modeling of solute redistribution in the mushy zone...

Modeling of Solute Redistribution in the MushyZone during Solidification of Aluminum-Copper Alloys

Q.Z. DIAO and H.L. T S A I

A mathematical model has been established to predict the formation o f macrosegregation fo r aunidirectional solidification o f aluminum-copper alloys cooled from the bottom. The model,based on the continuum formulation, allows the calculation o f transient distributions o f tem-perature, velocity, and species in the solidifying alloy caused by thermosolutal convection andshrinkage-induced fluid flow. Positive segregation in the casting near the bottom (inverse seg-regation) is found, which is accompanied by a moving negative-segregated mushy zone . Theeffects o f shrinkage-induced fluid flow and solute diffusion on the formation o f macrosegre-gation are examined. It is found that the redistribution o f solute in the solidifying alloy is causedby the flow o f solute-rich liquid in the mushy zone due to solidification shrinkage. A higherheat-extraction rate at the bottom increases the solidification ra te , decreasing the size o f themushy zone, reducing the f low o f solute-rich liquid in the mushy zone and, as a result, lesseningthe severity o f inverse segregation. Comparisons between the theoretical predictions from thepresent study and previous modeling results and available experimental data are made , and goodagreements are obtained.

I. INTRODUCTION

SOLIDIFICATION o f alloys is characterized by theexistence o f a mushy zone , in which the solid phase andliquid phase coexist. It has been reported that transportphenomena (i.e., heat and mass transfer and fluid flow)in the mushy zone are the major cause for the formationo f casting defects, such as segregation, porosity, and hottears, tL2] In particular, the formation o f macrosegrega-tion (i.e., a large-scale nonuniformity in composition) incastings is understood to be caused by two mechanismsoccurring in the mushy zone.Lm] The f i r s t mechanism cor-responds to the floating or settling o f precipitated phasesduring solidification. The precipitated phases could bethe equiaxed grains and/or broken dendrites in the mushyzone. The other mechanism is related to the flow o f solute-rich o r solute-poor liquid in the mushy zone . The fluidflow can be caused by solidification contraction a n d / o rthermala n d / o r solutal gradients. A higher concentrationo f solute is frequently found near the bottom surface o fa casting unidirectionally solidified from the bottom,which is commonly called inverse segregation. Inversesegregation is a kind o f macrosegregation that has beenreported to be caused by solidification contraction. [1]

In the past, many experimer, tal and theoretical studieson the formation o f inverse segregation have been re-ported. Scheil t3] developed an expression which can pre-dict the "maximum segregation" at the chill surface asa function o f alloy composition in a unidirectionally so-lidified ingot. Kirkaldy and Youdelis t4] extended Scheil'sequation to predict not only the maximum segregationbut the positional variation o f the segregation. Theirmodel, however, is limited to one dimension, and the

Q.Z. DIAO, Graduate Student, and H.L. TSAI, Associate Professorof Mechanical Engineering, are with the Department of Mechanicaland Aerospace Engineering and Engineering Mechanics, Universityof Missouri-Rolla, Rolla, MO 65401.

Manuscripts submitted September 17 , 1992.

momentum, energy, and species equations are not com-pletely coupled. Perhaps the first rigorous model to pre-dict the formation o f macrosegregation was described inthe pioneering articles by Flemings and co-workers.[5,6.7]The well known "local solute redistribution equation"was derived, which predicted successfully the formationo f inverse segregation, centerline segregation, and seg-regation resulting from changes o f cross section for uni-directional solidification o f AI-Cu alloys. However , thesolute diffusion was neglected, and the thermal gradientsand velocity distributions used in the equation were mea-sured o r assumed.

Kato and Cahoon t81 studied the inverse segregation indirectionally solidified A1-Cu-Ti alloys with equiaxedgrains based on the theory proposed by Kirkaldy andYoudelis and reported that the theoretical predictions wereconsistent with the experimental measurements. Ohnakaand Matsumoto t9,1°] have analyzed the unidirectional so-lidification o f A1-Cu alloys based on the conservation o fmomentum, energy, and species and have incorporatedthe solute diffusion effects. However, they did not showthe transient solute distribution, and the importance o fshrinkage-induced flow on the formation o f macrosegre-gation was not discussed.

Recently, Bennon and Incropera [1~,12] and Beckermannand Viskantatl3j developed the "continuum formulation"based on the classical mixing theory or volume aver-aging technique, which is valid fo r the entire casting do-main, including the solid phase, mushy zone , and liquidphase. Employing the continuum formulation, manymultidimensional solidification problems involving acomplete coupling o f momentum, energy, and speciesequations have been solved. Neilson and I n c r o p e r a[141 usedthe continuum formulation to study the unidirectional so-lidification o f aqueous NH4C1 and the effects o f thecompositionally-induced fluid motion. As aqueous N H 4 C Iliquid in the mushy zone is enriched by the lighter H20species when solid NHaC1 is precipitated, a density in-version is created, leading to pronounced freckle phe-nomena. [14] In the above-mentioned continuum

METALLURGICAL TRANSACTIONS A VOLUME 24A, APRIL 1993--963

formulation, with the exception o f the buoyancy terms,a constant density was assumed throughout the entire do-main, and as a result, the shrinkage effect was neglected.

