A model of binary alloy solidification

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 20, 1067-1084 (1984)

A MODEL OF BINARY ALLOY SOLIDIFICATIONT

D. G. WILSON, A. D. SOLOMON AND V. ALEXIADES Mathematics and Statistics Research Department, Computer Sciences Division, Union Carbide Corporation,

Nuclear Division, Oak Ridge, Tennessee, U. S.A.

SUMMARY

A linear model for the solidification of a dilute binary alloy is presented. In this model the solidus and liquidus curves are linear. As a consequence internal energy depends linearly upon temperature and concentration. The formulation is a generalization of the well-known enthalpy method to treat a phase change problem involving coupled heat and mass transfer. Both analytic and numerical formulations are given. Results from the latter are presented and compared with an explicit solution of Rubinstein for a Stefan-like problem posed in a semi-infinite slab. Some remarks on the behaviour of the explicit solution are given.

INTRODUCTION

In this paper we describe a model of a binary alloy solidification process. Unlike the classical Stefan problem, alloy solidification involves a coupled system of equations for both heat and mass transfer. In addition a relation exists between freezing temperature and relative component concentration at the freezing front.

There have been a number of attempts to formulate and simulate this alloy solidification process. In 1967, Rubinstein‘ gave a similarity solution for a problem representing the crystallization of a binary alloy in a semi-infinite domain. For this solution the melt temperature and the concentration of the solidified alloy are constants. Fix,’ in the proceedings of the 1977 meeting in Gatlinburg, gave an analysis of numerical methods for alloy solidification.

More recently, Crowley and Ockendon3 and Bermudez and Saguez4 have considered binary alloy solidification problems which are similar to ours. Both of these analyses are limited by the necessity of the assumption that the heat capacities in the solid and liquid phases are the same. Because of the way we construct our weak formulation, we are able to avoid this unrealistic assumption and address problems beyond the reach of previous analyses.

In what follows we give a very brief introduction to two-component phase diagrams, whose relations we assume hold at the solidification front; we state the mathematical problem, which we assume represents the solidification of a simple binary alloy; we described our ‘phlogiston formulation’ of the problem, which is derived by analogy with the enthalpy method for simple problems; we describe the finite difference scheme, with which we compute the various constituents of our phlogiston formulation; we discuss Rubeinstein’s explicit solution and comment on the occurrence of ‘mushy regions’, and finally we compare our numerical results with values computed using this explicit solution.

t Research sponsored by the Applied Mathematical Sciences Research Program, Office of Energy Research, U.S. Department of Energy under contract W-7405-eng-26 with the Union Carbide Corporation.

0029-598 11 841061067-1 8$01.80 Received 1 7 August 1982

1068 D. G. WILSON, A. D. SOLOMON A N D V. ALEXIADES

AN INTRODUCTION TO TWO-COMPONENT PHASE DIAGRAMS

A simple two-component phase diagram of the kind we consider shows three things: the freezing temperature of a liquid of specified concentration (the liquidus line); the melting temperature of a solid of specified concentration (the solidus line); and the equilibrium concentrations of a solid and liquid which may co-exist at a specified temperature.

W LT 3 I-

(r W a 3 W I-

a

TA

0 CONC E N T RAT ION

1

Figure 1. A simple two-component phase diagram for the case in which the two constituents form solid solutions in all proportions. T, and T, are the melting points of components A and B, respectively

Figure 1 shows a two-component equilibrium phase diagram for the case in which the two constituents form solid solutions in all proportions. The upper curve is the liquidus line and the lower curve is the solidus line. It is customary to assume that these curves are smooth and monotone increasing. The co-ordinate axes are labelled 'Concentration' and 'Temperature'. The former is to be measured in some dimensionless fraction of component B (the higher melting). Both mole fraction and weight fraction seem to be in use for this purpose.

The interpretation of this diagram is as follows. If one has a volume of this A / B mixture in thermodynamic equilibrium with uniform temperature equal to T* and overall concentration of component B equal to C", then exactly one of the following three alternatives must hold:

1. If the point on the diagram corresponding to (C*, T*) lies on or above the liquidus line, then the entire volume of material will be in the liquid phase. 2. If the point of the diagram corresponding to (C*, T") lies on or below the solidus line, then the entire volume of material will be in the solid phase. 3. If the point on the diagram corresponding to (C", T") lies between the solidus and liquidus lines, then both solid and liquid phases will be present in the volume. In this case the concentration of the liquid will be C,, where the point (CL, T*) lies on the liquidus line, and the concentration of the solid will be C,, where the point (Cs, T*) lies on the solidus line. If

BINARY ALLOY SOLIDIFICATION 1069

one assumes, as we shall, that the two phases have the same density, then one could compute the volume of each phase present from simple conservation of mass relations.

The phrase ‘at thermodynamic equilibrium’ is a crucial part of this explanation. It must be emphasized that these phase diagrams are statements about equilibrium conditions. If one has a volume of the A / B mixture at a particular C* and T* lying between the liquidus and solidus curves and alternative 3 above does not hold, then the system is not in equilibrium. In this case one must have supercooling or supersaturating or some other non-equilibrium condition.

