Macrosegregation patterns in multicomponent Ni-base alloys

,. . . . . . . .

CRYSTAL G R O W T H

Journal of Crystal Growth 177 (1997) 145 161

ELSEVIER

Macrosegregation patterns in multicomponent Ni-base alloys S.D. F e l i c e l l i a'~, D . R . P o i r i e r b'*, J.C. H e i n r i c h ~ Department o/'Aeros'pace and Mechanical Engineering, The UniversiO' qf Arizona. Tucson, Arizona 85721, U5;4 b Department ~?[Materials Science and Engineering, The Universi O' of Arizona, Tucson, Arizona 85721, USA Received 15 July 1996: accepted 18 October 1996

Abstract

A mathematical model of the dendritic solidification of multicomponent alloys, that includes thermosolutal convection and macrosegregation, is presented. The model is an extension of one previously developed for binary alloys. Numerical simulations are given for ternary and quaternary Ni-base alloys, and the evolution of macrosegregation during solidification is studied. The results show that the segregation patterns vary greatly with cooling conditions, adopting several shapes and levels of intensity. Calculations of segregation in rectangular molds and in molds with smooth and abrupt variations of the cross sections exhibit significant differences in the distribution of macrosegregation due to the change in geometry. In addition, the segregation patterns are found to be particularly sensitive to the values of the equilibrium partition coefficients of the alloy components.

1. Introduction

A considerable number of works on numerical simulation of alloy solidification, with emphasis on convection and the formation of macrosegregation defects, have appeared since 1987. Most of these works have dealt with binary alloys and have provided ample support that segregation defects are formed because of thermosolutal convection occurring in the melt and in the mushy zone during

* Corresponding author. Fax: + 1 520 621 8059. 1 Visiting Scientist, on leave from Centro Atomico Bariloche, 8400 Bariloche. Argentina, and fellowship holder of Consejo Nacional de lnvestigaciones Cientificas y Tecnicas de la Reptiblica Argentina.

solidification [1 6]. Experimental reports with metallic systems and ammonium chloride solutions have also concluded that convection is the major cause of essentially all forms of observed macrosegregation [7 9]. When cooling a binary alloy from below, even though the temperature profile is gravitationally stable, strong convection may develop, especially when solute that is lighter than the solvent is rejected to the liquid, producing channels in the mushy zone that leave behind segregates known as freckles [2]. In a multicomponent alloy, however, each alloying element contributes its change to the density. Some components decrease the alloy density, while others increase it, depending on their respective solutal expansion coefficients. In addition, the phase diagram of the alloy is much more complex, dictating different rates of solute rejection or

0022-0248/97/$17.00 Copyright i ~ 1997 Elsevier Science B.V. All rights reserved Pll S0022-0248196)0 1069-X

146

XD. Felicelli et ul. / .hmrnal of Oystal Growth 177 (1997) 145 161

absorption for each one of the components during solidification. These characteristics make the formation of segregation less predictable than in binary alloys, and can produce a typical convection modes that lead to segregation patterns not observed before in the binary systems. Predictions of macrosegregation in alloys with two or more solutes date back to 1970, which considered simplified models of convection in the mushy zone only [10, 11]. Models for multicomponent alloys that include thermosolutal convection, both in the liquid and the mushy zone, are just starting to emerge [12, 13] and will probably be widely applied due to the fact that most alloys of practical interest comprise many elements. The modeling of multicomponent alloys requires different algorithmic strategies than the ones used for binary alloys. In particular, knowledge of the temperature in the mushy zone does not automatically provide the equilibrium liquid concentration, a feature on which most binary models rely. In a multicomponent alloy, many combinations of the liquid concentrations can have the same liquidus temperature, making it more difficult to calculate the solidification path. The situation becomes further complicated if the topological details of the phase diagram, like valleys, invariant points, secondary solid phases, etc., are considered. in the present work, an algorithm to model the solidification of alloys with any number of components is devised. Emphasis is made in the capability to calculate the macrosegregation produced by the convection of several alloy elements, rather than in the topological complexities of the phase diagram. The model is applied to Ni-base alloys, motivated by their practical use in aircraft engines.

2. Solidification model

The mathematical model for the solidification of multicomponent alloys is an extension of an earlier model of binary solidification developed by the authors [1]. The dendritic mushy zone of the alloy is treated as a porous medium of variable porosity. The porosity is a function of the volume fraction of liquid, which varies from zero (all-solid region) to one (all-liquid region). A unique set of equations,

governing the conservation of mass, momentum, energy and solute, is solved in the whole domain, with no tracking of internal interracial conditions. The equations in the mushy zone automatically reduce to the governing equations for the all-liquid or all-solid regions as the fraction of liquid varies from one to zero, respectively. This type of a model, known as a continuum model, has been used extensively in simulation of solidification

[3 6]. In what follows, the conservation equations and thermodynamic relations used in the model are briefly described, extending only on those aspects relevant to multicomponent solidification. The reader is referred to Refs. [1] and [14] for further details on topics common to the binary model. The following basic assumptions are made: (a) Only solid and liquid phases may be present, i.e., no pores form. (b) The liquid is Newtonian and incompressible, and the flow is laminar and two dimensional. (c) The solid and liquid phases have the same thermal properties and densities. (d) There is no diffusion of solute in the solid phase. (e) The thermal properties are constant, and the Boussinesq approximation is made; hence the density is constant except in the body force term of the momentum equation. (f) The solid is stationary. Additional assumptions are invoked during the course of the paper. 2.1. M a s s and m o m e n t u m conservation

With the above simplifications, the equations of continuity and momentum can be written as follows [15, 16]: V'u =0,

~=

(1)

+----u

Po K.,.

