2 Geometrical details of each step in the designed cast part.

Element

C

Si

P

S

Mn

Cu

Amount (wt.%)

3.4–3.6

1.2–1.3

0.13–0.15

0.101

0.64

0.48

geometrical information of the cast part was presented by author elsewhere [10]. However, most of the presented works are for the nodular cast iron, and based on the chemical composition. Moreover, the cooling rate (R) which has the most important impact on the final properties of the cast parts, was not taken into consideration in the previous works [10]. 3. Experiments Gray iron specimens with the composition shown in Table 1 were cast at 1420 °C. To obtain different cooling rates, a step form sample designed which is illustrated in Fig. 2a. To calculate real cooling rates, five thermocouples embedded in the middle of spatial geometry of each step. These K-type thermocouples connected to data acquisition system (DAQ) in personal computer by an analog-to-digital (A/D) transformer (schematically showed in Fig. 2b). The cast body is molded in CO2 mold box. After casting, the bulk cut into metallographic samples. These samples are cut in places which thermocouples are embedded. Fig. 2a shows the location of samples in the body. All these samples then are analyzed by commercial image analyze software (aquinto) to obtain phase amounts, and also hardness Brinell test was conducted. Details of each step are summarized in Table 2. 4. Results and discussion 4.1. Thermal histories The first output of sample modeling is the thermal distribution and cooling rates. The thermal distribution of sample for the proposed model is illustrated in Fig. 3, and an example of the cooling curve predicted by the developed model with corresponding experimental one is shown in Fig. 4. It is clear that the middle of S1 step is the last point to solidify in comparison to other sections. Note that in the case of cooling rates, it is assumed that the results are for the geometric center of each section in the modeling; and also in the experiment, the tips of thermocouples are in the same place. The cooling rates results obtained from the modeling and experiment are as illustrated in Fig. 5. It was really a segment of thermal distribution and resultant cooling rates during the freezing

Step number

Label

Geometric module (cm)

Dimensions (L mm  W mm  H mm)

1 2 3 4 5

S1 S2 S3 S4 S5

1.5 1 0.75 0.5 0.25

150  83.5  55 150  52.5  24.2 150  51.5  15.8 150  51.5  10 150  48.5  5

stage from the liquidus temperature (Tst and Tmst) to the end of eutectic solidification, which was determined by the solidification procedure and related to the solidified microstructure of sample melt. 4.2. Phase distributions Fig. 6a shows the contour of graphite distribution in the sample. For the thick section with the high module, the volume fraction of graphite is the most. Because of the low cooling rate in this section, carbon has enough time to diffuse and forms the graphite. The contour of the volume fraction for the cementite phase distribution is also shown in Fig. 6b. Clearly, for the thin sections with the little geometric module, cementite content is higher. For these sections, the high cooling rates contribute with the high amount of unstable phase (here, the cementite phase). The volume fraction distribution of austenite phase is illustrated in Fig. 6c. The amount of the austenite for the thick section with the low cooling rate is more than the other sections. Note that in this paper, the amount of austenite was the sum of primary and secondary austenite. Because of this assumption, the volume fraction of austenite contained a fraction of pearlite. Primary austenite is decomposed during the eutectic transformation. When the melt passes the eutectoid line in the phase diagram, the secondary austenite starts to nucleate. Depending on the cooling manner and the cooling rate, the secondary austenite can be nucleated in the form of reneged austenite or pearlite (ferrite and cementite lamellas). In this paper, the proposed model calculated the amount of whole austenite. Therefore, the thick section had the high amount of austenite in the form of pearlite; for the thin sections, the amount of austenite contained the reneged austenite (ferrite). 4.3. Phases vs. the cooling rate The image analysis was carried out on the different metallographic pictures for each step at the tip of thermocouples to obtain

Fig. 2. (a) Designed sample for getting different cooling rates and (b) schematic illustration of A/D conversion and data acquisition system (DAQ).

M. Jabbari, A. Hosseinzadeh / Computational Materials Science 68 (2013) 160–165

163

Fig. 3. Contour of temperature distribution obtained from the numerical modeling.

1400

Temperature (.C)

Experiments Numerical

1200

1000

800

600 0

400

200

600

time (s) Fig. 4. Predicted cooling curve from numerical modeling and obtained experimental one for the S5 step (highest cooling rate).