Observe that at thermodynamic equilibrium there must be a discontinuity in concentration across the phase front. This follows from the liquid and solid in equilibrium at the same temperature having different concentrations. Furthermore, this discontinuity is not necessarily constant. It corresponds to the horizontal gap between the liquidus and solidus lines which varies with the freezing temperature.

We limit our attention to a neighbourhood of the point on the diagram labelled T., the melting point of component A (the lower melting). We have several reasons for doing this. It justifies the assumption we make that the solidus and liquidus curves are linear. It also avoids the possibility that this approximation will become progressively worse as the process continues. For, from physical considerations we expect the liquid to be depleted in component B (the higher melting) as solidification proceeds, and hence we expect the freezing temperature to decrease with time.

In a neighbourhood of the point (0 , TA) both phases are dilute solutions of component B in component A. We assume that the solidus and liquidus curves are linear with slopes ys and y J , respectively, as indicated in Figure 2. In this case the jump in concentration across the phase front varies linearly with the freezing temperature.

CONC

Figure 2. A linear phase diagram for a dilute binary alloy

STATEMENT OF THE MATHEMATICAL PROBLEM

The problem which we consider consists of two diffusion problems, one for heat and the other for material, posed in a finite slab, O=sx=GL, which is divided into two regions by a free boundary, x = S ( t ) . In this simplest mathematical model of alloy solidification the equations governing heat and mass transfer are assumed to couple only at the free boundary. Conservation relations for energy and mass determine the movement of the free boundary. The solidification temperature and the limiting values of solute concentration at the free boundary from the solid and liquid phases are related by linear phase diagram relations. Neither material nor heat can escape at the right-hand boundary. At the left-hand boundary no material can enter or escape and a temperature T o ( t ) s T . is imposed. Initially the material is in the liquid phase.

1070 D. G . WILSON, A. D. SOLOMON AND V. ALEXIADES

That is, the initial temperature, TI, and concentration, C,, satisfy T,(x) 2 y L C I ( x ) + TA, 0 S X G L .

The diffusion equations in the solid, 0 < x < S ( t ) , annd liwuid, S ( t ) < x < L, regions are:

pcsT,( t ,x)=(ksT,( t ,x)) , for t > 0 , and x € ( O , S ( t ) )

C,(t ,x)=(DsC,( t ,x)) , for t>0, and x ~ ( o , S ( t ) )

~ c , T ~ ( t , x ) = ( k , T ~ ( t , x ) ) ~ for t>0, and x € ( S ( t ) , L )

C, ( t, x) = ( DLC, ( t, x)), for t > 0, and x E ( S ( t ) , L)

(1)

(2)

(3)

(4)

Here, p denotes the mean density of the alloy; c,, cs the specific heats; kL, ks the thermal conductivities and DL, Ds the material diffusivities in the liquid and solid phases, respectively. For simplicity we assume that these are all constants.

The initial conditions are:

S(0) = 0 ( 5 4

T(0, x) = Tr(x ) , C(0, x) = Cr(x) , both for X E (0, L ) ; (5b)

(6)

and

where

TI (x) a YLCI (x) + TA The conditions at the two fixed boundaries are:

T(t,O)= T,,(t) C,(t,O)=O ( 7)

(8)

where To( t ) S TA; and

T, ( t, L ) = 0 C, ( t, L) = 0

all for t > 0. Under the assumption that a sharp interface x = S( t ) exists, the boundary conditions at S are:

p H S ( t ) = k,T,(t, s-) - k,T,(t, S + ) (9)

[C(t , s - ) - C ( t , S+)]S(t)=D,c,( t , S+)-D&,(t, s-) (10)

T( t ,S- )=T( t ,S+)=ySC(t ,S- )+TA=yLC(t ,S+)+TA (11)

and

The latent heat, H, in (9) may be a linear function of concentration H = H A + a C , where HA is the latent heat of pure component A and ‘u’ is a constant. Equations (9) and (10) are statements of conservation of energy and material, respectively, at the phase front. Equations (1 1) assert that at the phase front the temperature is continuous, the concentration has a jump and the temperature and concentration are related by the linear phase diagram relations.

The essential problem consists of (1)-(5), (9) and (10) and the assumption that the phase diagram relations hold at the phase front; these particular exterior boundary conditions, the linearity of the phase diagram relations and the linear dependence of the latent heat upon concentration are convenient but not essential assumptions.

In Reference 5 we pointed out a shortcoming of what is essentially this same formulation of alloy solidification in a semi-infinite slab. Namely, the implicit assumption that the material


on the ‘solid’ side of the solid-liquid interface is all solid and the material on the ‘liquid’ side of the interface all liquid may not be satisfied. Thus the mathematical interface may not have the physical significance that one is naturally tempted to associate with it.

We expect that the preceding statement of the problem suffers from the same shortcoming. To avoid this shortcoming and to provide equations more suitable for computation, we derive an alternative, weak formulation of the problem.

