Macro Micro Modeling of Solidification

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Proc. Natl. Sci. Counc. ROC(A)

Vol. 23, No. 5, 1999. pp. 622-629

Macro-Micro Modeling of Solidification

LONG-SUN CHAO AND WU-CHANG DU

Department of Engineering Science

National Cheng Kung University

Tainan, Taiwan, R.O.C.

(Received November 3, 1998; Accepted March 31, 1999)

ABSTRACT

To solve solidification problems, macro-models are generally applied. Macro-micro models, by

considering the formation of microstructures in terms of nucleation and growth, can obtain better results

than can macro-models. Except for temperature distributions, macro-micro models can offer more in-

formation about the solidification process, including undercooling, grain size, grain density etc. Thesedata can be used to predict the mechanical properties of materials directly. In this paper, two macro-

micro models are built to investigate the equiaxed solidification of eutectic. One macro-micro model

considers the nucleation step, and the other does not (assuming that nucleation occurs instantaneously).

An experimental example of a cylindrical casting is used to test these models. The finite difference method

is utilized to solve the heat transfer problem, and the source term method is employed to handle the released

latent heat. From a comparison with the experimental result, the computed cooling curve is found to be

very close to the experimental one. From the computational results, it is found that a higher cooling rate

yields the greater undercooling, which leads to a smaller grain size or higher grain density.

Key Words: macro-micro modeling, solidification, nucleation, cylindrical casting

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I. Introduction

For binary alloys, macro-models based on phase

diagrams are generally used to solve solidification

problems. These models can give only rough predic-

tions of the solidification time, isotherms etc., which

do not have any direct relation with the microstructures

and physical properties of solidified alloys.

Recently, the micro-viewpoint has gradually been

incorporated into the solidification models. In these

models, the microstructure evolution during the solidi-

fication process is considered. The whole process

includes three different steps: nucleation (an increase

in the number of nuclei), growth (an increase in thevolume of the grain) and impingement. These three

steps are illustrated in Fig. 1.

In the solidification process, as the temperature

of the liquid metal falls below the melting point,

nucleation begins, as shown in Fig. 2. Crystal clusters

(or embryos) are formed. These clusters may melt or

grow. When the clusters are big enough, they will not

melt any more. At this time, they are called nuclei.

At the beginning of nucleation, the number of nuclei

increases very slowly. After a critical undercooling

value is reached (∆T n in Fig. 2), the number increases

rapidly. Nucleation proceeds until the decreasing tem-

perature starts to increase, i.e., recalescence occurs. At

Fig. 1. Equiaxed solidification of the eutectic system. [Reprint from

Rappaz (1989)]

this point, the number of nuclei reaches its maximum

value.

After nucleation, there is a long period of growth.





L.S. Chao and W.C. Wu

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ρ c p∂T

∂t

= k [∂2T

∂r 2

+ 1

r

∂T ∂r

+ ∂2T

∂ z2

] + ρ L∂ f s

∂t

, (1)

where c p is the specific heat, ρ is the density,

k is the thermal conductivity, L is the latent heat,

and f s is the local volume fraction of the solid.

(2)Initial and boundary conditions:

(i) The initial condition is

T (r , z,t =0)=T 0, (2)

where T 0 is the pouring temperature of the

liquid metal.

(ii) The boundary conditions are,

(a) at the center line (r =0),

∂T ∂r r = 0

= 0 , and (3)

(b) at the air/metal interface ( z=16 cm),

q"=hconv(T −T ∞), (4)

where q" is the heat flux, hconv is the

convective heat transfer coefficient of

air, and T ∞ is the environment tem-

perature.

(c) At the metal/mold interface:

Since the temperature distribution of the

sand mold is of less interest and requires

a great deal of computing effort, the

equivalent heat transfer coefficient is

used for the sand mold. The boundary

condition at the metal/mold interface can

be given by

q"=heff (T −T ∞). (5)

The primary difference between macro and macro-

micro models is in how the local volume fraction of

solid, f s, is computed. In the macro model, Voller’s

method (Voller and Swa, 1991) is used to calculate f s.

In macro-micro models, since equiaxed grains are as-

sumed to be spherical, f s can be expressed as

s(t ) = 4

3π R3(t ) • N (t ) , (6)

where N (t ) is grain number per unit volume and R(t )

is the grain radius. The derivative of f s can be written

as

df sdt

= 4π R2(t ) • N (t ) •

dRdt

+ 43π R

3(t ) •

dN dt

. (7)

Because the radius is very small in the nucleation step,

the second term on the right-hand side of Eq. (7) can

be ignored, and the equation can be rewritten as

df sdt

= 4π R2(t ) • N (t ) •

dRdt

. (8)

Equations (6) and (8) are the basic equations of the

micro-macro models. The detailed computing methods

used to obtain f s for the two macro-micro models built

in this paper are described in the following.

