Negative Segregation due to Bulk Liquid Flow

Solidification of Alloy*

By Kenji MURAKAMI,** Hiroyuki AIHARA** and Taira

during

OKAMOTO**

Synopsis

A formula has been derived which predicts the degree of negative segrega-tion due to bulk liquidfow during solidification of alloys. The formula includes the alloy composition, the primary dendrite-arm spacing and the width o f the mushy zone as well as the solidification condition and the bulk flow rate. It is ascertained that the formula can be applied both to aluminum-copper alloys solidified unidirectionally from a copper chill face under flow conditions and to continuously cast high carbon steels electromagnetically stirred during their solidification.

I. Introduction

Electromagnetic stirring has been widely employed

in continuous casting of steels for purposes of reducing centerline segregation, developing equiaxed grain

structure, and so on. A white band is a zone depleted in solute elements and is known to arise from the

stirring. There have been many works'-9) on the cause of the formation of the white band and the

degree of negative segregation in it. These works can be classified roughly into two categories.

In the first category,'-3,9) it is proposed that a turbulent flow of bulk liquid due to the electro-

magnetic stirring penetrates into the mushy zone and

sweeps away the liquid enriched in the solute elements in the mushy zone. Takahashi et a1.31 developed a model along this line assuming that bulk liquid

completely replaced the liquid enriched in the solute elements in the mushy zone whose fraction liquid

was larger than a critical value. The critical fraction liquid depended on the solidification conditions and

the flow rate of the bulk liquid. However, it has been shown that, when there is a turbulent flow over

a porous layer, the flow rate gradually decreases from the top surface of the layer toward the inside.1o,11)

It is, therefore, unreasonable to postulate that the composition of liquid in the mushy zone within some

critical distance from the solidification front is equal to that of the bulk liquid. Moreover, as Bridge and

Rogerss) have pointed out, the model cannot explain the formation of a slightly solute enriched zone

corresponding to the end of stirring. Mechanisms involved in the second category base

on the unsteadiness of solidification resulting from

the onset and cessation of electromagnetic stirring. According to Kor,5) basing on a formula12) derived

for a planar solidification front, solute depletion and enrichment can be accounted for by the changes in

the solidification rate as a result of the onset and cessa-tion of stirring, respectively. Bridge and Rogerss)

calculated positive and negative segregations in terms

of the changes in the relative moving rate of liquidus and solidus isotherms. These last two models are attractive and seem to be able to predict both the

positive and negative segregations in and adjacent to a white band. However, it will be shown later that they cannot adequately describe the macro-segregations. The purposes of the present work are to provide a model on the formation of the negative segregation due to a bulk liquid flow, and to reveal the factors affecting the negative segregation.

II. Model

In analysis of the negative segregation due to a bulk liquid flow, the following assumptions are made.

(1) The flows both in the bulk liquid region and in the interdendritic region are fully turbulent.

(2) The flow caused by solidification shrinkage is neglected.

(3) For the transports of the solute elements in the liquid, only the eddy transport mechanism is taken into account and the molecular diffusion is neglected.

(4) The diffusion of the solute elements in the solid in the direction parallel to the growth direction is neglected.

(5) The solute concentrations in the bulk liquid are homogeneous.

(6) The solid grows as columnar dendrites. (7) The solidification proceeds under a steady

state.

The differential equation describing the net overall

change in a solute content in a volume element in

the mushy zone schematically shown in Fig. 1 can

Fig. 1. A sc hematic illustration of a mushy zone.

* Based on the paper presented

received on February 28, 1985 * * The Institute of Scientific and

to the 108th ISIJ Meeting, October 1984, 5909, at Hiroshima University in Hiroshima.

; accepted in the final form on June 14, 1985. © 1985 ISIJ Industrial Research, Osaka University, Mihogaoka, Ibaraki 567.