For unidirectional solidification of A1-Cu alloys cooledfrom the bottom, as the heavier copper species is re-jec ted when the solid aluminum is precipitated, stablesolute and temperature gradients are created, and no nat-ural convection can be induced. Hence , the inverse seg-regation in directionally solidified A1-Cu alloys cannotbe predicted by the above-mentioned models. Chiang andTsai usa61 have modified the continuum formulation to in-clude the fluid flow and domain change caused by so-lidification contraction. In their study, they concludedthat shrinkage-induced fluid flow is dominated in themushy zone compared with the natural convection dueto temperature gradients. As the fluid flow in the mushyzone is the major cause o f the formation o f macrosegre-gation, the model will be extended in the present studyto include the species equation to investigate the soluteredistribution in the mushy zone during unidirectionalsolidification o f A1-Cu alloys cooled from the bottom.

II. MATHEMATICAL FORMULATION

The model, based on the continuum formulation in-cluding the effect o f shrinkage developed by Chiang andTsai ,u5,161 is used in the present study. In the derivationo f the governing equations, the following assumptionsare made: (1) the properties o f the solid and liquid phasesare homogeneous and isotropic; (2) the solid and liquidphases in the mushy zone are in local thermodynamicequilibrium; (3) the density fo r each phase is constantbut can be different for the liquid and solid phases; (4)the Boussinesq approximation for natural convection canbe invoked; (5) no pore formation is present; and (6)binary alloys. Based on the continuum formulation, thecontinuity, momentum, energy, and species equationsfor the solidification o f a two-dimensional casting aregiven as follows:

Continuity

0Ot (p) + V. (pV) = 0 [1]

where t is the time, p is the density, and V is the velocityvector.

Momentum

Ot(Ou) + V-(pVu) V txt Vu Ox

/*t P- - - ( u - u , )

K Pz

Ca2K , / % lu - u , l ( u - u,)

- V . ( p f ~ V , u , )

[2]

0- ( p v ) + V.(oVv) = V.dt

/m_PVv~ 0p m p ( v - v , )Oy K Pl

C•2Kl /=p l IV -- V,I (V -- V,)

+ Pg(~r ( T -- To) + ~ , ( f f -f~,o))

[3]

where u and v are the velocities in the x- andy-directions, respectively, and Vr (= Vz - V,) is the rel-ative velocity vector between the liquid phase and thesolid phase. The subscripts s and l refer to the solid andliquid phases, respectively; p is the pressure; /~ is theviscosity; f is the mass fraction; K, the permeabilityfunction o f the casting, is a measure o f the ease withwhich fluids pass through the porous mushy zone; C isthe inertial coefficient; fir and t , are the thermal expan-sion coefficient and solutal expansion coefficient, re -spectively; g is gravitational acceleration; and T is thecasting temperature. The subscript 0 represents initialcondition, and the superscript a represents species. Thethird and fourth terms on the right-hand side o f Eqs. [2]and [3] represent the first- and second-order drag forces,respectively, for the flow in the mushy zone. The second-to-last term on the right-hand side o f Eq. [2] and thethird term from the last on the right-hand side o f Eq. [3]represent an interaction between the solid and the liquidphases. These two terms are zero except in the mushyzone , in which case neither f, nor f/is zero. The last termon the right-hand side o f Eq. [2] and the second-to-lastterm on the right-hand side o f Eq. [3] represent the effecto f shrinkage, and both terms are identical to zero whenthe density difference between the solid and liquid phasesis not considered. The last term on the right-hand sideo f Eq. [3] is based on the Boussinesq approximation fo rnatural convection. It is noted that Eqs. [2] and [3] re-duce to appropriate single-phase limits ( i . e . , K is zeroin the pure solid and K is infinite in the pure liquid, seealso Eq. [8]).

Energy

-ot(Ph) + V . ( p V h ) = V. ~Vh + V- V ( h , - h )

-- V . ( p ( V - V , ) ( h i - h ) )

[4]

where h is the enthalpy, k is the thermal conductivity,and c is the specific heat. The f i r s t two terms on thefight-hand side o f Eq. [4] represent the net Fourier dif-fusion flux. The last term represents the energy flux as-sociated with the relative phase motion.

964--VOLUME 24A, APRIL 1993 METALLURGICAL TRANSACTIONS A

Species

0(pf f ' ) + V . (pVf ~) = t7. ( p D T f'~) + 17. (pD17(f7 - f ' ~ ) )

- 17. (p(V - Vs) (fit - f~)) [5]

where D is the mass diffusion coefficient. The first twoterms on the right-hand side o f Eq. [5] represent the netFickian diffusion flux, while the last term represents thespecies f lux due to relative phase motion. In Eqs. [1]through [5], the continuum density, specific heat, ther-mal conductivity, mass diffusivity, solid mass fraction,liquid mass fraction, velocity, enthalpy, and mass frac-tion o f constituent a are defined as follows:

P = g~P~ + g~Pt; c = f s c s "~- f lCl ; k = g,k~ + gtkt

gsPs gtPlD = f~19~ + f~Dt; f ~ = , f t = - -

P P

V = f~Vs + j~Vt; h = f~hs + f thD f'~ = f~J~ + f t f f

[61

where gs and gt are the volume fractions o f the solid andliquid phases, respectively.

If the phase specific heats are assumed constant, thephase enthalpy for the solid and liquid can be expresseda s

hs = csT; hI = cIT + ( cs - Cl)Te + H [7]

where H is the latent heat o f the alloy and Te is the eu-tectic temperature.