1. Model I

Nucleation is assumed to occur instaneously. This

means that the nucleation step does not need to be

considered in this model, and that N (t ) is a constant.

Fig. 3. A schematic diagram of the (testing) cylindrical casting.




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Accordingly, Eqs. (6) and (8) can be rewritten as

s(t ) = 4

3π R

3(t ) • N , (9)

df sdt = 4π R

2

(t ) • N •dRdt . (10)

In the growth step, by using the Johnson-Mehl

model (Johnson and Mehl, 1939), the growth rate V can

be given by

V =µ •∆T 2, (11)

where µ is a growth constant and ∆T is the undercooling,

which is equal to the difference between T e and T . T eis the eutectic temperature. The expression for R can

be obtained by integrating Eq. (11):

R = r 0 + µ [T e – T (t )]

2dt

0

t

, (12)

where r 0 is the critical radius (Kurz and Fisher, 1989).

For the impingement step, Eq. (10) is modified

by multiplying by a factor F :

df sdt

= 4π R2(t ) • N • dR

dt • F . (13)

Here, F is taken as 1− f s (Avrami, 1940). Equation (13)

can be rewritten as

df sdt

= 4π R2(t ) • N • dR

dt • (1 – f s) . (14)

By integrating Eq. (14), the expression for f s becomes

s(t ) = 1 – exp[ – N • 4

3π R

3(t )] . (15)

2. Model II

In the nucleation step, the Gaussian distribution

(Zou and Rappaz, 1991) is used to capture the variation

in the trend of the nucleation rate. The equation for

dN / dt can be written as

dN dt

= –N max

2∆T σ exp[

– (∆T – ∆T 0)2

2∆T σ ]dT dt

, (16)

where ∆T 0 is the undercooling at the tip point of theGaussian distribution, ∆T σ is the standard deviation of

the distribution, and N max is the total grain density of

the whole distribution (integrating Eq. (16) from zero

undercooling to infinity).

When ∂ T / ∂ t >0 (i.e., the recalescence occurs),

dN =0. This is the end point of the nucleation step. In

the growth step, Eq. (11) is also used to calculate the

grain radius. The two-step Close-Pack model

(Alexandre et al., 1991) is applied in the impingement

step, and the f s is given by

Table 1. Constants Used in the Two Micro-Macro Models

Constant Value

N (in Model I) 6.0×109 m−3

µ 3.0 ×10 −8 m/s °C

∆T 0 20 °C

∆T σ 4.75 °C

nmax 1.20×1011 m−3

Fig. 4. The computed cooling curves obtained by the macro and

macro-micro models. (a) Cooling curves; (b) locally enlarged

cooling curves.




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df sdt

= φ • 4π R2(t ) • N (t ) •

dRdt

, (17)

φ =

– 10.32log10

( f s) 1 ≥ f s ≥ 0.8

1 f s < 0.8 .(18)

In this paper, the finite difference method is used

to compute the temperature distribution. In formulat-

ing the finite difference equations, the central differ-

ence is used for the space derivative, and the backward

difference is used for the time derivative. The algebraic

equations are solved iteratively by using the line

S.O.R. method (Anderson et al., 1984).

III. Results and Discussion

In this paper, two macro-micro models have been

built to study solidification processes. The testing

material is gray cast iron, whose microstructure is

equiaxed eutectic. The constants used in these two

models are listed in Table 1. The node number of the

uniform grid used in the computation is 61×26. In

Model I, the time step can be 1 second. However, in

Model II the step can not be larger than 0.1 second,

and the relaxation factor is adjusted up to 1.2 since it

is more difficult to make it converge than it is in Model

I. For convenience, the center point of the casting isused as a reference point (Fig. 3).

Figure 4 shows the computed cooling curves of

the reference point for the macro and macro-micro

models. In Fig. 4(a), there is no big difference between

these two curves. However, the undercooling phenom-

enon can not be obtained from the macro model. In

Fig. 5. The cooling curves from the experiment and the macro-micromodel.

Fig. 6. The computed cooling curves obtained by the two macro-

micro models. (a) Cooling curves; (b) locally enlarged cooling

curves.