PHLOGISTON FORMULATION

The enthalpy formulation for Stefan-like problems is well known. Although it comes in several variations, the basic ideas are: (1) the enthalpy, E, is defined in terms of the temperature, T, so that in each phase the heat capacity of the material is the partial derivative of the enthalpy with respect to the temperature; (2) a jump in the value of the enthalpy equal to the latent heat of the phase transition occurs at the critical temperature; ( 3 ) instead of modelling the heat equation in each phase and tracking the phase front, the equation pE, = div ( kp grad T ) is modelled everywhere. The subscript p, on the thermal conductivity here, indicates the phase. A diagram of a typical enthalpy versus temperature relation is shown in Figure 3.

E

Figure 3. Enthalpy as a function of temperature for a pure material. H is the latent heat of the phase transition. The slope of the E vs. T line in each phase is the heat capacity of the material in that phase

We mimic this prescription in defining our weak formulation of the binary alloy solidification problem. Following Fix,2 we define a new variable W, which coincides with the critical temperature at the phase front, throughout the region of interest by

yLC for T > TA+ yLC (liquid) ysC for T < TA + ysC (solid)

w={

Because we assume that the linear phase diagram relations hold at the phase change front, W is continuous just as T is; and C has a jump across the phase change front just as E does. As a consequence, if we drew a new phase diagram with C replaced by W, then the solid and liquid regions would be divided by the single line T - W = TA.

1072 D. G. WILSON. A. D. SOLOMON AND V. ALEXIADES

Inverting (12), the concentration is given in terms of W by

yLIW when T > TA+ W yS‘W when T < T A + W

C = {

Figure 4 shows C as a function of T and W. It is evident that this relation is intimately connected with the phase diagram. Furthermore, if one considers C as a function of W for constant T, then the resulting relation is of the same form as the enthalpy-temperature relation shown in Figure 2.

C

JUMP IN C FROM PHASE DIAGRAM

Figure 4. Concentration as a function of temperature and W. The line T- W = T, is in the T / W plane. The quarter-planes marked ‘solid’ and ‘liquid’ are cut directly over this line

We next define the phlogiston, @, incorporating the linear phase diagram into the definition and preserving the characteristics of the enthalpy method stated above. The equation defining @ in terms of temperature and concentration is

(14) pcL(T( t , x ) - T A - y L c ( t , x)) in the liquid I pcs( T( t, x) - TA - ySC( t, x)) - pH in the solid

@( t, x ) =

Figure 5 depicts this relation. The phlogiston is represented as a pair of quarter-planes over the (C, T ) plane separated by a gap corresponding to the discontinuity in the concentration across the phase front. In each phase the inclination of the appropriate quarter-plane to the (C, T ) plane, along a line of constant concentration, is just the heat capacity in that phase.

In terms of the two continuous variables, temperature and W, the relation is

(15) PcL( T - TA - w> pcS(T-TA- W ) - p H

when T - W > TA when T - W C TA

@ = {


IDUS

Figure 5. Phlogiston as a function of temperature and concentration

There is no requirement that cL = cs nor that H be constant but, to avoid a nonlinear partial differential equation in the solid phase, we have assumed that H may vary at most linearly with concentration (and hence with W ) .

We point out that liquid which is about to precipitate solid material, i.e. has T and W satisfying T - W = TA, always has zero phlogiston and that newly formed solid always has phlogiston equal to -pH. Furthermore, T - W > TA if and only if Q, > 0 and T - W < TA if and only if @ < -pH. Thus the two phases may be characterized using either T - W or Q,.

Values of Q, in the interval -pH < Q, < 0 correspond to T - W = TA. We have now defined the elements of our phlogiston model. @ and C have jumps at the

phase front and correspond to enthalpy; T and W are continuous across the phase front and correspond to temperature.

Proceeding formally we define the equations which determine the evolution of these elements. Using (1)-(4) and (12)-(15) we write equations for 0, and C, as follows:

@ , = ( k ( T , W)Tx-a (T , W)WX)X (16) and

Ct = (P(T7 W ) Wdx (17)

both for t > 0 and x E (0, L). Here

{: k ( T, W ) = for T - W > TA for T - W < TA

PDLCL for T - W > TA p(Dscs+a/ys) for T - W < T ,

a(T, W ) =

and

P ( T, W ) = { DL' yL Ds / YS

for T - W > TA for T - W < T A

1074 D. G. WILSON, A. D. SOLOMON AND V. ALEXIADES

Until now we have not addressed the possible occurrence of values of @ in the interval ( -pH, 0). We wish to accommodate this possibility consistent with (14)-( 17). To accomplish this we define a frozen fraction, f, by

if @ ( t , x) < -pH

if O < @ ( t , x )

t, x)/ ( - p H ) if -pH s @( t, x) 0

We then extend the definitions of the coefficients k, (Y and P by

k = f k s + (1 - f ) k,

(Y = f ( Y s + ( 1 - f ) ( Y L

P =fPs+(l-f>PL

and assume that (16) and (17) hold everywhere with the extended coefficients k, (Y and P. Corresponding to the boundary conditions (7) and (8) we require

T(t,O)= T,(t) W,(t,O)=O (18)

and

T,(t,L)=O W,(t,L)=O

all for t > 0.

obtained from these using relations (12), (14) and (6). The resulting conditions are: The initial conditions for T and C are taken directly from (5b) and those for @ and W are

all for x E (0, L ) and again with the restriction TI 2 yLCI + TA. The interface conditions (5a) and (9)-( 11) are not stated separately. These will be automati-

cally fulfilled by a solution of (16)-(20), provided that the interface, {( t, x) 1 -pH 5 @( t, x) S O), is a smooth curve, just as in the enthalpy formulation of the Stefan problem. Our phlogiston formulation, (16)-(20), is a weak formulation of the problem stated in the distributional sense. A proper weak formulation may easily be obtained by multiplying the equations by a test function and integrating over [0, L] (see References 3 and 4 for details of the method).