Manuscript

(1212) Research Article

Transactions ISIJ, Vol. 25, 1985 (1213)

be written as

aC a aCL Ed f L ..................(1) a

t ax ax

where, C: the average concentration of solute in the volume element at time t

~d: the turbulent diffusivity

f L . the fraction liquid CL : the concentration of the interdendritic

liquid x : the coordinate parallel to the growth

direction. For steady state solidification,

z =x-Rt

where, z : the distance measured from the moving solidification front into the bulk liquid

R : the growth rate. Therefore, Eq. (1) becomes

dC d dCL

_

- aL ......... ......(2) dz dz ~ dz

The boundary conditions are

JC=GLO, f L =1 at z= 0 C=C(-L), fL=0 at z=-L

where, L : the width of the mushy zone. Integrating Eq. (2) under the boundary conditions, we obtain

- dCL CLO-C(-L) _ _id Iz-o• .........(4)

R dz z=0

Thus, the effective distribution coefficient ke of the solute element can be expressed as follows;

k = C( L)

e

CLO

1 dCL =1+ •Ed z=0• ............(5) RC LO dz z=O

The equation of the solute conservation can also be written as

a (CLfL) = a ~dfL aCL +koCL a fL _ W,(x, t) at ax ax at

...........................(6)

where, W' : the amount of the solute which diffuses into the solid per unit time

w, = 0 : in the case where there is no diffusion in the solid

w, = k0fs aCL : in the case where the diffusion in at the solid is complete (f s is frac- tion solid)

k0: the equilibrium distribution coefficient. For steady state solidification, Eq. (6) becomes

-Rdd(CLfL) -_ d ~d L dCL -k°CLR dfL z dz f dz dz

-W(z) ..............................(7)

and W = 0: in the case where there is no

a d id R+id z-°• dfL +- z z=O dz z=0

Yamada and Kawabata,l0) who studied the resis-tance law of the turbulent flow over a porous layer, showed that the mixing length l in the layer was

proportional to the scale of the porosity. The porous layer they studied had a uniform porosity distribution. On the other hand, when an alloy solidifies from the mould wall into the bulk liquid, the porosity is not uniform, being maximum at the solidification front and zero at the roots of dendrites. Hence, the following expression for a mixing length in the mushy zone can be written

z l=e•d• l+L .....................(10)

where, d: the primary dendrite-arm spacing e : a constant.

Yamada et a1.10,11) showed that the velocity U of a fluid in the layer with uniform porosity could be given as follows;

U = Uoer .........................(11)

where, U° is the velocity of a fluid at z=0 (slip velocity) and a a coefficient, which is regarded as a function of z in the mushy zone of solidifying alloys. In the vicinity of the solidification front, Eq. (11) can be approximated as follows;

U=r~U1(1+aoz) .....................(12)

where, r? : the ratio of the slip velocity to the bulk flow rate UB

a0: a constant. The eddy kinematic viscosity Er is given11"3) as

it - d 12 dU ........................(13)

z Because the turbulent diffusivity nearly equals the kinematic viscosity,14) we obtain the following formula by combining Eqs. (10), (12) and (13),

2

id =Od2UB l+ z ..................(14) L

diffusion in the solid.

W = -ko f~R dCL : in the case where the diffu- dz sion in the solid is complete.

Therefore, the concentration gradient in the liquid in the mushy zone is given as

-Ed fL L _-(1 -k0)CLR d L +W dCL 2 dz2 dz dz _ dfL did Rf

L+id dz +f L dz

...........................(8)

Substituting Eq. (8) into Eq. (5) and ignoring the term including the second derivative of concentra-tion, the effective distribution coefficient becomes

(1--ko)•id z=O• dfL

(1214) Transactions ISIJ, Vol. 25, 1985

where, O=aoe2r~. Combining Eqs. (9) and (14), and putting K=(1ke)/ (1-ko), we finally get

dfL

K - dz z=o ............(15) R +2+dfL Od2U

B L dz z=o

When an alloy contains solute elements pi (i=1, 2, ...) which diffuse very rapidly in the solid and elements qi (i= 1, 2, ...) which hardly diffuse in the solid, the concentrations of the elements CLpi and CLgi in the interdendritic liquid can be expressed as follows, respectively;

`"

G _ opi Lpi - 1 ...............(16) \ kOpi)fL+kOpi

C - C fckooi-1) (17)

where subscripts pi and qi refer to the elements pi and q27 respectively. The liquidus temperature TL of the interdendritic liquid is a function of the com-

position of the liquid, and we can write

.

dTL = aT :dcLPl+ iaTL dCLg2 aC aC $i L 4ti Lgi

........................(18)