The assumption o f permeability function in the mushyzone requires consideration o f the growth morphologyspecific to the alloy under study. In the present study,the permeability function analogous to fluid flow in theporous media is assumed, employing the Carman-Kozenyequationt174s~

gts 180K - c i = [8]cl (i - gt)2' d2

where d is proportional to the dendrite dimension, whichis assumed to be a constant and is on the o rde r o f 10 -2cm. The inertial coefficient, C, can be calculated fromt~91

C = O. 13gl 3/2 [9]

Closure o f the system o f conservation equations re-quires supplementary relations. With the assumption o floca l thermodynamic equilibrium, the required relationsmay be obtained from the equilibrium phase diagram.Neglecting the curvatures in the solidus and liquidus linesin the phase diagram, the solid mass fraction and solid-and liquid-phase compositions can be expressed as

f " = l " ~ - ~ p LT-----Z~mJ ; ~ = '1 + f ~ ( l c p - l ) '

[ 1 }.,,,f f = 1 + f ~ ( k p - 1) [10]

where T I is the liquidus temperature corresponding t o f ~,

Tmis the fusion temperature w h e n f~ = 0, and the equi-librium partition ratio kp is the ratio o f the slopes for theliquidus and solidus lines. It is noted that the assumptiono f loca l equilibrium does not preclude the existence o fa nonequilibrium condition o n a macroscopic scale.

In the previous governing differential equations(Eqs. [1] through [5]), there is a solid-phase velocity.The solid-phase velocity is associated with the move-ment o f precipitated solid equiaxed grains a n d / o r brokendendrites in the mushy zone. Generally, there are certainrelations between the solid-phase and liquid-phase ve-locities; t2°1 however, the exact relations are not knownat this t ime. A l s o , under the condition o f unidirectionalsolidification cooled from the bottom, the solid phase islikely to be stationary. Hence, we assume the solid-phasevelocity is zero in the present study, and the equationsare simplified accordingly.

III. N U M E R I C A L M E T H O D

The governing equations are in the general format , assuggested by Patankar, t211 for the numerical solution o fheat and fluid flow problems ( i . e . , they contain a tran-sient term, a diffusion term, a convection term, and sourceterms). Hence , any established numerical procedure forsolving coupled elliptic partial differential equations canbe used, with slight modifications fo r the source terms.In the present study, the equations were solved itera-tively at each time step using the control-volume-basedfinite difference procedure described by Patankar. t211 Afully implicit formulation was used for the time-dependentte rms, and the combined convection/diffusion coeffi-cients were evaluated using an upwind scheme. TheSIMPLEC algorithm was applied to solve the momen-tum and continuity equations to obtain the velocity field.At each time step, the momentum equations were solvedf i r s t in the iteration process using the most updated vol-ume fractions o f solid and liquid fo r the mixture density,the permeability function, and the mass fractions o f solidand liquid. T h e n , the energy equation was solved to ob-tain enthalpy, with which the temperature can be cal-culated. Similarly, the species equation was solved toobtain the concentration distribution. Next, the volumefractions o f solid and l iquid, the permeability, and themass fractions o f solid and liquid can be updated. Thisprocess was repeated for each iteration step. Fo r eachtime step, iterations were terminated when the maximumresidual source o f m a s s , momentum, energy, and spe-cies was less than 1 × 10-5. A line-by-line solver, basedon the tridiagonal matrix algorithm (TDMA), was usedto solve iteratively the algebraic discretization equations.The last five terms on the fight-hand side o f Eq. [2], thelast six terms on the fight-hand side o f Eq. [3], and thelast two terms on the fight-hand side o f Eqs. [4] and [5]represent the source terms and are treated according tothe procedure suggested by Patankar.mJ More details aboutthe computational procedure used in the present studycan be found in the previous articles, t~5,~61

As the governing equations are valid in the pure liquidand solid regions and the mushy zone , there is no needto t rack the geometrical shape and the extent o f each


region. Hence, a fixed-grid system was used in the nu-merical calculation. Figure 1 shows the grid system ofthe casting domain. A grid system of 46 x 61 pointswas utilized for the casting domain of 4 × 20 cm. Dueto the symmetry of the domain, only half of the gridpoints ( i .e . , 23 x 61) were used in the actual calcula-tion. The grid was slightly skewed to provide a h ighe rconcentration of nodal points near the bottom of the cast-ing, where larger temperature and solutal gradients werepresent. A finer grid system will certainly give more ac-curate computational results, but the CPU time neededfor the computation will also be increased. Hence, themesh size used in the present work is a compromise be-tween the accuracy and the computational cost . Duringsolidification, due t o density change, the total castingdomain decreases with time. The domain change is han-dled by the front tracking method, and a detailed de-scription can be found in previous publications. [~5,~6] Itis noted that the numerical method used in the presentstudy has been extensix, ely tested by the authors in manyother problems,t22] Hence, the results obtained from the

4.0 cmI-- =l

20.0 cm

i t i on

Fig. 1--Schematic representation of the physical domainand the meshsystem.

present study should be reliable. A variable time-stepsize is used in the calculation to provide optimum so-lution accuracy while maintaining numerical stability. Theinitial time-step size is 0.0001 seconds, and the maxi -mum time-step size is 0.025 seconds. The calculationswere executed on Apollo DN10000 and HP*-730 work-

*HP is a trademark of Hewlett-Packard Company, Colorado Springs,Co.

stations. Typical CPU time required for the calculationof each case is about 100 hours.

IV. R E S U L T S AND D I S C U S S I O N

A . Temperature, Velocity, and Solute Distributions

As shown in Figure 1, a unidirectional solidificationcan be induced by passing some coolant at a desired tem-perature, To through a chill placed at the bottom of thecasting while the other sides of the casting are insulated.Heat transfer between the chill and the casting is accom-plished via convection through an effective convectiveheat-transfer coefficient he. An A1-4.1 wt pet Cu alloyis used in the present s tudy, and its thermophysical prop-erties and the casting conditions are summarized inTable I.