Po

V~u + - - - -

Po ~x

,,

-

4, . - v

+--

Po

q~,q~,

(2)

147

S.D. Felicelli et al. / J o u r n a l o f C o ' s t a l G r o w t h 177 ( l 997) 145 - 161

Z

\¢/+

-

ouv

v

+ -Po- ~Y

Po

+

(3)

The variables are defined in the nomenclature. The density in the body force term varies with the temperature and the liquid concentration of the alloy components, in the form

p = p o I l + flT(T-- TR)+

j=IL[~Jc(CJ--cJ) 1

(4)

a small role in the formation of segregation defects [1, 2]. Therefore, even if the individual concentrations are large, the assumption is believed to remain valid. Eq. (5) must be complemented with an expression to calculate Cj in the mushy zone. Using the definition of mixture concentration, C, for each one of the solutes: C j = ~bC~+ (1 - ~b)C~.

The average concentration of the solid, C~, is computed integrating the microsegregation path with a trapezoidal rule: 1 -

where _

fir

1 ap

Po ~T

=

1 ap

2.2. Conservation of the alloy components It is assumed that the diffusion of each alloy element in the liquid is simply Fickian; that is cross-effects are not considered. Hence, an equation of solute conservation can be written independently for each one of the alloy solutes. Recalling that the diffusion in the solid phase is neglected and the solid is stationary, each solute has a conservation equation comprising the diffusive and convective transport in the liquid phase, and the accumulation:

U C j d 0 -= 1

l.i

~b'

(7)

(8)

where n indicates the time level, and the partition coefficients k~ are assumed to be constant. In case of remelting, Eq. (8) is replaced by lj.,+ 1 = H ( 0 , + 1),

(9)

where H is a table function that gives the value of I j for old values of 0- This function is constructed during solidification, saving the history of P at selected values of 0, for all nodes in the mesh that have undergone solidification. In Eq. (9), linear interpolation is used to compute F'" + 1 for ~b's different than the saved values. Combining Eqs. (6)-(8), the following relation can be derived for the liquid concentration during solidification:

~jn+X Cj,,+I = _ /

--

Ij,,+l

!bjFj.nlAn - - 2 ~. ~ 1 ~,W

_

_

(~n+1)

qS"+1 + ½kJ(~b, _ 0,,+ ,)

-

(10)

while during remelting it is computed as

(j,,,+ , _ H(O,+ 1) C~,"+1 =

) - u.

f,1

I ,i''+x = I .i'" + ½U(Cj'" + Cj ""+ 1)(0" - 0 "+ '),

P~o~C--~/"

In Eq. (4), the superscript " j " indicates an alloy component, and TR and C~ are reference values for which p = Po. Note that while fit < 0 (in general), the individual solutes can either decrease (/?~: < 0) or increase (fl{: > 0) the density of the alloy.

OCj - v . (DJOVC ~t

(6)

0,+1

(11)

(5)

where the superscript "j" refers to an individual alloy component. The assumption of negligible interdiffusion is best supported if the concentrations of the elements are small relative to the amount of solvent, which includes most cases of practical interest. Futhermore, it is known that in metal alloys, transport of solute by diffusion is very small when compared to convective transport, and plays

Both Eqs. (10) and (11) require the knowledge of the fraction of liquid at the current time level. A way to calculate ~b"+1 is described in the next section.

2.3. Calculation of the.fraction of liquid At this point, the information provided by the phase diagram of the alloy in consideration is used.

S.D. Felicelli et al. .,'Journal of Co,stal Growth 177 (1997) 145 161

148

For this purpose, the liquidus surface is expressed as N

TI-= TM + E mJcJ"

at the eutectic temperature, where solidification at constant T = TEUT is assumed, i.e., no troughs or valleys are considered.

(12)

2.4. Energ), equation

r= l

where TL is the liquidus temperature, TM is the extrapolated melting temperature of the pure solvent, and the tnr are coefficients which in general are functions of Cj In the particular case of constant mr, they represent the change in liquidus temperature with solute/, that is mr = aT/aC-i/. In this work, all the numerical examples are done with constant mr, however the model can accommodate nonconstant mr as long as the liquidus surface is expressed in the form given by Eq. (12). From Eqs. (6) and (7) of the previous section. C r = ~C~ + I r.

(13)

Assuming constant thermal properties, and both diffusive and convective transport with latent heat release, results in the energy equation [14]: aT at

- -

--

~V2T

L aq5 c at

-

u.

V T .

(17)

As it was done in the binary model [1], this equation must be reformulated to make the latent heat term implicit, in order to avoid stability problems in the final algorithm. To eliminate the fraction of liquid from Eq. (17), differentiate Eq. (13) with respect to time, yielding

Multiplying Eq. (13) by mj, summing over all the solutes and substituting Eq. (12), yields:

aC j

~/?= ~ - 1 mi(Ci - I i) T L - TM

The time derivative of the integral F can be approximated, using Eq. (8), by

(14)

When Eq. (14) is evaluated at time level n + 1, Eq. (8) can be substituted into Eq. (14) to estimate the value of F] + i and an improved equation for the fraction of liquid is obtained:

A,,+ 1 ~bn + l

__ B n + 1 ¢ n

Tin+l_ T M - B "+1

(15)

with N

A "+1

~ mr(c r''+1 - U'");

OCj .a d) al r - 0 ~[- + C] ~t + ~[.

air at -~

krCj a6 ~-"

(18)

(19)

Although this is not true for remelting, numerical experimentation has shown that the use of Eq. (19) during remelting does not introduce a significant error, as long as the time step is not too large. On the other hand, keeping the term al/& explicit substantially deteriorates the convergence properties of the algorithm. Substituting Eq. (19) into Eq. (18) and rearranging yields

j= 1

(1 - k r) C{ 04)

1 N

B"+l =

E

mrU(C/""+l+ Ci'").