As with the enthalpy formulation, the great advantage of (16)-(20) over (1)-(11) is that it allows us to compute a solution without tracking the free boundary. This is the topic of the next section.

FINITE DIFFERENCE APPROXIMATION

We have constructed a finite difference approximation to the problem defined in the previous section. The scheme is a straightforward explicit discretization of (1 6)-(20).


The space step size is chosen by specifying the number of intervals, I , into which the slab length, L, is to be divided. Then A x = L/I . Since the scheme is an explicit one, the time step size is limited by the stability requirement D" A t / A x 2 < 0.5, where D" is the maximum of the four 'diffusivities' of the problem. We have taken At = 0.4Ax2/ D", where D" is the maximum of k J ( p c r . ) , k s / ( pcs), DL and Ds. Thus in the absence of an interface each of the underlying discrete diffusion problems is stable.

The parameters entering the explicit time difference equations are evaluated at each node. Values appropriate for the solid and liquid regions are used when a node is definitely in either a solid or liquid region. The liquid region is defined by the condition @ 2 0 , the solid region is defined by the condition @ G - p H ; and the condition 0 > @ > -pH defines the interface region.

After @ and C are computed at an ( n + 1)th time step, T and W are computed at the advanced time from them. This is done for nodes in the solid or liquid by using the relations between T, @ and C and between W and C given in the preceding section and depicted in Figures 4 and 5. For nodes where -pH < 0 < 0 we have used -@/( p H ) as the frozen fraction, f; we have extended the relation between W and C and used W = C / [ ( f / ~ ~ ) + ( l - f ) / ~ ~ , ] ; and we have taken T = TA + W.

At the fixed boundaries we have taken centred difference approximations to the derivatives T, and C, in applying the boundary conditions T, = 0 and C, = 0.

The equations for the difference scheme are as follows:

for i = l , . . . , , 1 - 1 , n = 1 , 2 ,...; and

Cy" =Cy +A[d2(n,i)(Wy+1 - W y ) - d 2 ( n , i - l ) ( W y - WY-'_,)] (22)

for i = 1 , . . . , I- 1, n = 1 , 2 , . . . . Here A stands for At/Ax2, and the coefficients k(n , i), dl(n, i) and d2(n, i) are given in terms of the frozen fraction, f y , and the parameters of the original problem by the relations:

if @yzO fr = - @ : / ( p H ) i f -pH<@y<O 1: if @y<--pH

for j = 1,2 , with d l s , d l L , d Z S and dZL defined by:

Here 'u' is the constant from the expression H = HA + aC.

1076 D. G. WILSON, A. D. SOLOMON AND V. ALEXlADES

The boundary conditions are:

TI: = To( n A t ) @:+I = pcs[ To( ( II + 1)At) - T . - ysC:+' ] - p H

C;+' = C: +2A[d2(n, 1)( W; - w:)] @;+I =@," -2h[k(11, I - 1)( T," - T7-1 ) - d l ( n , I - 1)( W ; - W ; 1 )]

C;" = C ; -2h[dZ(n,I-l)(W; -W;-j)]

for n = l , 2 , . . . . The initial conditions are:

Cf = C,(iAx)

Ti = T , ( i A x )

Wl = y L C I ( i A x )

for i = O , l , ..., I

for i = O , l , . . . , I for i = l , . . . , I ;

@ p 1 = pcL( ~ , ( i ~ x ) - T~ - w,' ) w:, = Y s ~ r ( 0 )

for i = I , . . . , I

@ A = pcs( To(0) - TA - W:, ) - pH

By requiring To( t ) < TA we assure that the first node, i = 0, will be initially solid.

advanced time step for each value of i as follows. After CP and C have been updated for a particular time step, W and T are computed at the

In the solid, defined by @:+'s -pH

w:+l = y s c : + ' ,

T:+' = TA+ .ysC:+' +( p H + @ : + ' ) / ( pcS)

In the liquid, defined by a:+' 3 0: ,:+I = yLc;++l T:+l = TA+ 3/LCrf1 +@:+'/( pcL)

In the region defined by -pH < @:+I < 0:

w:+l = c:+I / [ ( f : + ' / YS) + t 1 -f;+' I / YLI

T:+l = TA+ W:+'

THE EXPLICIT SOLUTION OF L. RUBINSTEIN

In Reference 1, Rubinstein gave an explicit, similarity solution for a binary alloy solidification problem posed in a semi-infinite slab. For small times one would expect a solution of (1)-( 11) to agree with Rubinstein's solution, and in the last section we compare some of our numerical results with values computed using this similarity solution. Before doing this we discuss the Rubinstein solution and point out some of its interesting features.