As the casting will start t o solidify from the bottom,there will be an upward positive temperature gradient inthe casting. Hence, it is expected that a stable temper-ature field exists in the casting, and no natural convec-tion due to inverse temperature gradients is present. Also,during alloy solidification, heavier copper species willbe rejected starting from the bottom. Hence, it is ex-pected that more copper species will be accumulated inthe lower portion of the casting, creating a stable neg-ative solutal gradient. In other words, neither tempera-ture gradients nor solutal gradients will induce natural

Table I. Thermophysical Propertiesof AI-4.1 Pet Cu and Casting Conditions

Symbol Value Unit Reference

Cs 1.0928 J g - l K - l 23Ct 1.0588 J g-~K-1 23Dt 3 × 10-5 c m 2 s --~ assumedDs ~0 assumedks 1.9249 W cm-lK 23kt 0.8261 W cm-lK 23kp 0.170 23p, 2.65 g c m - 3 23Pt 2.40 g cm-3 23/3T 4.95 × 10-5 K-~ assumed/3, -2.0 assumed/z 0.03 g cm-~s-1 23H 397.5 j g-1 23Te 821.2 K 23Tm 933.2 K 23ff.o 4.1 PctTo 970.0 KTc 293.0 Khc 0.02 W cm-2K-1


convection in the casting. Hence, under the solidifica-tion conditions used in the present study, the only pos-sible driving force for the fluid flow in the casting issolidification contraction.

Figure 2(a) shows the isotherms in the casting at atime of 180 seconds. As expected, all isotherms are planar,and the temperature increases upward. Correspondingtemperature distribution along the vertical direction isgiven in Figure 2(b) . The liquidus and solidus fronts arealso shown in the figure, indicating the regions of solidphase, mushy zone, and liquid phase. T h e positions ofthe liquidus front and solidus front and the size of themushy zone as functions of time are shown in Figure 3.It is seen that the size of the mushy zone increases withtime as it moves away from the chill. Figure 4(a) showsthe velocity distribution in the casting at a time of 180seconds. As explained above, the fluid flow is causedonly by solidification contraction; hence, the directionof the velocity is toward the chill. The length of the arrowin the figure represents the magnitude of the velocity.T h e velocity is generally small compared with typicalvalues found in forced convection or natural convection.With the exception of the no-slip condition at the wall,the velocity in the mushy zone becomes nearly uniformin the horizontal direction, which is caused by the factthat an isotropic permeability function is assumed in thepresent study. It is noted that the velocity shown in thefigure is the continuum velocity, as defined in Eq. [6],which is weighted by the liquid fraction (the solid-phasevelocity is assumed to be zero in the present study). Amore detailed velocity profile along the centerline of the

953.116

937.042

920,967

904,893

' 888.819

- 872.744

856.67

- 840.596

- 824.521

808.447

P u r e L i q u i d

M u s h y Z o n e

P u r e S o l i d

15.0

10.0g&

e -

E

t215 . 0

0.0700 800 900 1000

T e m p e r a t u r e (°C)

(a) (b)

Fig. 2 - - ( a ) I s o t h e r m s : ( b ) temperature distribution along the verticaldirection at t = 1 8 0 s .

casting is given in Figure 4(b) . It is seen that the con-tinuum velocity is constant in the pure liquid p h a s e anddecreases abruptly toward the chill in the mushy zone.However, the actual liquid-phase velocity remains veryhigh in most portions of the mushy zone, compared tothe continuum velocity in the pure liquid phase .

15 ' ' ' ' I ' ' ' ' I ' ' ' ' l ' ' ' ' I . . . .

Pure Liquid

e . -

5 \O,d ~ Mushy Zone _

o0 50 100 150 200 250

Time (sec)

Fig. 3 - -Pos i t ions of the liquidus and solidus fronts as functions oft ime.

o

o~5

' l l | l | l l l l l l | l aI I I I ~ I I | I I I | I I

~ l l l l l l l l l l l l l '' l l l l l l l l l l l l l ~l l l l l l l l l l l l l l

I l l l l l l l l l l l l l t' l l l l l l l l l l l | l ol I | l I | l l l l l l l l

I I | l l l l l l l l l l l l' I | l l l l l l l l l l l jl l l l l l l l l l l l l l

4 1 1 1 1 1 1 1 1 l | I I I '~ I I l I L I I I I | I I I *l l l l l l l l l l l l l lJ l | l l l l l l l l l l l ~

' l | l l l l l l l l l | l II I I I I I I I I I I I I I

l l l l l l l l l l l l l l '~ | | | | l l l l l l i l | lI | l l l l l l l l l | l l

l l l l l l l l l l l i | l i[ i l l l l l l l l l | I I

| l l l l l l l l l l l lU L I 2 U L I 2 H L ~

P l l l l l l l l l | | I | l

IIiiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiiiii II | I I I I I i t 1 1 1 1 1 1

' ' ' ' I ' ' "

C o n t i n u u m V e l o c i t y

V = lsVs + fiVt

L i q u i d P h a s e

V e l o c i t y V]

I ' ' ' '

Pure L i q u i d

Mushy Z o n et

I/

.=_=.:.=.-_.~. . . . . . . . . . . . . . . . . . . . . . .

P u r e S o l i d

1 5 . 0

10.0

5.0

i i i , I , , i i I , , J , 0.0

0,00 2.50 5.00 7,50V e l o c i t y (10"3cm/sec)

(a ) (b)

6E£:oo

r-~

Fig. 4 - - ( a ) Ve loc i ty vectors: (b) velocity distribution along thecenterline i n the vertical direction at t = 1 8 0 s .