(16)

-

OC / at

OC~ ,;b a t

(20)

2r = 1

Since the model assumes equilibrium solidification (no undercooling), the liquidus temperature in the mushy zone must be equal to the temperature resulting from solving the energy conservation equation, to be described in the next section. In this way, the model is closed and all variables are calculated. The same strategy employed in the binary model [1] was used here to calculate the fraction of liquid

In the mushy zone the temperature must equal the liquidus value, T = TL(CJ/), therefore, aT aC~ ;=1 aC/j at

aT at"

(21)

Multiplying Eq. (20) by aT/aCL summing over all the solutes, and substituting Eq. (21) and the resulting expression into Eq. (17), the final form of

S.D. Fe]ice]li el aL / J o m w a l (?f C~ystal Growth 177 g1997j 145 161

the energy equation is obtained: 1 -

C

E j :NI ( I

-- kJ)mJCj

~

~V2r

~t

-

c ~:

1(1 - kJ)m~C~ - u" AT,

(22)

where m j = ~T,/~C~[c,~ is now the local slope of the liquidus surface in the direction of C~. Note that Eq. (22) is not uniformly valid t h r o u g h o u t the domain, because it does not reduce to Eq. (17) when ¢ = 1. Therefore, the terms proportional to L/c apply only to the m u s h y zone. 2.5. Algorithm and numerical method

The equations described in the previous sections are solved sequentially, and an iteration is performed to achieve convergence at each time step. To advance from time t", at which all conditions are known, to time t" +1 = t" + At, the following steps are taken, where the dependent variables are computed using the latest available values of all other variables on which they depend. 1. At time t", variables u", v", T", etc., are known. 2. Advance to time step t "+ 1. Set i = 0, u,-"_+~ = u ~, ,n + 1 z Un, -Ti n=+o 1 = T", etc., where i is an iteration ri=o index. 3. C o m p u t e Ui+l, " + 1 ~/+1 "" + 1 from Eqs. (1 ?(4). T"+ 1 from Eq, (22). 4. C o m p u t e -~+1 5. F o r nodes in which li+m" +it ~< TL(cj. ~,+ 1), calculate v,i+l'a"+lfrom Eq. (15). 6. For nodes in which v,i+l ,j,,+l < 1, calculate cj., /.i++ ~ from Eqs. (10) and (11). --j.n+ l(__t,-,i,n+ 1 7. C o m p u t e Ci+l , - , ~ . i + 1 in the liquid region) from Eq. (5). 8. If ]kb~':lx --~bn+ 1[] < g and ~"~i+lmJ'n+11--C'/'n+ 1H < g , then Calculate I i''+ 1 and C~.'"+ 1 from Eqs. (7)-(9). Set u" + 1 t~i+l, ,,, + 1 v" + 1 ~--- Ui+I, , + 1 T" + 1 ~ Tn ~ i ++l ,1 etc. Setn=n+ Otherwise Set i = i + 1 G o to step 4.

1; go to step 2

149

Steps 4 8 are repeated iteratively during each time step until convergence in the fraction of liquid and in the solutes' mixture concentrations are achieved. The velocities are calculated only once per time step because they have negligible sensitivity to small changes in the rest of the dependent variables [14]. More details a b o u t other considerations in the algorithm, such as the treatment of remehing, the solidification at eutectic temperature, and the selection of the time step and mesh size, are given in Ref. [1]. The model was implemented using a finite element method based on the bilinear Lagrangian isoparametric element. Since the pressure is not needed in the calculations, it was eliminated using a standard penalty function formulation. A Petroy Galerkin technique, in which the weighting function is perturbed in the convective terms, was used in the convective terms of the transport equations. A comprehensive description of the numerical method can be found in Ref. [1].

3.

Results

and

discussion

The m u l t i c o m p o n e n t model was used to simulate the solidification of Ni-base alloys. A systematic study was carried out to predict the macrosegregation patterns in the following cases: 1. Change in segregation as a function of growth rate in a small mold. 2. Change in segregation as a function of growth rate in a large mold. 3. Change in segregation as a function of the partition coefficient of AI. 4. Change in segregation as a function of the cross section of the mold. The calculations for cases 1 3, were performed for a ternary N i - A I - T a alloy of composition 5.8 w t % A1, 15.2 w t % Ta, and balance Ni. For case 4, the simulations were done using a quaternary of composition 6 w t % AI, 6 w t % Ta, 8 w t % Cr, and balance Ni. The physical properties used in the calculations are given in Table 1. The same properties were a d o p t e d for AI and Ta in both alloys, with the exception of the equilibrium partition coefficients. The permeability functions K,., K,. given in Refs. [2, 14] were used in this work.

150

S.D. Felicelli et al. /Journal of Crystal Growth 177 (1997) 145 161

Table 1 Estimates of thermodynamic and transport properties used in the simulations Property

Ret2

Reference concentrations (Cases 1-3): CRA~= 5.8 w t % AI; C~" = 15.2 w t % T a

Reference concentrations (Case 4): C M=6wt%At;C~"=6wt%Ta;C

cr=8wt%Cr

Reference temperature (Cases 1 3): TR = 1685 K Reference temperature (Case 4): TR = 1691 K

[2O] [20, 21]

Eutectic t e m p e r a t u r e : TE = 1560 K

[22]

Extrapolated melting point of solventa: TM = 1754 K Equilibrium partition ratios (Cases 1 3): k A' = 0.54; k ~" = 0.48

[22]

Equilibrium partition ratios (Case 4): k AI = 0.89; k x" = 0.61; k cr = 1.0

[23]