Combining Eqs. (16), (17) and (18 ), the following equation is obtained

dfL = [ I m J -(1 k0pi)C°pi dYL L $i p1 l(1 kOpi)fL+kopi~2

+~ {m01(k001_ 1)C'ogifikooi-2) ......(19) where, m p1 a T L ) UC

Lpi a TL

mgi aCLgi

Letting G be the temperature gradient,

dz dTL

We have the following equation by combining Eqs. (19) and (21)

dfL G dz z=o micoz(koi-1)

where i refers to all solute elements. For a binary alloy with straight liquidus and solidus, the following equation holds

dfL - G ...............(23) dz z=o mGLo(ko-1)

where, m : the slope of the liquidus. Since the temperature differences 4 T1 and 4 T2 shown in Fig. 2 are expressed as follows,

4 T1= GL,

4 T 2 = mcLo(ko- l ) . .....................(24)

and 4 T2 = ko 4Ti

Eq. (15) yields

1 k°LR +(1 +2k

Od2UB o)

III. Experimental Procedure

The experimental apparatus used in the present work was the same as that used in the previous work.l5) Aluminum-copper alloys (2.1, 4.0 and 8.8 wt% Cu) were prepared from a 99.9 wt% Al and an Al-50wt%Cu master alloy. Each alloy weighing about 80 kg was melted in a crucible in an electric furnace. The superheat of the melt in the crucible was 25 K above its liquidus temperature. The melt flew down through a hole of 7 mm in diameter at the bottom of the crucible onto a channel, which was made of heat insulating materials and had a width of 2.5 cm. As part of the bottom of the channel, a copper brick chill 20 cm long, 8.5 cm wide and 10 cm thick was set. During the flow of the melt, a solidification layer developed on the chill. The flow rate was varied by changing the slope of the channel. The mean flow rate in the channel was measured through dividing the amount of the flowing melt per unit time by the cross sectional area of the flow, being of 13 to 90 cm/s. During the flow of the melt, the temperatures of locations at 10, 17, 24, 31 and 38 mm from the chill face were measured with five chromel-alumel thermocouples of 0.1 mm in diameter. The thermocouples were thin coated with ceramics except the hot junctions.

Macrostructures of the alloy ingots solidified on the chill were examined on the sections parallel to the flow direction and perpendicular to the chill surface

(longitudinal sections). Microstructures were exam-ined on the sections perpendicular to the growth direction of the columnar grains (transverse sections). Since, the growth direction of a columnar grain was

parallel to that of the columnar dendrites composing the columnar grain in the present crystal growth mode,15) the transverse sections were also perpen-dicular to the growth direction of the columnar den-drites. From the microstructures on the transverse sections, primary arm spacing was measured as the average distance between adjacent primary arms of columnar dendrites. Samples for the analysis of copper concentration were taken from the alloy

Fig. 2. Portion of a binary phase

temperature differences A T1

diagram

and A T2.

showing the

Transactions Is", Vol. 25, 1985 (1215

ingots at locations of 5 N 8, 10 .13, 15-l8, 2l-24, 27-'3O and 33-'36 mm from the chill face using a shaper. In some specimens, the solidus isotherm hardly advanced beyond some distance from the chill face. Macrosegregation data of such specimens were not adopted.

Iv. Results

Photograph 1 shows some macrostructures on the longitudinal sections of the alloy ingots of various copper contents solidified in flowing melts on the copper brick chill. The flow direction was from the right to the left and the flow rate was 67 cm/s. As has been shown by many workers,'-3,15,16) colum-nar grains grew with their axes deflected toward the upstream direction from the normal to the chill face. Increases in copper content and flow rate

produced more fresh grains in a columnar zone and, hence, the macrostructure became more equiaxed-like. This is because, with an increase in copper content, growing crystals became more dendritic and slender, and their arms would be easily broken off or melted off by the flowing melt.'5~

The primary dendrite arm spacings in the ingots at various flow rates and cooling rates are shown on logarithmic graph papers in Figs. 3 and 4 for the alloys with 2.1 wt% Cu and 4.0 wt% Cu, respectively. The cooling rate adopted here was the one at the moment when the solidification front passed the

position concerned, and was given by measuring the slope of the cooling curve just below the liquidus temperature. The relationships between the primary arm spacing and the cooling rate on a stagnant melt condition19~ are shown by broken lines in the figures, the slope being -0.5. In the present work, the melt flow made the primary arm spacing larger and less sensitive to the cooling rate than when the melt was stagnant. For a given cooling rate and a copper

Fig. 3. Primary

solidified

arm spacing of an

in a flowing melt.