M E T A L L U R G I C A L T R A N S A C T I O N S A V O L U M E 2 4 A , A P R I L 1 9 9 3 - - 9 6 7

Figure 5(a) shows the isoconcentrations in the castingthat are planar, similar to the isotherms. However, thecopper concentration does not decrease monotonicallyfrom the bottom as the temperature field; instead, it de-creases, up to some location in the mushy zone , to aminimum concentration lower than the initial value, andthen increases to the initial copper concentration. Thecopper concentration along the vertical direction is shownin Figure 5(b), where it can be clearly seen that a positive-segregated (i.e., above the initial copper concentration)region near the chili and a negative-segregated (i.e., belowthe initial copper concentration) mushy zone are present.The higher copper concentration in the solidified castingnear the chilled surface is commonly called inverse seg-regation. There is an abrupt change in solute concentra-tion at both the solidus and liquidus fronts, and the copperconcentration in the pure liquid phase remains the sameas the initial value. Except in the region very near thesolidus front, the copper concentration in the mushy zoneis below the initial value. It is noted that the total soluteaccumulated in the positive-segregated region must beequal to the total solute depreciated in the negative-segregated region in order to satisfy the conservation o ftotal solute. It is also noted that a positive solutal gra-dient exists between the location with minimum soluteconcentration in the mushy zone and the liquidus front.However, the positive solutal gradient is caused by thedownward shrinkage-induced flow and, hence, cannotcreate natural convection. In fact, a negative solutal gra-dient exists for the liquid phase in the entire mushy zone ,which will be discussed next .

4 . 1 0 0 0 0

. . . . 4 ~ ; 0 ;3 . 9 5 9 4 8

Low

3 , 9 s g 4 8

4 , 0 4 6 0 3

4 . 1 7 5 8 5

4 . 2 6 2 4

1 , 4 7 8 7 8

. . . . I ' ' ' 1 . . . . I . . . .

Pure Liquid

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .~ . Mushy Zone

, , , . . . . , , , , 7 1 " ; , , 0.03.75 4.00 4.25 4,50 4.75

Solute Concentration (%)

15.0

10.0

5.0

t -O

E

8t -.tot~

(a) (b)

Fig. 5 - - ( a ) Isoconcentrations: (b) solute distribution along the ver-tical direction at t = 180 s.

In o rde r to explain the phenomena observed inFigure 5(b), detailed solute distributions in the castingare given in Figure 6. In Figure 6(a), the loca l soluteconcentration in the solid phase, ff~, is defined as themass fraction o f solute in a unit mass o f solid phase.Similarly, the loca l solute concentration in the liquidphase, i f , is defined as the mass fraction o f solute in aunit mass o f liquid phase. During solidification, copperspecies is rejected into the liquid phase, and as a result,the local solute concentration o f the liquid phase in themushy zone increases continuously toward the chill, asshown in Figure 6(a). As the solute concentration o f solidphase in the mushy zone is equal to the partition ratiotimes the solute concentration in the liquid phase, it alsoincreases toward the chill. In the pure liquid region, if/is constant, but in the pure solid region, ff~ increases to-ward the chili. The corresponding solid fraction, fs, andliquid fraction, f~, in the casting, as defined in Eq. [6],are shown in Figure 6(b). The liquid fraction increases,starting from zero at the solidus front, to units at theliquidus f ront and beyond. Similarly, the solid fractionincreases, starting from zero at the liquidus front, to unitsat the solidus front and toward the chill. The productf~ffl, the total amount o f solute in the solid phase per

10.0

7.5

5.0u,._=

2.5

0.0

: . . . . I ' . . . . ' ' ' I ':Pure So l id .~ Mushy Zone i

i \ ,i ii i

,i i i

. . . . ] 1 , , , , I , , i ,

(a )

' I . . . .

Pure Liquid

I , = = =

30.0

22.5

15.0=._-

7.5

0.0

1.0

0.5

0.0

7.5

I . . . . I ' " I "

Iit

fs i

(b)

. . . . I I . . . . I " ' 1 ' " Iii i! I

•.-= 5.0 ~ s f l ~ ~" ~= 5.0

2.5 iiii ~ _ _ ~ , , i[iii 2.5

0.0 , , i i . . i . . . . 0.00.0 2.5 5.0 7.5 10.0

Distance from Chi l l (cm)

( c )

Fig. 6 - - ( a ) Local solute distributions in the so l id and l iqu id phases:(b) distributions of so l id and l iqu id fractions: (c) total so lu t e in theso l id and liquid phases, all at t = 180 s.

1.0

l=

0.5

. . . . 0.0

. . . . 7.5


unit mass o f mixture (liquid and solid mixture in the mu-shy zone), increases toward the chili. On the other hand,the product, j ~ / , the total amount o f solute in the liquidphase per unit mass o f alloy mixture, decreases towardthe chill. The final solute concentration, f~, as given inEq. [6], is the sum o f these two products, i.e.,

f~ = f s f ~ + f ~ f f [11]which is plotted in Figure 5(b). Hence, from the abovediscussion, it is understood that the formation o f anegative-segregated mushy zone is caused by the flowo f solute-rich liquid in the mushy zone due to solidifi-cation contraction. However, near the chill at the bottomo f the casting, the velocity o f the fluid carrying soluteslows down (and becomes zero at the wall), causing thesolute to pile up, which leads to the inverse segregationin the casting.

Figure 7 shows a sequence o f solute concentrationprofiles during solidification. The higher solute concen-tration in the region near the chill increases with timeuntil the region is solidified, then the solute concentra-tion in the solid phase remains unchanged, because themass diffusion coefficient in the solid phase is negligible.