Change of liquidus temperature (m r = OTL/~CS~)

[21] [21] [21]

mAJ= --5.17K(wt% AI) I mcr=-2.11K(wt%Cr) 1 mra = - 2.55 K ( w t % T a ) 1

Thermal expansion coefficient:/~r = - 1.15 x 10 - 4 K Solutal expansion coefficients: []~l = 2.26 x 10 2 ( w t % AI)- 1

[24] [24] [24] [24]

/IcCr= 2.26 X 1 0 - 3 ( w t % Cr) 1 tq[" = + 3.82 x 10 - 3 ( w t % T a ) -1

[25]

Viscosity:/~ = 4.90 x 10 - 3 N s m -2 Specific heat: c = 660 J k g ~ K - 1 Latent heat: L = 2.90 x l0 s J k g Thermal conductivity: ~,- = 80 W K J m Density (Cases 1-3): Po = 7365 k g m - 3 Density (Case 4): P0 = 6900 k g m - 3 Diffusion coefficient in liquid: D = 5 x 10 9 m z s - 1

[26] [27] [28] [24] [24]

Extrapolated melting point of solvent is based on the liquidus temperature given b y TL=[1685--2.11C}r--2.55(C~"

15.18) -- 5.17(CAI -- 5.81)] K

where 1685 K is f r o m Ref. [ 2 0 ] for Ni - 15.18 w t % T a 5.81 w t % AI, and the coefficients are f r o m Ref. [ 2 1 ] . b E x t r a p o l a t i o n of the thermal conductivity of solid nickel to its melting point (Ref. [28]).

The simulation procedure, common to all cases modeled, consisted in cooling from below a mold cavity filled with an all-liquid alloy of the nominal composition. Initially, there is a linear temperature profile, cold at the bottom and hot at the top (Fig. 1). Cases 1-3 were simulated in a mold of rectangular shape, while in Case 4 the lateral walls of the mold were not straight. Geometries and dimensions of the domains of the four cases are shown in Fig. 2. At time zero, a constant cooling rate, r, is applied at the bottom boundary, and a temperature gradient, G, equal to the initial tem-

perature gradient, is imposed at the top. Convection is forced to start right away by initially prescribing a small random perturbation to the solute concentration fields. The evolution of the solidification is followed until considerable segregation has developed or, in some cases, until all the liquid has solidified. A finite element discretization of 20 elements in the horizontal direction and 30 elements in the vertical direction was used in the calculations with small molds (Cases 1, 3 and 4). The mesh density was quadrupled to 40 x 60 elements in the large

S.D. Felicelli et al. , J o u r n a l o f C w s t a l G r o w t h 177 (1997) 145

3T ~ C/ u=v=0oro.n=0--=G--=0 'by 'c3y

u(x, y,O) = v(x, y,O) = 0 T(x, y,0) = TL(C~) + Gy C,' (.~,y,O) = C/, eO(x, y,O) : 1 U=v=O

u=v=0

--=0 On

aT

or= 0 cln

C~ = 0

a C/ = 0 On

g

g .=v=0. or=~ oC/=o at ' by F i g . 1. S c h e m a t i c boundary

calculation

domain,

showing

initial

and

conditions.

mold (Case 2). The computer runs were carried out on several machines to test the performance and portability of the code: a Convex C4620 minisupercomputer, a Sun Sparc 4, a Silicon Graphics Crimson, and a cluster of six HP-735. In one HP-735, it took about 10 rain of CPU time to simulate 1 min of solidification in the 40 x 60 mesh. Case 1: Segregation versus 9rowth rate in a small mold. A rectangular mold of dimensions 7 mm wide and 20 mm high was used for this case. Two sets of calculations were done. In the first set, the temperature gradient was held constant and the cooling rate varied. In the second set, different gradients were used at a constant cooling rate. The values selected for the two sets are listed below: First set: G = 2000 K/m; 0.17 < r < 0.57 K/s. Second set:

l0 3

< G _< 104 K/m; r = 0.28 K/s.

Both sets of calculations gave similar results, in the sense that the patterns and strength of segregation seemed to respond to the growth velocity

161

151

rather than to either the temperature gradient or cooling rate alone. Fig. 3 shows shaded contour plots of the volume fraction of liquid and the mixture concentration of AI at two different times during the soliditication of the ternary alloy. The applied cooling rate at the bottom boundary was 0.28 K/s and the initial ten> perature gradient was 2000 K/m, which yielded an average growth rate of approximately 8.9 x 10 s m/s. At time 80 s the plot of fraction of liquid (Fig. 3at indicates the onset of formation of a channel within the lnushy zone, near the center of the mold. The channel region remains liquid because it has a higher AI content, which decreases the melting temperature. This is clearly seen in Fig. 3b, which shows a jet of Al-rich liquid flowing upward into the all-liquid zone. The upward flow is produced by the buoyancy of the light element A1 which, for this alloy, overcomes the effect of tile heavier element Ta (A1 decreases the alloy density while Ta increases it). Similar solute-rich plumes are observed in transparent aqueous ammonium chloride solutions, which have been often used to study the solidification of alloys [ 17, 18]. At 200 s, the mushy zone has filled over ,~ of the mold (Fig. 3c), a second channel developed, and both turned to the left wall, merging into a single channel. Another channel is also beginning to grow on the right wall. Due to its lower melting point, the alloy within the channels remains liquid for a longer period of time, and consequently accumulates higher levels of solute before solidifying. In Fig. 3d it can be observed that even though the channels still have a high fraction of liquid, the AI content has already gone up to over 8 wt%. from an initial value of 5.2 wl%. When solidification is complete, these regions leave behind localized segregation defects, some of them known as "freckles" because of the spot-like appearance in horizontal cross scctions of the casting. A comparison of how the segregation pattern changes with the growth rate is shown in Fig. 4. Different gray scales are used for each figure to appreciate better the segregation patterns; the intensity of segregation is given by the corresponding gray bar. For a low solidification velocity (Fig. 4at, the segregation is strong within the channels along the vertical walls. The channels lead to

152

S.D. Felicelli et al. /'Journal o/Co,stal Growth 177 (1997) 145 161

CASE 1 and 3

CASE 2

ll00

'.0

I

7

<

40

<

>

>

CASE4 14

<

< 7 >

>

/t o J

7 > 14

<

7

] I

/

o 14

< >

It'°

<

7 ~

7

..