Cooling rate (.C/s)

20

Experiments Numerical

15

10

5

0

0

0.5

1

1.5

2

Geometric Module (cm) Fig. 5. Comparison of cooling rates with different geometric module for numerical modeling and experiments. The lines are guide to the eye. Fig. 6. Results of numerical modeling for phase distribution of (a) graphite, (b) cementite and (c) austenite.

the volume fraction of each phase from the experimental, and comparing them with the corresponding numerical model. Results are illustrated in Fig. 7a–c graphically. The variation trend of modeling and experimental are the same in all these three phases, but there is a small difference between the results of the modeling and the experiments for the amount of the austenite phase, and these differences are the most in that of the graphite phase. This may happen since the image analysis tools are on the basis of two distinct color with the dark and bright, which stands for each phase.

Therefore, the results of the image analysis of the graphite phase not only contains the graphite itself but also the cementite (dark phase) in the matrix, which causes an overestimation. Consequently, the amount of graphite phase measured in experiment is higher than that of modeling, and the amount of cementite phase is lower.

M. Jabbari, A. Hosseinzadeh / Computational Materials Science 68 (2013) 160–165

15

(a)

Experiments Numerical

300

Hardness (HB)

Fraction of graphite (%)

164

10

5

0

0

5

10

15

250 200 150 10

.

fG (%wt)

(b)

50

0

Cooling rate (.C/s)

30 300 20 0

5

10

15

20

.

Cooling rate ( C/s) Fraction of austenite (%)

0

40

10

90

(c)

250 200 150 40 20

Experiments Numerical

80

20

fC (%wt)

10 0 0

Cooling rate (.C/s)

Fig. 10. Proposed surface for the influence of the cementite fraction and the cooling rate on the hardness.

70 60 50

0

5

10

15

20

.

Cooling rate ( C/s) Fig. 7. Comparison of the volume fraction obtained from the numerical modeling and experiment data for (a) graphite, (b) cementite and (c) austenite.

gray cast iron products (on the basis of cooling rate), the amount of graphite phase can be estimated by the hardness value, and vice versa. To approach a good correlation between the hardness and volume fraction of graphite, the hardness values are plotted based on the obtained fitted data from this study and the equation proposed by the author elsewhere [10]. This means that, the hardness of the gray cast iron will be dependent on the volume fraction of graphite and the cooling rate as follow:

HB ¼ A  f ðGÞ þ B  f ðRÞ

Experiments

220

Hardness (HB)

10

Fig. 9. Proposed surface for the influence of the graphite fraction and the cooling rate on the hardness.

Experiments Numerical

Hardness (HB)

Fraction of cementite (%)

Cooling rate ( C/s) 60

20 5

20

where A and B are constants, f(G) is a function of graphite phase which is also dependent on cooling rate, and f(R) is a function of cooling rate. The proposed equation for the correlation of the hardness with both the fraction of the graphite and the cooling rate is given below, and such surface is illustrated in Fig. 9.

200 180 160 140

HB ¼ 230:78  87:5  120

ð11Þ

2

4

6

8

10

Fraction of graphite (%) Fig. 8. Effect of graphite volume fraction on the hardness.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:0716 þ 0:07316  fG  0:0335  R

þ 0:0735  R2

ð12Þ

The same procedure was conducted to find a correlation between the hardness and the amount of cementite phase together with the cooling rate, which proposed as to be below, which is represented in Fig. 10.

4.4. Hardness vs. the phases The correlation between the hardness Brinell (HB) and the graphite phase obtained from experiments is shown in Fig. 8. Results showed that increasing the graphite fraction causes in the decrease of hardness. This is reasonable that the graphite phase acts as a hole in the metal matrix. Even by increasing the fraction more, the graphite flotation occurs, which may cause the start of surface cracks. The benefit of such diagram is that, in the same condition of

HB ¼ 108:492 þ 0:006  fC3  0:432  fC2 þ 10:5  fC  0:0402  R þ 0:0882  R2

ð13Þ

The hardness increases with the increase of cementite phase amount. This phenomenon can be expected by the increase of hard phase amount. It is known that the increased amount of carbide makes the material more brittle, but the hardness property is exactly related to the amount of hard phase. Besides, during the

M. Jabbari, A. Hosseinzadeh / Computational Materials Science 68 (2013) 160–165

Hardness (HB)

300

200

100 20 60 10

40 20

f (%wt) G

0

0

fC (%wt)

Fig. 11. Proposed surface for the influence of the cementite and graphite fractions on the hardness.