Al-2. lwt%Cu alloy

Fig. 4. Primary arm spacing of an

solidified in a flowing melt.

Al- 4.Owt%Cu alloy

Photo. 1. Macrostructures on longitud rate of about 67 cm/s. The from the right to the left.

inal sections of

copper content

aluminum-copper was (a) 2.1 wt%,

alloys solidified

(b) 4.0 wt% and

under the flowing

(c) 8.8 wt%. The

condition at a

flow direction

flow

was

(1216) Transactions ISIJ, Vol. 25, 1985

content, the primary arm spacing hardly depended on the flow rates within the range of flow rate from 13 to 67 cm/s. At flow rates higher than 83 cm/s, the primary arm spacing was smaller than that at lower flow rates. Table 1 lists the data which are necessary to calcu-late the value of K from Eq. (25). Since steady state solidification was not attained in the present work, some data in the table were determined as follows. The value of the effective distribution coeffi-cient ke of copper was the ratio of the average copper concentration in the region of 20 to 30 mm from the chill surface to the initial copper concentration. From the cooling curves, the positions of the liquidus and solidus of a solidifying ingot could be known as a function of time. Letting the time be t*, at which the position of 25 mm from the chill surface was in the middle of the mushy zone, the width of the mushy zone at time t was adopted as L. The average temperature gradient G was obtained through dividing the temperature difference between the liquidus and solidus by the width of the mushy zone at time t. The growth rate R was the average velocity of the liquidus and solidus isotherms at time t*. The

primary arm spacings in Table 1 were the ones at a position of 25 mm from the chill face. Specimens containing 8.8 wt% Cu had equiaxed-like structures and the primary arm spacings of them could hardly be measured on the transverse sections. In some

grains, they could be measured and were around 180 pm.

The effective distribution coefficient ke of copper is plotted against the flow rate on the specimens with various melt concentrations in Fig. 5. When the copper concentration of the flowing melt was the same, the effective distribution coefficient decreased with increasing flow rate. At a given flow rate, the

effective distribution coefficient was larger in the speci-

mens with lower copper concentration of the flowing melt.

The effective distribution coefficient smaller than unity means that a part of the interdendritic liquid

enriched in the solute element has been swept away

by the flowing melt penetrating into the mushy zone.

In Fig. 6, the value of 1/K is plotted against kOLR/(d 2 UB). The straight line was drawn by the linear regression so that the intercept on the ordinate might be 1 +2k0. The equilibrium distribution coeffi-cient of copper in aluminum-copper alloy is 0.17.17) Since the macrostructures of specimens containing 8.8 wt% Cu were equiaxed-like, only the data on the specimens with 2.1 wt% Cu and 4.0 wt% Cu were taken into account in the linear regression. This fact will be discussed later. The line is expressed as follows;

Table 1. Experimental data of aluminum-copper al-

loys solidified unidirectionally in flowing

melts. These data were taken at the loca-

tions of 25 mm from the chill face.

Fig. 5. Effective distribution coefficient of copper

minum-copper alloys solidified in flowing

These data were taken at the locations of

from the chill face.

in alu-

melts.

25 mm

Fig. 6. A

for

variation of 1/K as a function aluminum-copper alloys.

of k0LR/(d2U13)

Transactions ISIJ, Vol. 25, 1985 (1217 )

-=35 .O. °LR +1.34 ...............(26) ~~ d2UB

Hence, the parameter 6 is obtained to be 0.029 (cm-1).