Fig. 4(b), the undercooling as well as the recalescence

phenomenon can be clearly seen from the cooling curve

of the macro-micro model.Figure 5 shows the computational and experimen-

tal cooling curves of the reference point. From this

figure, it can be found that the computational data of

the cooling curve and undercooling are quite close to

the experimental data (Kanetkar et al., 1988). Figure

6 illustrates the cooling curves of Model I and Model

II. Though these two models are quite different from

each other, the computational results are very similar

(Fig. 6(a)). Taking a close look (Fig. 6(b)), we can

find that Model II has a smaller degree of recalescence

(or the maximum undercooling), which is closer to the

experimental result.

Figure 7 shows f s vs. the grain radius at the ref-




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erence point for three different models, Model I, ModelII and Model II, without modification of φ (in Eqs. (14)

and (15)). Because of the modification of φ in Model

II (after impingement occurs, f s>0.8), the curve become

smoother than the one without modification. In Model

I, since f s increases slowly and smoothly with the grain

radius, the larger time step can be used, and the con-

vergence rate is faster than that in Model II. However,

in Model I, f s will not be one until the radius reaches

infinity. This is not reasonable. Accordingly, the end

point of solidification is set at the radius when f s=

0.999.

At the reference point, the relationship between

the nucleation rate (or grain density) and time is that

Fig. 8. The nucleation rate and grain density distributions vs. time.

Fig. 7. f s vs. the grain radius for different models.

Fig. 9. The distribution of the highest level of undercooling.

shown in Fig. 8. In the figure, it is shown that the

Gaussian distribution can successfully simulate the big

change of the nucleation rate in a short time. After

the nucleation step, the grain density reaches a constantvalue, i.e., the final grain density. It can also be found

that the nucleation time (about 18 seconds) is very short

compared to the local solidification time (about 300

seconds). This proves that the assumption of instan-

taneous nucleation is reasonable.

In general, it is thought that a higher cooling rate

(dT / dt ) yields a higher grain density since the higher

cooling rate leads to greater undercooling, which re-

sults in a larger number of nuclei (Kurz and Fisher,

1989). This is consistent with the computed results

described in the following. In Figs. 9-11, the distri-

butions of the maximum undercooling, grain density

and radius (of Model II) are shown. From these figures,

it can be found the grain radius and density are strongly

related to the undercooling. Closer to the center point

(where the cooling rate is lower), the maximum level

of undercooling is lower, the grain density is lower and

the grain radius is larger.

From this study, it can also be seen that the

advantages of Model I are that the formulation is simple,

and that the undercooling prediction is not bad.

However, Model I can not obtain the nucleation rate

or the grain density which varies with time, and the

grain radius should be infinite, which would make f s

equal to one, which is not reasonable. Therefore,Model I is only suitable for rough evaluation of the




− 628 −

Fig. 11. The grain size distribution after solidification.Fig. 10. The grain density distribution after solidification.

solidification process. On the other hand, Model II can

obtain more information about solidification and a

better undercooling prediction than can Model I.

However, the computation for Model II is not verystable, so a small time step is needed. The convergence

rate of each time step is also low. Consequently, Model

II uses much more computation time than does Mode

I.

IV. Conclusions

In this paper, two macro-micro models have been

built in order to study the equiaxed solidification of

eutectic. Model I ignores the nucleation step by as-

suming that it occurs instantaneously; however, it is

considered in Model II. These two models can obtainthe undercooling (or recalescence), which cannot be

obtained by the macro models. From the computational

results, the following conclusions can be drawn:

(1)By using Model I or II, cooling curves as well

as undercooling can be well predicted, but the

computed results obtained by Model II are closer

to the experimental results.

(2)The formulation of Model I is simple, and the

convergence rate is fast. This model is suitable

for preliminary evaluation of the solidification

process.

(3)Model II can obtain the nucleation rate, grain

density and radius which varies with time, which

cannot be obtained by Model I. Since the varia-

tion of f s vs. the grain radius is not as smooth

as in Model I, Model II needs a smaller time step

for the computation to be stable.(4)During the solidification process, the cooling

rate and undercooling are two important factors

influencing the formation of the microstructure

and heat transfer.

(5)A higher cooling rate yields a higher level of

undercooling, which results in greater grain

density and a smaller grain radius.

Acknowledgment

This research was supported by the National Science Council,

R.O.C., under Contract NSC 85-2212-E006-025.

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Macro Micro Modeling of Solidification