B . Factors Affecting the Solute Distribution

As the solid-phase velocity is assumed zero in thepresent study, solute redistribution in a solidifying cast-ing can be caused only by fluid flow and diffusion mech-anisms. If there is no fluid flow and no solute diffusion,the solute concentration should remain the same as theinitial value. Since shrinkage-induced flow is the onlypossible flow for the unidirectional solidification consid-ered in the present study, the effects o f shrinkage-inducedflow and solute diffusion on the solute redistribution arediscussed in the following.

1. I f the shrinkage-induced flow is neglectedThe shrinkage effect can be neglected in the mathe-

matical modeling by nullifying the corresponding terms

4.75 . . . .

f I . . . . I . . . . I '

1: 20 sec4.50 -

: 60 see

: 120 sec

=" 4 : 1 8 0 seco

~ 4 . 2 5e , -

t -OO

o 4 . 0 0 - 1c0

43

3.75 . . . . i . . . . i . . . . i ,0 . 0 2 . 5 5 . 0 7 . 5

Distance fromChill (cm)

F i g . 7 - - S o l u t e d i s t r i b u t i o n at d i f f e r e n t t i m e s .

1 0 . 0

in Eqs. [2] and [3] (i.e., the last term and the second-to-last term on the right-hand side o f Eq. [2] and Eq. [3],respectively)• The density is constant throughout thecasting, and the average value o f the solid and liquiddensities given in Table I is used in the modeling. U n d e rthis condition, there is no fluid flow in the alloy, andthe solute redistribution is caused only by the diffusionmechanism. As the copper species will be rejected intothe liquid phase during solidification, the solute concen-tration in the mushy zone is slightly h ighe r than that inthe pure solid phase, as shown in Figure 8. A maximumsolute concentration occurs in the mushy zone near theliquidus front. However, the difference in solute distri-bution for the entire casting is relatively small comparedwith that in Figure 7, because the species diffusion is avery slow process, evidenced by a small value o f massdiffusion coefficient given in Table I. It is noted that aslightly lower solute concentration in the pure solid phaseand the lower portion o f the mushy zone is compensatedby the h ighe r concentration in the upper portion o f themushy zone near the liquidus front, so that the overallconservation o f copper species is satisfied.

2. I f the solute diffusion is neglectedSolute diffusion in the casting can be neglected if the

solute diffusion coefficient D in Eq. [5] is assumed tobe zero. U n d e r this condition, Figure 9 illustrates thesolute distribution in the casting at a time o f 180 sec-onds. By comparing Figure 5(b) with Figure 9, nearlyidentical solute distribution in the casting is found. Thisimplies that u n d e r the conditions o f the present study,the solute redistribution is controlled mainly by the fluidflow due to solidification shrinkage. The result is alsoconsistent with the previous discussion concluding thatthe diffusion mechanism is not the primary factor con-tributing to the formation o f macrosegregation. Hence,it is concluded that the formation o f inverse segregation

Ao~=.,_

c"o

e'-Q)t -o(0

4.75

4.50

4 . 2 5

4.00

3.75 . . . . ' '0.0 1 0 . 0

= 1 ' ' ' ' I ' ' ' ' Ii ii ii ii ii ii ii iI Ii M u s h y Z o n e ii ii ii i! I

• !! Ii ii ii f c = f s f s ¢ + f t f t~ I!

• I= ii ii ii ii i

i iIi ii ii ii l , , , , I , , , , I

2.5 5.0 7.5

D i s t a n c e f r o m Chi l l (cm)

F i g . 8 - - S o l u t e d i s t r i b u t i o n w h e n sh r inkage - induced f l o w i s n e g l e c t e dat t = 1 8 0 s .

M E T A L L U R G I C A L T R A N S A C T I O N S A V O L U M E 24A, A P R I L 1 9 9 3 - - 9 6 9

4 . 7 5. . . . i i . . . . i , , i , , i . . . .

i ii ii ii ii ii ii i

4 . 5 0 i i~ [ MushyZone [

. i it - i i0 i i

" ~ 4 . 2 5 i i

o i io i ct a ct io#- f = I s i s + t l f l i

" " i i4 . 0 0 i i

o 9 i i

I Ii ii iii i

3 . 7 5 . . . . ; I , , ~ I , ,= ~ , I , , , ,

0 . 0 2 . 5 5 . 0 7 . 5 1 0 . 0

Distance from Chill (cm)

F i g . 9 - - S o l u t e d i s t r i b u t i o n w h e n s o l u t e d i f f u s i o n i s n e g l e c t e d at t =1 8 0 s .

t -O

t -

O

c-

O

00

--'s

"50 9

, , a Ii

4 . 5 0

4 . 2 5

4 . 0 0

4 . 7 5. . . . I ' I ° ' I

i

i= ii ii ii ii ii iii Mushy Zone ii ii i

I Ii ii ii ii i

f= = fs fO + f l f l t z iii

ii ii ii ii ii ii i• iI

i I3 . 7 5 , [ I , , , I , , i , , I . . . .

0.0 2.5 5.0 7.5 10.0Distance from Chill (cm)

Fig. 10--Solute distribution when both shrinkage-induced flowandsolute diffusion are neglected at t = I80 s.

near the bottom o f the casting is caused by solidificationcontraction. It is noted that solute diffuses from locationso f higher concentration to locations o f lower concentra-tion, while solute transportation by fluid flow dependson the direction o f the fluid flow. In the present study,the fluid flow is caused only by shrinkage, and the solute-rich liquid flows toward the bottom o f the casting. How-ever , the solute diffuses upward from the higherconcentration at the bottom o f the casting. The net resultis that the maximum segregation at the bottom o f thecasting in Figure 9 is slightly higher than that in Figure5(b).