We consider only the phase diagram shown in Figure 1. The liquidus and solidus lines, given by TI, = fL , ( C) and Ts = f s ( C), respectively, are continuous and monotone increasing and intersect only at their endpoints (0, TA) and (0, TB) , where TA and Ts are the melting points of the pure materials. The problem is essentially that of the earlier section on the 'Statement of the mathematical problem', with the slab length, L, replaced by +a. There are, however,


a few variations and instead of citing (1)-( 1 1 ) and pointing out changes we state the problem anew. It is assumed that for t > 0 the solid and liquid regions are separated by a sharp interface curve x = S ( t ) with S a differentiable function. The diffusion equations in the solid, 0 < x < S ( t ) , and liquid, x > S( t ) , regions are:

pcsT,(t, x) = (ksT,(t, x)), for t > 0, and x E (0, S ( t ) ) (23)

C,(t,x)=(D,C,(t,x)),fort>O,andn~(O,S(t)) (24)

pcr-Tt ( t , x) = ( kLT, ( t , x)), for t > 0, and x > S( t ) ( 2 5 )

C, ( t , x) = ( DLCx ( t , x) ) , for t > 0, and x > S( t ) (26)

Here the positive constants are as in that earlier section. The initial conditions are:

S ( 0 ) = O T(0, X) = TI C(0, X) = Cl

where TI and CI are constants satisfying

O<Cr<1 and Tr>f,(Cr)>TA

At the fixed boundary, x = 0, we require

T(t,O)=TO C,(t,O)=O for t > O

where 7;) is constant satisfying

TOG TA.

The conditions at infinity are:

lim T ( t, x) = TI lim C( t, x) = C, X’IU 1’00

for each t > 0. The relations at the free boundary are:

p H S ( t ) = ksT,(t, S-)- k,T,(t, S’)

[C(t, S-)-C(t ,S+)]S(t)=D,Cx(t ,S+)-D,C,( t ,S-)

and

T(t , S--) = T(t, S+) =fs(C(t, S-)) =fL(C(t , S+))

Here the latent heat, H, is a constant. The well-known similarity solution for this problem has the form:

C( t, x) = C,, for 0 s x < S( t )

(32)

(33)

(34)

T(t,x)=T,,+(T,,-T,) erf (x/2d(ast)>/erf A for O s x < S ( t )

~ ( t , x) = C, +(CL-Cr) erfc ( x / 2 J ( ~ ~ t ) ) / e r f c ( A J ( ~ ~ / D ~ ) ) for x > ~ ( t )

T(t , x)=TI+(T,,-Tr) erfc(x/2d(a,t))/erfc(AJ(a,/aL)) for x > S ( t ) (36)

(35)

where as = ks( pcS), aL = k L / ( pcL). (T,,, C, and C, are constants whose values are to be determined.) The interface is given by S ( t ) = 2Ad(ast) where A also is a constant to be determined.

1078 D. G. WILSON, A. D. SOLOMON A N D V. ALEXIADES

Substituting the expressions for T and C into the interface conditions gives two transcendental equations involving the four constants T,,, Cs, CL and A. The form of these equations is

and

( CI - CL>/ ( c s - CL) = H2( A J( as/ DL)) (38) where

Al(A) = kS/(%Hl(A))

~ , ( x ) = Jn-x exp (x') erf (x)

H,(x) = Jn-x exp (x') erfc (x)

= kL/[CYLH2(h J ( % / ( Y L ) ) I

and

(39)

Tcr =f . (CS) =fL(Cd (40)

These equations are supplemented by the phase diagram relations

to give a system of four equations in the four unknowns T,,, Cs, CL and A. The following theorem is a slight generalization of the result given in Reference 5 . The proof

is essentially the same.

Theorem. The system of equations (37), (38) and (40) has a solution (T,,, Cs, CL, A ) with TA < T,, < TI, 0 < CL < Cs < 1 and 0 < A for each set of positive thermophysical parameters and initial and boundary values satisfying (28) and (30).

Proof It is an elementary exercise in differential calculus to verify that W,(A), given by equation (37), is continuously differentiable and monotone increasing for A E (0, +a). Further- more, lim,,()+ Wl(A)=TOGTA, and lim,,,, W l ( A ) = T , + H / c L > T I ~ f f r ( C I ) > T A . Thus there exist unique non-negative numbers p and v, independent of DL., such that v > p, W1 ( p ) = TA and Wi(v)=fL(CI) .

For A E (p, v) we define W,(A) by

cf. equation (38). Observe that W, is independent of DL. We shall establish the existence of A * E (p, v ) such that

W,(A*) = ~ , ( A * J ( % / D L ) )

The corresponding values of T,,, Cs and C, will be W, ( A *), fs' ( Wl( A *)> and fL1 ( Wl ( A *)), respectively. In brief the argument is that W,(A) and H,(AJ(cys/ DL)) are continuous on (p, v) with H2(AJ(as/DL))> W,(A) for A near p and H,(Ad(as/DL))< W,(A) for A near v.