V. Discussion

The effect of liquid flow on a primary dendrite-arm spacing has been investigated by Okamoto et al.l8~ using cyclohexanol, a transparent organic material, of the purity 99.7 wt%. They found that the liquid flow made the primary arm spacing larger and spoiled the influence of cooling rate on it. Since a primary arm spacing depends on the solute diffusion in the interdendritic liquid,19~ the liquid flow which promotes the solute mixing causes the primary arm spacing to become larger. The reason why the primary arm spacing of dendrites solidified in a flowing liquid is less sensitive to cooling rate than in a stagnant liquid may be attributed to that the solute mixing due to the liquid flow is much greater than that by the diffusion. Takahashi et a1.3} showed that columnar dendrites had smaller primary arm spacings when

grown in a flowing melt than in a stagnant melt. They measured the arm spacings as functions of flow rate and distance from the chill, and not of cooling rate. Fluid flow should change the local solidifica-tion variables such as temperature gradient and cooling rate from those prevailing in a stagnant melt condi-tion, which might lead to a decrease in dendrite arm spacings.

The proposed model on negative segregation due to a fluid flow stands only when the mushy zone is composed of columnar dendrites. For equiaxed

grain or branched-columnar grain structures, some measure of the mixing length other than the primary arm spacing should be adopted. Moreover, equiaxed

grains near the solidification front might move due to a shearing stress coming from a bulk liquid flow. Such movement would require an additional mecha-nism in the transport of solute elements from the mushy zone to the bulk liquid.

Although many works have been done on the for-mation of negative segregation due to electromagnetic stirring in continuous casting, the work done by Ayata et al.' is the only one published so far, in which all quantities listed in Table 1 are known. They studied the influence of electromagnetic stirring of steels in a mould on the negative segregation of carbon in continuously cast high carbon steel blooms. The primary arm spacing of the steels was 260 pm nearly independent of the bulk flow rate ranging from 0 to 35 cm/s.20' However, the primary arm spacing they measured was the spacing between parallel secondary arms developing from neighbouring pri-mary arms on a section perpendicular to columnar dendrites. The measurement method was different from that used in the present work. Therefore, in order to compare the experimental results obtained in the present work with those given by them, their value of the primary arm spacing must be corrected to the value which would have been obtained by the method used in the present work. For this purpose,

the primary arm spacing measured by Ayata et al. should be multiplied by a factor of For the

evaluation of the value of df L , the li uidus tem- dz z=0

perature of steels proposed by Hirai21j and the equilibrium distribution coefficients of carbon, silicon and manganese obtained by Kagawa and Okamoto,22~ being 0.31, 0.67 and 0.79, respectively, were used.

In Fig. 7, the value of 1/K for carbon is plotted

against the value of R/d2 UB df determined dz z=0

from the corrected primary arm spacing. The solid line in the figure was drawn by the linear regres-sion so that it might pass through the point of 1 +

2/ L df L on the ordinate. The line is expressed dz z=o

by the following equation,

-4=38.3._ R dfL +1.56 .........(27) K d2UB dz z=o

In the regression analysis, the data point with the largest 1/K value was neglected. From Eq. (27), the parameter 0 was obtained to be 0.026 (cm-1), which was nearly equal to that for the aluminum-copper alloys in the present work. In Fig. 7, the regression line for the aluminum-copper alloys is also shown by a broken line. Figure 8 shows the relationship between the effective distribution coeffi-cient of carbon and the bulk flow rate measured by Ayata et al. The solid line in the figure represents the relationship of Eq. (15) with 8=0.029 (cm-1). The agreement between the experiment and the calculation is good.

Recently, Bridge and Rogers6~ proposed a mecha-nism on the occurrence of negative segregation in a white band and positive segregation in the zone corresponding to the end of stirring. According to them, the segregations can be explained on the basis of the relative velocities of the liquidus and solidus isotherms. Assuming that the fluid flow is due only

Fig. 7. A variation of 1/K as a function of R/(d2UB(dfL/ dz) z_o) for continuously cast high carbon steels. Solid and broken lines represent the regression lines for the steels and aluminum-copper alloys, respectively.

(1218) Transactions Is" Vol. 25, 1985

to solidification shrinkage and that an alloy

ing from the chill face according to the

relationships,

b

and

where, xL, xs

b, aL, a

Then the solut

liquid is given

of no diffusion

is solidify-

following

xs = nstb+as .....................(28-b)

: distances from the chill face to the liquidus and solidus isotherms, re- spectively

t : time

s : constants. e concentration GL of the interdendritic by the following equation23~ in the case in the solid,

Lnga+l E GL _ Go fL (1-ko) f _ ...........(29) - n

qa+ 1 .......