3. I f both the solute diffusion and shrinkage-induced.[low are neglectedIn this case, both the solute diffusion coefficient in

Eq. [5] and terms related to shrinkage effect in Eqs. [2]and [3] are set to zero. U n d e r this condition, it is ex-pected that there is no solute redistribution, and the sol-ute concentration in the entire casting should be equal tothe initial value.

Although microscopically the solute will be rejectedfrom the solidifying dendrites o r solid grains into theinterdendritic liquid, there is no mechanism to transportthe solute; hence, the average solute concentration (i.e.,from the macroscopic point o f view) should remain thesame as the initial value. Figure 10 shows the calculationresult from the mathematical model proposed in the pres-ent study, which is as expected. In fact, the result shownin Figure 10 provides an excellent opportunity to test thecorrectness o f the mathematical model and its associatednumerical technique established in the present study, atleast in the limiting case.

C. Comparison with the Model by Flemingsand Nereo

For comparison, the "local solute redistribution equa-tion" derived by Flemings and Nereo[s] is rewritten inthe following:

vOg, 1 - fl . VT~ g_,- - = - 1 + ! [ 1 2 ]

where /3 is the solidification contraction, e is the rate o ftemperature change, and VT is the temperature gradient.Equation [12] was derived by combining the conserva-tion o f solute mass and the conservation o f total mass.The solute diffusion was neglected in the derivation. Itis noted that in order to use Eq. [12], the data regardingvelocity distribution, temperature gradients, and the rateo f temperature change must be obtained first. Flemingse t a l . [ 5 ' 6 ' 7 ] obtained the data through experimental mea-surements o r by assumptions, and they used the equationto successfully predict the formation o f inverse segre-gation and banding segregation. In the present study, thevelocity and temperature distributions calculated fromthe momentum and energy equations (i.e., Eqs. [1]through [4]) without considering the species caused theconvection term in Eq. [3], and the species equation,Eq. [5], will be used to substitute into Eq. [12]. Forconvenience, the above equation is integrated to obtainthe following form:

[ _ ( l - k p ~ gt(l+V'_VeT)-Idg,]f f = ~ ° exp L \ l - f l / f ~ g,J [13]

Hence, the transient solute distribution in the castingcan be obtained by substituting the transient velocity,temperature gradients, and the rate o f temperature changecalculated from Eqs. [ t ] through [4]. Figure 11 showsthe solute distribution calculated by Eq. [13] and by thecomprehensive model proposed in the present study at atime o f 100 seconds. It is seen that good agreementsbetween these two methods are obtained. The small dif-ference could be caused by the neglecting o f solute dif-fusion and the decoupling o f the species equation fromthe momentum and energy equations in Eq. [13].

9 7 0 - - V O L U M E 24A, A P R I L 1993 M E T A L L U R G I C A L T R A N S A C T I O N S A

4 . 7 5

4 . 5 0

t -O

~ 4.25e--IlJ

t-O

~ 4.00

. . . . I . . . . I . . . . I "

T ime = 100 s e c

- - Present model

. . . . F lemings ' equat ion (s}

3 . 7 5 . . . . I . . . . i . . . . i . . . .

0 . 0 2 . 5 5 . 0 7 . 5 1 0 . 0

D i s t a n c e f rom Chi l l (cm)

Fig. 11--Comparison between the present s tudy and the results ob-tained from the "local solute redistribution equation" by Flemingsand Nereo[5] at t = 100 s.

5 . 2 5 . . . . I . . . . I . . . . I . . . .

fo~ = 4.6% :

Present wo rk5.00 . . . . Calculated, Flemings &Nereo(7)-

• Measured, F lemings & N e r e o(z)

8 4 .75

e -

c .m fo*X= 4.1% :8 4 . 5 0t \~ ,~ - - - Present wo rk

"5 . . . . . Calculated, Kato &Cahoon(B)"6 •~ [ . . ~ . " Measured ' Ka t ° & Cah°°n(B)425I . . . . . . . . . . . . . . . . . . . . . . . .

400 / i i...... / i , i / ....... , 7 ....0 . 0 2 . 5 5 . 0 7 . 5 1 0 . 0

Distance from Chill (cm)

Fig. 12--Calculated results and experimental data for the solutedis-tributions in the completely solidified castings.

D. Final Solute Distribution in the Solidified Casting

The final solute distribution after the casting is com-pletely solidified is g iven in Figure 12. It is noted thatonly the solidified portion of the casting near the chillis plotted in the figure, and still a negative-segregatedmushy zone exists at a distance further away from thechill. The experimental data and the calculated results ofFlemings and Nereo m for 4.6 pct initial copper concen-tration and the results of Kato and Cahoon[8] for 4.1 pctinitial copper concentration are also shown in the figure.It is seen that excellent agreements between the predic-tions of the present study and the results of previous studiesare obtained. It is noted that the previous theoretical pre-dictions are based on much simpler and restricted models,and they do not show the transient phenomena as thepresent m o d e l does. From Figure 12, it can be concludedthat the mathematical m o d e l and its associated numericaltechnique established by the present study are appropri-ate. As the solid-phasevelocity is assumed t o be zero inthe present s tudy, it can also be said that under the uni-directional solidification cooled from the bottom, the as-sumption of a stationary solid phase is acceptable.