I I it1° 10

'°

f ~

7 ....

>

<

14

>

Fig. 2. Mold shapes and sizes used in the simulations. Measures are in millimeters.

the formation of freckles on the surface of the casting, a fact verified experimentally in Ni-base superalloys 1-19]. In the casting interior, however, the segregation is rather weak, adopting the form of short and wandering streaks. When the growth rate increases, the internal streaks become stronger in solute content, they are less numerous, and tend to be more aligned with the direction of gravity (Fig. 4b and Fig. 4c). Also, the segregation on the

vertical walls begins to lose importance. The effect is accentuated for higher velocities (Fig. 4d and Fig. 4e), where the wall freckles practically vanish, and the segregation is mostly concentrated in the mold interior, in the form of a few weaker streaks and plume-rising channels. Finally, at a solidification rate of 1.2 x 10 -4 m/s (Fig. 41), only two very weak spots of segregation remain, which disappear completely at higher solidification rates.

S.D. Felicelli et al. /Journal o/'Crystal Growth 177 (1997) 145 161

0.020

153

0.020 0.97 0.94 o.01 0.88 0.85 o.52 0.76 0.76 0.73 0.70 0.67 o.84 0.81 o.58 o .55

0.015

0.010

8.50 8.43 8,36 8.29 8.23 6.16 6.09 8.02 5.95 5.88 5.81 5.74 5.67 5.80 5.53

0.015

0.010

0.005

0.005

0.000 0.000

0.000 0.000

0.007

0.007

(a)

(b)

0.020

0.020 0.95 0.91 0.86 0.81 0.78 0.72 0.67 0.82 0.57 0.53 0.48 0.43 0.38 0.34 0.29

0.015

0.010

0.006

8.21 8.00 7.78 7.87 7.36 7.15 (i.83 6.72 6.51 6.30 6.08 5.87 5.68 5.45

0.01

0.01,

5.23

0.00

0.000 0.000

0.007

(c)

0.00_ 0.000

0.04)7

(d)

Fig. 3. S o l i d i f i c a t i o n of N i - 5 . 8 % AI 15.2% Ta. G = 2000 K/m, r = 0.28 K"s. (a) a n d (b) F r a c t i o n of liquid a n d m i x t u r e c o n c e n t r a t i o n o f AI at t = 80 s. (c) a n d (d) S a m e v a r i a b l e s at t = 200 s.

-~

c~

('D

(~

.-..]

,

×~

)<

©

I

i

~"

X

I

~'~

oo

~_>

×

o

tii!iii~ '

o

§

.o

p 8

o

P

o

o o ol

tm

p o o

o

o ~

.o o o

o

i

C~

S.D. Felicelli et al. ,'Journal c?f Co'stal Growth 177 (1997) 145 161

there, leading to a higher solute accumulation in those regions [2]. When the solidification velocity is very high, the liquid next to the walls solidifies before it has time to accumulate solute. In this case, the channels form due to convective instabilities only (random local solute accumulation due to convection), which can occur at any point within the casting, but preferentially in the interior, where convection is stronger. Although only the concentration of A1 is shown in the figures, it must be mentioned that the distribution of Ta presents similar patterns, with different levels of segregation because AI and Ta have different equilibrium partition ratios. Both solutes are convected with the fluid, therefore resulting in similar segregation patterns. The similarity in the segregation patterns can be expected because convection is the dominant mechanism of solute redistribution during solidification. The last plot in this case shows the amount of segregation versus growth rate (Fig. 5), where the intensity of segregation is represented by the difference between the maximum and minimum values of the mixture concentration of AI. Two curves are shown, one corresponding to the set of runs with constant temperature gradient, and the other to the set with constant cooling rate. Although the plotted points do not correspond exactly to the same volume of solidified alloy, the decrease in segregation intensity for higher growth rates is evident. Case 2: Segregation versus 9rowth rate in a large mold. This set of simulations was done on a rectangular mold of 4 0 m m in width and 100mm in height. The same ternary alloy used in Case 1 was used here. All other conditions remain the same as in Case 1. Fig. 6 shows results after 400 s of solidification under a fast growth rate of 1.6 x 10 ~ m/s. It is observed that in the lower 40 mm of the mold, almost no segregation has occurred, in agreement with the calculations for the small mold at this velocity. At about 50 mm, however, a channel segregate has begun to form, ejecting a plume of solute-rich liquid into the all-liquid region (Fig. 6a). The contour lines of fraction of liquid (Fig. 6b) show that the channel is not completely open as we had observed in the smaller mold, probably because the mesh refinement is not enough to resolve

6

155

~

r = 02.8 K/s, 1000
E

L~ i

4

E 2

0

. . . .

2.0E-5

t

. . . .

4.0E-5

I

. . . .

6.0E-5

i

,

8.0E-5

,

,

,

I

i

i

1.0E-4

i

i

I

. . . .