It is well known that H2 is continuous and monotone increasing for x ~ ( O , + m ) with lim,,,+ H2(x) = 0 and limx++oo H2(x) = 1. Each of W,, f L 1 and f;' are continuous and monotone increasing, and hence so are the composite functions f L 1 ( W,) and f;' ( Wl). Since the solidus and liquidus curves intersect only at their endpoints, we have f L 1 ( Wl(A)) less than fs ' ( W , ( A ) ) and fL1 ( W , ( A ) ) G fL1 ( W , (v)) = C,. Therefore W2 is continuous and non-negative


on (p , v). Finally, since

and

we have l imA+v- W,(A) =O. The conclusion is that the curves y = W,(x) and y = H2(x) must cross and the existence of

A, T,,= Wl(A), Cs=f i l (Wl(A)) and CL=f;'(W1(A)) satifying (37), (38) and (40) is thus established. It is evident that TA < T,, < TI, 0 < C, < Cs < 1 and 0 < A.

Corollary 1. A is bounded away from zero independently of DL.

Proof. Since W, is contiuous on (p , v) and lim,,,+ W2(A) =+a, there exists p*> p such that W,(p*) = 1 and A < p* implies W2(A) > 1. Since W, is independent of DL, p* is also. Since W2(A) =H,(Ad(cws/DL)), which is less than one, we have and A z p * .

Corollary 2. The limiting problem consisting of (23)-(34) with DL=O does not have a similarity solution.

Proof. The crucial item is that (34) cannot be satisfied. The only candidate for a solution has C(t , x) = C, everywhere, A = p* and T,, = Wl(p*). But fi.(C,) # f s (CI) and thus the phase diagram relations (34) cannot hold.

In physical terms the problem (23)-(34) with DL = 0 is not well posed. If DL = 0, then C ( t, x) = C, for x > S ( t ) is the only reasonable solution. This is exactly what one expects from (35) for small DL. What happens for very small but positive DL is that C(t , x) dips down just in front of S ( t ) so that the phase diagram relations hold. This is not possible when DL = 0.

Corollary 3. The limiting problem (23)-(33) with DL=O has a similarity solution with C(t, x) = C, everywhere, A = p*, and T,, = Wl(p*) =fs( C,). (Here the phase diagram relations have been dropped.)

Proof. Equation (33) is satisfied trivially. Taking T,, =fs(Cl) defines a two-phase Stefan problem for T with (37) the usual nonlinear equation for A.

Corollaries 2 and 3 and the accompanying commentary are of interest because they help to make more nearly understandable the unexpected results from both the Rubinstein solution and our numerical simulation of our phlogiston model.

The approach of Cs to C, as DL + 0 is interesting and has an interesting consequence. Table I shows values of A, T,,, C, and C, for a sequence of problems with H = p = ks = kL = c, = cL = Ds = 1, To =0, Ti = 2.6, C, =0.8,fs(C) = 1 + C,fL(C) = 1 + 2 C and DL = 1~0,0~1,0-01,0~001, 0*0001. These were obtained as numerical solutions of (38), (39) and (40).

The interesting consequence, which we pointed out in Reference 5 , is that as problems with progressively smaller DL are considered a 'mushy region' appears just in front of the interface x = S( t ) . In this region the temperature and concentration values satisfy fs( C ) < T < fL( C) .

1080 D. G. WILSON, A. D. SOLOMON AND V. ALEXIADES

Table I. Values of A and critical temperatures and concentrations for several successively smaller values of DL

DL A CL

1.0 0.620 1.99 0.990 0.495 0.1 0.570 1.85 0.847 0.424 0.01 0.555 1.81 0.806 0.403 0.001 0353 1.80 0.801 0.400 0.0001 0.553 1.80 0.800 0.400

This is inconsistent with the implied assumption of thermodynamic equilibrium in each phase as discussed in the second section of the present paper, and it calls into question exactly how x = S ( t ) is to be interpreted as the phase boundary.

RESULTS OF SOME NUMERICAL EXPERIMENTS

In order to test the effectiveness of the computational method of the section on ‘Finite difference approximation’, numerical experiments were carried out in cases corresponding to the explicit solution of the last section. The runs made included cases where cs # c,, a situation which has not been dealt with previously. The runs were performed on a finite slab with vanishing fluxes, T, = C, = 0, at the right-hand boundary. At any fixed time t, three regions could be distinguished: a region consisting of solid alloy where - p H ; a liquid phase region where @ > 0 and an intermediate zone where -pH<@<O. As we have shown, such a zone may occur for the explicit solution as well. The experiments were not aimed at particular binary alloys, but were chosen to test the simulation model itself. Hence, while S.I. units are implied, the thermophysical parameters were not assigned dimensions.

Case 1. Heat and mass transfer of comparable propagation speeds; initial state above the liquidus line

The parameters describing this case are as follows: cs = 0.1, cL = 0.5, ks = 1, k, = 1, ys = 20, yL=40, p = 1, H = 5 0 , = 1, DL= 1, L (slab length)= 1 and 2, TA=83, T,,=50, TI = 115, Cl =0-1. Note that the liquidus line is f L ( C ) = T A + +yLC = 83+40C, whence f L ( O . l ) = 87 < TI or (Cr, T I ) lies above the liquidus line. We find that A =0.143, Tc,=95.64, Cs =0.132, Cl =0-066 for the semi-infinite slab.