E = - (1-a)(l -ko)

a

n nL-ns a =1- q-- 'is 2

where, Go : the initial solute concentration

f L : the fraction liquid p: the solidification shrinkage.

From Eq. (29), the macrosegregation is obtained as a function of nq for a given alloy. Bridge and Rogers claimed that the negative segregation could be caused by a liquidus arrest (nq N -1) due to electromagnetic stirring, and the positive segregation by a subsequent liquidus acceleration (nq>0).

In a high carbon steel, the minimum effective distribution coefficient of manganese calculated from Eq. (29) at nq=-1 is about 0.98, where j3=0.04 and k0=0.79,22) However, the effective distribution co-efficient of manganese as low as 0.94 was obtained by Bridge and Rogers6~ in a continuously cast high carbon steel.

In the present work on aluminum-copper alloys, the mushy zone widened with time, and hence, nq was positive. Therefore, contrary to the experimental

results, the mechanism of Bridge and Rogers6~ predicts

positive segregation of copper. It is concluded from the above argument that their mechanism is inade-

quate for describing the segregation due to the bulk liquid flow.

Kor5~ proposed that the solute depletion and en-richment accompanied by electromagnetic stirring were due to sudden changes in the growth rate brought about by the onset and cessation of the stirring. The author calculated the macrosegregation from the formula derived by Smith et a1.12~ for a planar solidification front, which is written as follows;

R1 GS (x1) 1 D xl

Go =1-- erfc 2 2

1 R

1-k 2R1 R 1-R + ( o) R exp -Ri Rl k

o- Ri

X Ri xi erfc R - 1 Ri xi D Ri 2 D

1 1 Ri + ko_ 2 R eXp [_ko(l-ko) k° R

i

X R1 xi erfc ko- 1 R1 xi ......(30) D 2 D

where, x1: the distance from the point at which the sudden change in the growth

rate occurred Gs(x1) : the solid concentration at x1

R, R1: the growth rates before and after the change D : the diffusion coefficient of the solute

in the liquid. In order to examine the model of Kor, a subsidiary

experiment was carried out. An A1-3.7wt%Cu alloy was set in a graphite crucible of 5.4 mm inside diam-eter and 10.0 mm outside diameter. The specimen was driven downwards and was solidified unidirec-tionally through a vertical electric furnace at a pre-selected velocity. During its solidification, the moving velocity of the specimen was suddenly increased. The growth rate was obtained using three thermo-couples inserted into the specimen and was 46 pm/s before the velocity increase and 183 pm/s after that.

The macrosegregation of copper was examined by use of an electron probe microanalyzer on a sectional

plane parallel to the growth direction. The result is shown in Fig. 9. It can be seen that positive segregation occurred at around the position of the velocity increase. In the figure, the solid concen-tration calculated from Eq. (30) is shown by a solid curve. In the calculation, the diffusion coefficient of copper in the liquid and the equilibrium distribu-tion coefficient of copper are 3 X 10-5 cm21s24~ and 0.17,17) respectively. It is seen that the model as-suming a planar solidification front overestimates the segregation resulting from a sudden change in the

Fig. 8. A variation of the effective distribution coefficient of carbon with the bulk liquid flow rate for continu-

ously cast high carbon steels. The solid line was drawn from Eq. (15) where the value of B was given

to be 0.029 cm-1 from the experiment for alumi- num-copper alloys.

Transactions ISIJ, Vol. 25, 1985 (1219)

growth rate. This overestimation occurs because, in the case of dendritic solidification, the solute rejected at an advancing solid-liquid interface diffuses more in the direction perpendicular to the growth direction into the interdendritic liquid than in the

growth direction. Thus, we can say that a concen-tration change resulting from Kor's model will play a minor role in the occurrence of the negative segrega-tion due to the bulk liquid flow.

It is well known that a slight solute enrichment occurs immediately adjacent to a white band.5-'~ This phenomenon may be attributed to the accelera-tion of the solidification. In the present work, since the model proposed assumes a steady state solidifica-tion, it cannot predict the solute enrichment.