1.2E-4

I~-~,

1.4E-4

Growth Rate (m/s) Fig. 5. Segregation versus growth rate for solidification of Ni 5.8%, AI- 15.2% Ta in the small rectangular domain (Case 1).

the narrow channels for this large lateral dimension. In spite of that, the solute-rich trail left by the channel during its growth, can be clearly detected. Note also that the liquid plume does not flow straight upward, but it turns to the left, looking like smoke coming out of a chimney, that is blown away by the wind. This is due to the highly unstable nature of the thermosolutal convection, which is easily perturbed by small variations in the concentration field, an effect that is accentuated in liquid metals with small solutal diffusivities and small viscosities. Fig. 7 shows the mixture concentration of A1 for two different solidification velocities, 9.6x 10 -~ m/s (Fig. 7a) and 1.2 x 10 4 m/s (Fig. 7b). Again, it is seen that when the solidification velocity increases, the number and strength of segregate channels decrease, and they locate preferentially in the mold interior, leaving the mold surface relatively free of segregation. Several numerical experiments performed in this case showed that the variation in segregation patterns and intensity with the growth rate follow similar trends to what was observed in a Case 1, supporting the fact that simulations in small molds, which are computationally less expensive, offer a good representation of the mechanisms involved in the formation of segregation.

S.D. Felicelli et al. /'Journal of C~stal Growth 177 (1997) 145 161

156

--~ 0.005 m/s 6.14 8.10 6.06 6.02 5.98 5.94 5.89 5.85

0.100

0.090 ......................................

iiiiiii!iiii!!!i!ii~.iiiiiiiiii!iiii iil

0.080 ...................................... :::::::::::::::::::::::::::::::::: : ........

- .u / a

5.81

, .............

==~-~,..,°

......

. . . . . . . . . . . . . . . . . . "~ ~ . . . .

.......... "::.':::" ........... ~ , : ' : : .

5.77 5.73 5.68

0.060

5.64 5.60 5.56

0.050 0.12

0.040

0.030

0.020

0.01(

0.000

0.020

0.040

(a)

I 0.00( 0.000

I

I

I

l

0.020

,

,

,

,

0.040

(b)

Fig. 6. Solidification of Ni 5.8% AI 15.2% Ta. Large mold (Case 2). G = 2000 K/m, r = 1.5 K/s, t = 400 s. (a) Mixture concentration of AI. (b) C o n t o u r lines of volume fraction of liquid and velocity vectors.

Case 3: Segregation versus partition coefficients. While performing the numerical simulations, it was noted that the segregation patterns were particularly sensitive to variations in the partition coefficients of the alloying elements. Additional numerical exploration in this subject was motivated by the fact that very few published equilibrium partition ratios of Ni-base alloys are available, and significant discrepancy is found in the few sources that provide them. A reference configuration based on the small domain used in Case 1 was utilized in this set of calculations. The ternary Ni A1 Ta alloy was solidified vertically in this mold under a temperature gradient of G = 5000 K/m and a cooling rate r = 0.28 K/s, which produced an average growth rate of 4.0 x 10-5 m/s. The distribution of mixture concentration of A1 for different values of the partition coefficient of this element was calculated.

The results for three values of kA1 a r e plotted in Fig. 8. Up to kAl = 0.7, the segregation pattern looked similar to what had been found in the previous calculations with the value of kA1 = 0.54. Above this value, a transition seems to occur. For kAl = 0.75, as observed in Fig. 8b, the wandering short streaks in the interior of the alloy tend to align vertically, and their number decreases. Solute-rich plumes rising into the liquid are still observed. At kA! = 0.8 (Fig. 8c), the segregation pattern has changed completely, showing instead four perfectly vertical segregates. The convection switched from a rather disorganized plume regime (Fig. 8a) to a very organized mode, in which each channel generates its own upward flow apparently without disturbing or interacting with its neighbor channels (Fig. 8d). The flow, however, is not strong enough to overcome the stabilizing temperature gradient, and no solute plumes rising into the

S.D. Fe/icelli et al. ,,'Journal o[C~,stal Growth 177 (1997) 145 161

6.38

6.30 6.22

157

0.100

6.31 6.24 6.17 6.11 6.04 6 .g8 5.gl 5.85 5.78 5,72 5.65

0.090

6.14 6,06

S.~ 5.89 5.81

5.73 S.65 5.57 5.49 5.40 s.:~

0.080

0.070 0.060

5.58 5.52 5.45 5.3~

0.050

,5,24

0.040

0.030

0.020

0.010

0.0~

0.000

0.020

0.040

(a)

.

0.000

.

.

.

0.020

0.040

(b)

Fig. 7. S o l i d i f i c a t i o n of Ni 5 . 8 % A1-15.2%. Ta. L a r g e m o l d (Case 2). G = 2000 K/re. M i x t u r e c o n c e n t r a t i o n of AI at: (a) r = 0.56 K/s, V = 9.6 x 1 0 - 5 m,,'s, (b) r = 0.83 K/s, V = 1.2 x 10 - 4 m/s.

overlying liquid are observed. The corresponding gray bar in each figure shows that the amount of segregation has decreased with increasing kAy, as expected. For even larger values of kAl, the segregates remain vertical, but their intensity fades, until eventually no segregation is present. The reason for this transition in the segregation pattern is not related now to the solidification rate, which remained approximately the same for the three examples of Fig. 8, but to the amount of A1 released to the interdendritic liquid. For large kAy, the A1 content in the liquid in the vecinity of the dendrite tips is hardly enough to compensate for the gravitationally stable Ta, which increases the density of the liquid. The buoyancy force is then very small, and the convection switches from a global unstable mode to a local mode, in which liquid flow occurs only within the channel and in a small layer at the tip of the dendrites, but nowhere else. Numerical experiments done with the same values of k a l , but with a lower temperature gradient,

showed that the transition is less marked, i.e., the channels also tend to straighten, but the segregation vanishes before a vertical alignment as observed in Fig. 8c is attained. Case 4: Segregation versus mold cross section. Because the solidification model was programmed as a finite element code, it has the capability of calculating in irregular geometries. To test this feature, molds of variable cross sections were used to study how the segregation pattern and intensity are affected with the change in geometry. Four different geometries were used; their shapes and sizes are shown in Fig. 2. The height of the molds was kept at 20 ram, while the width varied between 7 and 14 mm. The objective was to study how the segregation changed when the width of the mold expanded or contracted, and particularly, when it did so gradually or abruptly. This set of calculations was done with the quaternary alloy (Ni 6%Al-8%Crq5%Ta). The initial temperature gradient was 5000 K/m, and the cooling