Using the scheme of the section on ‘Finite difference approximation’, a simulation was performed for slabs of lengths 1 and 2 with Ax = 0.025, At = 2.5 x The output of each run consisted of temperature, concentration and phlogiston values at the spatial mesh points at times t = 0*02,0.04, . . . . At each time step up to t = 0.34, there was exactly one point with -pH<cD<O, for both slab lengths. Beginning at time t =0.36 the simulation with L = 1 exhibited a large intermediate zone. Taking L = 2 and computing until the unit interval was frozen resulted in sharp interface with exactly one intermediate node for all time steps.

In Table I1 we compare the interface location from the explicit solution with those of the simulations with slab lengths 1 and 2. For the latter, if an intermediate zone of more than one point was present, we have listed the interval on which the computed phase change occurs. Thus if the last ‘solid’ point was the mth node point and the first liquid point was the nth node point, then the interface is assumed to lie on the interval ( ( m + l)Ax, ( n - 1)Ax). For a single intermediate node, we have simply listed its location.


Table 11. Interface locations as functions of time for the explicit solution and simulations with slabs of lengths 1 and 2 (data

from Case 1)

Explicit Difference Difference t solution scheme L = 1 scheme L = 2

0.0 0.04 0.08 0.12 0.06 0.20 0.24 0.28 0.32 0.36 0.40 0.44 0.48 0.52 0-56 0.60 0.64 0.68 0.72 0.76 0.80 0.84 0.88 0.92 0.96 1.00

0.0 0.181 0.256 0.313 0,362 0,404 0.443 0.479 0.512 0.543 0.572 0.600 0,627 0.652 0.677 0.701 0.724 0.746 0.767 0.788 0.809 0.829 0.848 0.867 0.886 0.904

0.0 0.175 0.250 0.325 0.375 0.425 0.475 0.525 0.575

(0.600,l) (0.625,l) (0.675,l) (0.725,l) (0.750,l) (0.775, 1) (0.825,l) (0*850,1) (0.875,l) (0.900,l) (0.950,l) (0.975,l) Frozen Frozen Frozen Frozen Frozen

0.0 0.175 0.256 0.325 0.375 0.400 0-456 0.475 0.525 0.550 0.575 0.600 0.625 0.675 0.700 0.725 0.750 0.775 0.800 0.825 0.850 0.875 0.900 0.925 0.950 0.975

As we see, (a) doubling the length of the interval eliminated the intermediate zone and reduced the effect of the right-hand boundary, (b) the accuracy of the interface location in the second simulation was good until about t = 0.5 when the right-hand boundary began to influence the solution. This reflects the finiteness of the interval versus the semi-infinite interval for the explicit solution.

In Table I11 we compare the explicit temperature and concentration distributions of the Rubinstein solution with their computed approximations on 0 s x s 1 for a typical time of t = 0.4 for the case L = 1. There is excellent agreement at all points.

Case 2. A material with a high thermal conductivity and a low material diffusivity

The parameters used are as follows: cs = 0.1, c, = 0-5, ks = 1,000, kL = 100, 'ys = 320, 'yL=640, p = l , Ds=O*l, D L = l , H = 5 0 , L (slab length)=l , TA=83 , T0=50, TI=115, C = 0.05.

For the explicit solution, these parameter values give the constant melt temperature T,, = 99.00 and the root A =0.200. In Table IV we compare the computed front location and the front location for the semi-infinite problem given by S( t ) = 2A d( a t ) over the total freezing time. For the simulation, the temperatures associated with the last frozen and first unfrozen


Table 111. Temperature and concentration profiles at t = 0.4 for the explicit solution and a simulation of a unit slab (data from Case 1)

Temperature Concentration Explicit Difference Explicit Difference

X solution scheme solution scheme

0.0 0.025 0.05 0.075 0.1 0.125 p.150 0.175 0.200 0.225 0.25 0.275 0.300 0.325 0.35 0.375 0.400 0.425 0.45 0-475 0.500 0.525 0.55 0.575 0.600 0.625 0.65 0.675 0.700 0.725 0.75 0.775 0.800 0.825 0.85 0.875 0.9 0.925 0.95 0.975 1.0

50.00 51.57 53.14 54.70 56.27 57.84 59.41 60.97 62.54 64.10 65.66 67-22 69-78 70.34 71.90 73.46 75.01 76.56 78.1 1 79.66 81.20 82.75 84.29 85.72 86.36 86.99 87.61 88.23 88.85 89.45 90.05 90-65 91.23 91.81 92.38 92.95 93.50 94.05 94.59 95.12 95.65

50.00 51.55 53.10 54.65 56.20 57.75 59.30 60.85 62.40 63.95 65.50 67.04 69.59 70.14 71.69 73.24 74.79 76.34 77.89 79.44 80.99 82.53 84.08 85.63 86.20 86.77 87.35 87.92 88-49 89.05 89.61 90.16 90.71 91.25 91.78 92.31 92.82 93.33 93.83 94.32 94.80