165

are in good agreement with the corresponding experimental data. However, the results of the hardness values predicted by Eq. (14) has the closer prediction to those of the experiments. The proposed formulas on the basis of numerical modeling shows a good agreement with the experiment data. This equations shows the effect of different phases, i.e. graphite and cementite volume fraction on the hardness. The inverse mode of this equation can be used for the estimation of volume fraction for each phase based on the hardness measurement. The proposed formulas can be easily used in the industrial range that was calculated by the authors in the previous work [10]. The good point is that, hardness measurement is the simplest test to carry out. Therefore, by using this kind of mechanical test, the amount of microstructural phases can be easily measured. 5. Conclusions

Hardness Brinell

240

[10] eq. (12)

220

eq. (13) eq. (14)

200

180

160

0

5

10

15

20

25

Cooling rate (.C/s) Fig. 12. Comparison of the proposed formulas in the present work with the corresponding experimental data from Ref. [10]. The lines are guide to the eye.

solidification, the amount of total austenite (primary c and pearlite) increases, which makes the material harder, since a mixture of iron carbide and pearlite is considerably harder than gray iron [10]. And finally the hardness of the cast iron was predicted by the new proposed equation based on the percentage of the graphite and cementite phases as the following equation, which is also shown in Fig. 11.

HB ¼ 160:2813  87:5 

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:0716 þ 0:07316  fG þ 0:006

 fC3  0:432  fC2 þ 10:5  fC

ð14Þ

To test the proposed formulas, the hardness of the cast part in the different steps are calculated by the Eqs. (12)–(14) and compared with those of the experimental data from Ref. [10], which is illustrated in Fig. 12. As it can be seen, the proposed formulas

Solidification behavior of the gray cast iron was modeled numerically. The nucleation and growth theory was used, and cooling rate was implemented as a key factor in the solidified phases. The amount of graphite, cementite, and total austenite modeled by approached FDM model on the basis of the cooling rate and level rule. As the cooling rate increases, the unstable phases increase. It is notified that increasing cooling rate results in the increase of cementite phase and the decrease of graphite and austenite phases. Hardness of gray cast iron decreases as the volume fraction of graphite goes up. Increasing the amount of cementite phase results in the improvement of the hardness of the gray cast iron. The proposed formulas based on the numerical modeling shows a good agreement with the experimental data. This equations shows the effect of different phases, i.e. graphite and cementite volume fraction on the hardness. References [1] D.M. Stefanescu, Mater. Sci. Eng. A 413 (2005) 322–333. [2] M.A. Gafur, M.N. Haque, K.N. Prabhu, J. Mater. Process. Technol. 133 (2003) 257–265. [3] L. Collini, G. Nicoletto, R. Konen, Mater. Sci. Eng. A 488 (2008) 529–539. [4] T.R. Vijayaram, S. Sulaiman, A.M.S. Hamouda, M.H.M. Ahmad, J. Mater. Process. Technol. 178 (2006) 29–33. [5] B.Q. LI, P.N. Anyalebechi, Int. J. Heat Mass Transfer 38 (1995) 2367–2381. [6] A. Shayesteh-Zeraati, H. Naser-Zoshki, A.R. Kiani-Rashid, J. Alloys Compd. 500 (2010) 129–133. [7] R. Venkataramani, R. Simpson, C. Ravindran, Mater. Charact. 35 (1995) 175– 194. [8] K.M. Pederson, J.H. Hattel, N. Tiedje, Acta Mater. 54 (2006) 5103–5114. [9] D. Maijer, S.L. Cockroft, W. Patt, Metall. Mater. Trans. A 30 (1999) 2147–2158. [10] M.M. Jabbari Behnam, P. Davami, N. Varahram, Mater. Sci. Eng. A 528 (2010) 583–588. [11] Grong, H.R. Shercliff, Prog. Mater. Sci. 47 (2002) 163–282. [12] H. Fredriksson, M. Hillert, The physical metallurgy of cast iron, in: Proceedings of the Third International Symposium on the Physical Metallurgy of Cast Iron, Stockholm, Sweden, 1984, p. 273.