E. Effect o f Heat Extraction Rate onInverse Segregation

The effect of the heat extraction rate on the formationof inverse segregation can be studied by varying the con-vective heat-transfer coefficient between the chill and thecasting. Figure 13 shows the inverse segregation for threedifferent heat-transfer coefficients. It is seen that a higherheat-transfer coefficient increases the solidification rateand decreases the severity of inverse segregation. Thisis because a higher solidification rate decreases the sizeof the mushy zone, and the time required for solute-richliquid in the mushy zone t o flow "against" the chill is

4 . 7 5

4 . 5 0

8~4 .25

8

~ 4.00i

' ' ' ' I ' ' ' ' I ' ' ' ' I ' '

- - he = 0.02 Wcm '2Kq, T ime = 210 s e c

. . . . he = 0.06 Wcm '2K"1, T ime = 4 0 s e c

~ - - - - .hc=0.18Wcm'2K "1, T i m e = 1 0 s e c

",I',, " ' ~ _ / /\./',,, ,1

3 . 7 5 , , , , I , . I , , , I

0 . 0 2 . 5 5 . 0 7 . 5 1 0 . 0

D i s t a n c e f rom Chi l l ( cm)

Fig. 1 3 - - E f f e c t of heat-extraction rate at the bot tom on the inversesegregation.

decreased. In other words, less solute can be carried bythe fluid flow in the smaller mushy zone t o pile up nearthe chill.

V. C O N C L U S I O N S

A numerical simulation of the formation of macro-segregation has been conducted for a unidirectional so-lidification of aluminum-copper alloys cooled from the


bottom. The mathematical model, based on the contin-uum formulation, allows a complete coupling betweenthe momentum, energy, and species equations and asimple treatment o f the casting domain , including thesolid phase, mushy zone, and liquid phase. Natural con-vection due to temperature and solute gradients andshrinkage-induced fluid flow are considered in the pres-ent model. Transient temperature, velocity, and speciesdistributions in the solidifying casting are calculated. Theresults from the present study indicate that the mathe-matical model and its associated numerical technique es-tablished in the present study can accurately predict theformation o f inverse segregation. The major conclusionscan be summarized as follows:1. A positive segregation o f species in the casting near

the chill (inverse segregation) is predicted. This seg-regation is accompanied by a moving and negative-segregated mushy zone.

2. The formation o f inverse segregation and negative-segregated mushy zone is caused by the fluid flow o fsolute-rich liquid in the mushy zone due to solidifi-cation contraction. The effect o f shrinkage-inducedflow on the solute redistribution in the mushy zoneis much more significant than the solute diffusionmechanism.

3. A higher heat-extraction rate at the bottom o f thecasting increases the solidification ra te , decreasingthe size o f the mushy zone , and lessens the severityo f inverse segregation.

4. The predicted inverse segregation compared favor-ably with the published experimental data and theprevious theoretical results using much simpler andrestricted models.

CCCldDfghHkk~KPtTrm

u

vVv~X, y

NOMENCLATURE

specific heatcoefficient, defined in Eq. [9]permeability coefficient, defined in Eq. [8]dendrite arm spacingmass-diffusion coefficientmass fractionvolume fraction o r gravitational accelerationenthalpylatent heatthermal conductivityequilibrium partition ratiopermeabilitypressuretimetemperaturefusion temperature at zero soluteconcentrationvelocity in the x-directionvelocity in the y-directionvelocity vectorrelative velocity vector (V~ - Vs)Cartesian coordinates

Greek Symbols

fls solutal expansion coefficientfir thermal expansion coefficient

/xP

rate o f temperature changedynamic viscositydensity

Subscripts

0 initial valuec chille eutecticl liquid phasem fusionr relative to solid velocitys solid phase

Superscript

a constituent o f al loy

ACKNOWLEDGMENTS

The authors are grateful for the partial support by theAir Force Office o f Scientific Research under ContractNo. F49620-88-C-0053/SB5881-0378.

REFERENCES

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vol. 239, pp. 1449-61.6. M.C. Flemings, R. Mehrabian, and G.E. Nereo: Trans. TMS-

AIME, 1968, vol. 242 , pp. 41-49.7. M.C. Flemings and G.E. Nereo: Trans. TMS-AIME, 1968,

vol. 242, pp. 50-55.8. H. Kato and J.R. Cahoon: Metall. Trans. A, 1985, vol. 16A,

pp. 579-87.9. I. Ohnaka and M . Matsumoto: Tetsu-to-Hagand (J. Iron Steel

Inst. Jpn.), 1987, vol. 73 , pp. 1698-1705.10. I. Ohnaka and M . Matsumoto: Trans. Iron Steel Inst. Jpn., 1986,

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1987, vol. 30, pp. 2171-87.13. C. Beckermann and R. Viskanta: PCH, PhysicoChem. Hydrodyn.,

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vol. 35, pp. 1771-78.17. P.C. Carman: Trans. Inst. Chem. Eng., 1937, vol. 15, pp. 150-66.18. K. Kubo and R.D. Pehlke: Metall. Trans. B , 1985, vol. 16B,

pp. 359-66.19. G.S. Beavers and E.M. Sparrow: J . Appl. Mech., 1969, vol. 36,

pp. 711-14.20. J. Ni and C. Beckermann: Transport Phenomena in Material

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21 . S.V. Patankar: Numerical Heat Transfer and Fluid Flow,Hemisphere. New York, NY, 1980, pp . 96-102.

22 . K.C. Chiang: Ph.D. Thesis, University of Missouri-Rolla, Rolla,MO, 1990.

23 . R.D. Pehlke, A. Jeyarajan, and H. Wada: Summary o f ThermalProperties f o r Casting Alloys and Mold Materials, Report No.PB83-211003, National Technical Information Service,Washington, DC, 1983, pp . 81-93.


modeling of solute redistribution in the mushy zone...

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