0,132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0-132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.066 0.067 0.068 0.070 0.071 0.072 0.073 0.074 0.075 0.076 0.077 0.078 0.079 0.080 0.080 0.081 0.082 0.083

0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.132 0.109 0.067 0.068 0.069 0.070 0.071 0.072 0.073 0.074 0.075 0.076 0.077 0.078 0.079 0.080 0.081 0.082 0.083

nodal points lie between 98.40 and 99.03. This shows very good agreement with the explicit solution value of T,,

Case 3. A degenerate phase diagram

Consider the case with yL = yS. Now the concentration should be continuous across the phase phase change front. The case was run for the following parameters values: cs = 1, cI. = 2,


Table IV. Computed interface compared with S = 2hd(at) for Case 2

t Computed 2 h J a t

1 x 0.400 0.400 2 x 0.550 0.566 3 x 1 0 - ~ 0.700 0.693 4 x 0.800 0.800

ks=l,OOO, kL=lOO, .ys=40, yL=40, Ds=O.l, DL=O.l, p = l , L(s1ab length)=l, TA=83 ,

In Table V we compare the interface locations for the explicit solution of the semi-infinite problem with the computed solution for the finite slab. Again, excellent agreement is observed throughout the simulation.

In this case the explicit solution exhibits the behaviour described in the last section. The solutions to (37)-(40), to two significant digits, are T,, = 99, A = 0.20, C, = 0.050 and C, = 0.025. To two significant digits the concentration C(x, t ) , given by equation ( 3 5 ) , is equal to 0.050 everywhere, and since the temperature at the interface is 99, there is a zone beyond x = S ( t ) which is neither liquid nor solid. On the other hand, the computer model predicts a finite interface zone between two fronts x = X ( t ) , x = Y ( t ) , Y ( t ) a X ( t ) . For x> Y ( t ) the material state is on the liquidus line with T = TI = 115, C = Cl = 0.05.

In Table VI we see the temperature and concentration predicted by the model at time t = 1 X

To=50, T1=115, C1=0*8.

Observe the finiteness of the intermediate zone.

CONCLUSION

In this paper we have exhibited a computational approach to the problem of modelling binary alloy solidification. This approach is applicable for any choice of thermophysical parameters and can be extended to much more general conditions than have been considered before now. The method rests upon the definition of a ‘pseudo-energy’ (phlogiston) and the use of a formulation which is equivalent to the ordinary weak solution formulation for Stefan-like problems with phlogiston replacing the specific internal energy. Upon comparison with the

Table V. Interface locations as functions of time for the explicit solution and the computer

simulation (data from Case 3)

Explicit Difference t solution scheme

0.5 x lop4 1 . 0 ~ 1 0 - ~ 1.5 x 1 0 - ~ 2.0 x 1 0 - ~ 2-5 x 3.0 x 3.5 x 4.0 x 1 0 - ~

0.304 0.430 0.527 0.608 0.680 0.745 0.805 0.860

0.292 0.420 0.521 0.603 0.672 0.736 0.798 0.855


Table V1. Temperature concentration and phase profiles at t = 1 x from the computer simulation (data from Case 3)

x Temperature Concentration Phase

0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.80 0.85 0.90 0.95

50.0 56.1 62.1 68.2 74.2 82.2 86.2 92- 1 98.0

103.8 109.2 113.2 114.7 115.0 115.0 115.0 115.0 115.0 115.0 115.0

0.0500 0.0500 0.0500 0.0500 0.0500 0.0500 0.0500 0*0500 0.0500 0.0489 0.0489 0.0489 0.0489 0.0489 0,0489 0.0489 0.0489 0.0489 0.0489 0.0500

Solid Solid Solid Solid Solid Solid Solid Solid Solid

Indeterminate Indeterminate Indeterminate Indeterminate Indeterminate Indeterminate Indeterminate Indeterminate Indeterminate Indeterminate

Liquid

explicit solution for corresponding problems on a semi-infinite line, we have seen that the model is highly accurate and computationally effective. We have also made observations on the explicit solution, the consequences of which will be expanded upon in forthcoming papers.

REFERENCES

1. L. 1. Rubinstein, ‘Crystallization of a binary alloy’, section 3 of chapter 2, pp. 52-60, in The Stefan Problem (English

2. G. J. Fix, ‘Numerical methods for alloy solidification problems’, pp. 109-128 in Moving Boundary Problems (Eds.

3. A. B. Crowley and J. R. Ockendon, ‘On the numerical solution of an alloy solidification problem’, Int. J . Heat

4. A. Bermudez and C. Saguez, ‘Etude Numerique d’un Probleme de Solidification d’un Alliage’, report of Institut

5. D. G. Wilson, A. D. Solomon and V. Alexiades, ‘A shortcoming of the explicit solution for the binary alloy

translation published by the American Mathematical Society), 1967.

Wilson, Solomon and Boggs), Academic Press, New York, 1979.

Mass Transfer, 22, 941-947 (1979).

Nationale de Recherche en Informatique et en Automatique (1981).

solidification problem’, Letters in Heat and Mass Transfer, 9, 421-428 (1982).

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A model of binary alloy solidification