VI. Conclusions

Aluminum-copper alloys containing 2.1 to 8.8 wt% Cu were solidified unidirectionally in flowing melts at flow rates ranging from 13 to 90 cm/s. The

primary arm spacing on a melt flow condition was larger and was insensitive to the cooling rate than that in the alloy solidified unidirectionally in a stag-nant state. The degree of the negative segregation due to the bulk liquid flow was larger as the copper content of the alloy liquid and the flow rate increased. For the purpose of predicting the negative segrega-tion, a formula was derived which included the alloy composition, the primary dendrite-arm spacing and the width of the mushy zone as well as the solidifica-tion condition and the flow rate. The formula stands only when the solidification proceeds as columnar

dendrites. It is ascertained that the formula can be applied both to the negative segregation in the alumi-

num-copper alloys in the present work and to that in

the white band occurred in continuously cast steels which have been electromagnetically stirred.

Acknowledgements

This work was partly supported by the research

project of The Institute of Scientific and Industrial Research on the development of new materials for energy.

Fig. 9. Measured and calculated copper concentrations along a longitudinal axis of a unidirectionally solidified A1-3.7wt%Cu alloy specimen. The origin

of the abscissa is the location of the solidification front at the moment when the growth rate was sud-

denly increased from 46 nm/s to 183 ~€m/s. Solid- ification proceeded from the negative side to the

positive side of the abscissa.

1)

2)

3)

4)

5) 6)

7)

8)

9) 10)

11)

12)

13)

14)

15)

16)

17)

18)

19)

20)

21)

22)

23)

24)

REFERENCES

K. Ayata, K. Narita, T. Mori and T. Ohnishi : Tetsu-to-Hagane, 67 (1981), 1278. K. Sasaki, Y. Sugitani, S. Kobayashi and S. Ishimura : Tetsu-to-Hagane, 65 (1979), 60. T. Takahashi, K. Ichikawa, M. Kudou and K. Shimahara: Tetsu-to-Hagane, 61 (1975), 2198.

J. P. Gabathuler and F. Weinberg: Met. Trans., 14B (1983), 733. G. J.W. Kor : Ironmaking Steelmaking, 9 (1982), 244. M. R. Bridge and G. D. Rogers : Met. Trans., 15B (1984), 581. A. A. Tzavaras and H. D. Brody : J. Metals, 36 (1984), No. 3, 31. D. J. Hurtuk and A. A. Tzavaras: Met. Trans., 8B (1977), 243. S. Kollberg : Iron Steel Eng., 57 (1980), No. 3, 46. T. Yamada and N. Kawabata : Proc. Japan Soc. Civil Eng., 325 (1982), 81. T. Yamada and N. Kawabata : Proc. Japan Soc. Civil Eng., 325 (1982), 69. V. G. Smith, W. A. Tiller and J. W. Rutter: Can. J. Phys., 33 (1955), 723. H. Schlichting : Boundary-layer Theory, McGraw-Hill, New York, (1979), 592. R. B. Bird, W. E. Stewart and E. N. Lightfoot: Transport Phenomena, John Wiley and Sons, New York, (1960), 629. K. Murakami, H. Aihara and T. Okamoto : Acta Met., 32 (1984), 933. R. Ichikawa, H. Taniguchi and 0. Asai: Z. Metallk., 71

(1980), 260. M. Hansen: Constitution of Binary Alloys, 2nd ed., Mc-Graw-Hill, New York, (1958), 85. T. Okamoto, K. Kishitake and I. Bessho : J. Cryst. Growth, 29 (1975), 131. T. Okamoto and K. Kishitake: J. Cryst. Growth, 29 (1975), 137. K. Ayata, K. Narita, T. Mori and T. Ohnishi : Private communication. Iron and Steel Handbook, Vol. I, Fundamental Theory of Iron and Steel, 3rd ed., ed, by ISIJ, Maruzen, Tokyo,

(1981), 205. A. Kagawa and T. Okamoto: J. Mat. Sci., 19 (1984), 2306. M. C. Flemings and G. Nereo : Trans. Met. Soc. AIMS, 239 (1967), 1449. M. H. Burden and J. D. Hunt: J. Cryst. Growth, 22 (1974), 109.