S.D. Felicelli et al. /Journal of C~stal Growth 177 (1997) 145 161

158

0.020

0.015

6.70

8.65

6.50

6.56

6,51

6.48

8.42

839

5.32

6.31

5.23

8.23

6.t4

6.14

6.04

8.Oe

5.g5

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5.58

0.010

5.77

5.81

5.67

5.73

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5.84

5.49

5.58

5.39

6,47

0.005

0.000 0.000

0.007

0.0(30

0.007

(b)

(a) 0.020

6,32 8.26 6,21 6.15 8.09 6.04

0.015

5.98 5,92

o,o3~I .......................................

5.85 5.81 5.75 5.69

0.010

0.018 .I t . ,

$| ~ |.,

5.63

|t,

5.58

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5.52

~

I ....... ', . . . . . . 0.018

,'

. .......... . . . . . . .

~ t f t s.~ ~e

~.l~tfe ~

x, tlr. t~t

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/

"'

t|ffl

~ ,ttt, X

~l?t/

. .~ I/*

/ I

4

....

,

.Iff, ~

x rI! ,tt

"I

~ T.~ . . . . . . ~ t T . l . . . . . . .

~r,,.

~*|s."

i P*'*

"'~'tl .......

t ~ ........ ,v.

I i .........

. . . . . .

t..

. . . . . . . . . .

, ! , - • .

0.005

0.000 0.000

0.007

(c)

(d)

Fig. 8. Segregation patterns versus partition coefficient of Al during solidification of N i - 5 . 8 % AI 15.2% Ta. (a) k = 0.7; (b) k = 0.75; (c) k = 0.8; (d) Velocity vectors at tip of dendrites for case (c).

S.D. Felicelli eta/. / Jo #'hal o/'C(vstal Growth 177 (1997) 145 161 0.0"20

6.13

6.11 6.10 6~06 6.00 6.05 5.03 e.o2 o.oo 5.98 5.97 5.95 5,93 5.92 590

0.015

0.010

0.020

0.015

0.010

0.000

0,000 -0.007

0007

(a)

0.000

0.007

(b)

0.020

6.19 0.16 6.13 6.10 6.07 s.04 6.Ol S.97 5.94 5.91 5.88 5.e~ 5.82 5.79 5.715

0.015

0.010

0.020

6.17 6.15 6.13 5.11 6.09 5.07 5.OS 6.03 6.01 5.99

0.015

5,97 5.95 5.93 5.90 5.88

0.010

0.005

0.000 0,000

5.15 6.13 6.11 6.10 O.Oe 6.07 6.05 6.03 6.02 6.OQ 5.98 5.g7 5.95 5.93 5,92

0.005

0.005

0.000 -0.007

159

0.005

0.007

0.014

(c)

0.000 0.000

0.007

0.014

(d)

Fig. 9. Solidification of Ni 6 % AI 8%, Cr 6 % Ta alloy in different mold shapes. M i x t u r e c o n c e n t r a t i o n of AI is shown.

rate applied at the bottom was 0.28 K/s. The growth rate varied because of the changing cross section. An open top (or stress-free) boundary condition was

applied at the upper end of the domain, to minimize the effect of this boundary on the convection, and to concentrate on the effect of the lateral walls.

160

S.D. Felicelli et al. /Journal of Co,stal Growth 177 (1997) 145 161

Fig. 9a and Fig. 9b show the segregation patterns obtained when the cross section of the mold expands or contracts smoothly, while Fig. 9c and Fig. 9d show the case of abrupt changes. It is immediately evident that there seems to be a tendency for segregation to concentrate on the surface of the mold during an expansion, and in the mold interior during a contraction. This is probably related to the variation in growth rate when the cross section changes. In an expansion, the solidification velocity decreases (more mass to solidify at the same cooling rate), and it was seen in the previous examples that segregation tends to accumulate along no-slip boundaries (the mold surface) in low growth rate conditions. As to the intensity of segregation, the step expansion (Fig. 9c) is the most severely segregated, with channels along the walls and several internal streaks. The smooth expansion (Fig. 9a), on the contrary, produces a relatively homogeneous interior, although channels along the walls are strong. The smooth contraction (Fig. 9b) also looks "cleaner" than its abrupt counterpart (Fig. 9d). These results suggest that abrupt changes are more prone to heavier segregation than smooth variations of the cross section.

4. Conclusions

A mathematical model of solidification of dendritic multicomponent alloys, that includes thermosolutal convection and macrosegregation, was devised. The model was implemented in a twodimensional finite element code, capable of simulating the complete solidification process in various geometries of two dimensions. The code was used to simulate the directional solidification of Ni A1-Ta and Ni A1-Cr-Ta alloys, Macrosegregation was calculated under several cooling conditions and different geometries. The results showed that the segregation patterns varied greatly with cooling conditions, adopting several forms: weakly segregated straight lines, wandering streaks, open channels on the casting surface, and internal channels. The same segregation behavior was observed in simulations done in small molds and in large ones. When changing geometry, the macrosegregation proved to be more severe in abrupt changes

than in smooth variations of the cross section. The segregation pattern was found to be particularly sensitive to the values of the partition coefficients of the alloy components. An accurate database of physical and thermodynamic properties is considered essential for realistic qualitative and quantitative predictions.

Nomenclature

specific heat capacity C,/

mixture concentration of solute j concentration of solute j in the liquid local average concentration of solute j in the solid reference concentration of solute j Dj diffusivity of solute j in the liquid Kx, K y permeability in the x and y directions magnitude of gravity in the x and y direcgx, gy tions initial temperature gradient G Ij auxiliary variable for average concentration of solute j in the solid k) equilibrium partition ratio of solute j latent heat L md change ofliquidus temperature with solute J n unit outward normal vector N number of allloy components or solutes pressure P t" cooling rate t time At time step T temperature eutectic temperature TE liquidus temperature TL melting temperature of solvent (extrap.) TM reference temperature TR U superficial velccity vector ( = 0u0 /dp pore velocity vector U x-component of the superficial velocity y-component of the superficial velocity V solidification velocity X, y coordinates thermal diffusivity thermal expansion coefficient fit

C J~

S.D. Felicelli el al. "Journal of Cm'stal Growth 177 (1997) 145 161

solutal expansion coefficient of solute,/

c tt p Po (/5

stress tensor convergence tolerance viscosity density reference density volume fraction of liquid

Acknowledgements This work was supported by the Advance Research Project Agency, under the Micromodeling Program of the Investment Casting Cooperative Arrangement, under contract MDA972-93-2-0001. The authors appreciate the many discussions with Dr. A.F. Giamei, from United Technologies Research Center, for sharing his expertise in directional solidification and computational software. The provision of financial support to Sergio D. Felicelli by the Consejo Nacional de Investigaciones Cientificas y Tecnicas (CONICET-Argentina) and by the Argentine Atomic Energy Commission, is gratefully acknowledged.

References [1] S.D. Felicelli, J.C. Heinrich and D.R. Poirier, Num. Heat Transfer B 23 (1993) 461. [2] S.D. Felicelli, J.C. Heinrich and D.R. Poirier, Metall. Trans. B 22 (1991) 847. [3] W.D. Bennon and F.P. lncropera, Int. J. Heat Mass. Transfer 3(1 {1987) 2161. [4] P.J. Prescott and F.P. lncropera, J. Heal Transfer 116 (1994) 735. [5] M.C. Schneider and C. Beckermann, Int. J. Heat Mass. Transfer 38 (1995} 3455. [6] (7. Beckermann and R. Viskanta, Appl. Mech. Rev. 46 [1993i 1. [7] S.N. Tewari, R. Shah and M.A. Chopra. Metall. Trans. A 24 {1993) 1661.

161

[8] J.R. Sarazin and A. Hellawell, Metall. Trans. A 19 (1988) 1861. [9] S.D. Ridder. S. Kou and R. Mehrabian, Metall. Trans. B I2 (1981) 435. [,10] R. Mehrabian and M.C. Flemings. Melall. Trans. 1 (1970) 455. [1 I] T. Fujii, D.R. Poirier and M.C. Flemings. Metall. Trans. B 10 (1979) 331. [12] M.C. Schneider and (7. Beckermann, Metall. Mater. Trans. A 26 (1995) 2373. [13] M.C. Schneider and C. Beckermann, ISIJ Int. 35 (1995) 665. [14] S.D. Felicelli, Simulation of freckles during vertical solidification of binary alloys, PhD Dissertation, University of Arizona, Tucson I 1991 ). [,15] S. Ganesan and D.R. Poirier, Metall. Trans. B 21 (1990) 173. [,16] P. Nandapurkar, D.R. Poirier and J.C. Hemrich. Num. Heat Transfer A 19 (199l) 297. [,17] S.M. Copley, A.F. Giamei, S.M. Johnson and M.F. Hornbecker, Metall. Trans. 1 (19701 2193. [,18] F. Chen and C.F. Chen, J. Fluid Mech. 227 (1991) 567. [,19] A.F. Giamei and B.H. Kear, Metall. Trans. 1 (1970} 2185. [20] P.W. Peterson, T.Z. Kattamis and A.F. Giamei. Metall. Trans. A I1 (1980)1059. [21] Metals Handbook, Vol. 8, 8lh ed. (American Society lbr Metals, Metals Park. OH, 19741 pp. 262, 288, 291. [22] R. Kadalbal, J.J. Montoya-Cruz and T.Z. Kattamis, Metall. Trans. A 11 (1980) 1547. [23] S.N. Tewari, M. Vijayakumal-, J.E Lcc and P.A. Curreri. Mater. Sci. Eng. A 141 (1991) 97. [24] D.R. Poiricr and E. McBride, Estimating the densities of the liquid transitional metals and Ni AI Cr Ta alloys, Internal Report for Investment Casting Cooperative Agreement, Dept. of Materials Science and Engineering, The University of Arizona. Tucson, AZ, Feb. 1996. [25] T. Iida and R.I.L. Guthrie, The Physical Properties of Metals {Oxt\~rd University Press, Oxford, UK, 19881 p. 183. [26] T. lida and R.I.L. Guthrie, The Physical Properties of Metals {Oxford University Press, Oxford, UK, 1988) p. gl. [27] T. Iida and R.I.L. Guthrie, The Physical Properties of Metals (Oxford University Press, Oxford. UK. 1988) p. 8. [28] A. Goldsmith, T.E. Waterman and H.J. Hirschhorn, Handbook of Thermophysical Properties of Solid Materials, Vol I {Macmillan, New York, 1961) p. 449.