One-dimensional solidification of binary mixtures

MECH. RES. COMM.

Vol. 4(2), 115-122, 1977.

Pergamon Press. Printed in USA.

ONE-DIMENSIONAL SOLIDIFICATION OF BINARY MIXTURES

To Tsubaki and B.A. Boley Technological Institute, Northwestern University, Evanston, lllinois, U.SoA. (Received and accepted for print 25 January 1977)

Introduction

The analysis of solidification of alloys and mixtures requires simultaneous consideration of heat conduction [I] and diffusion [2]. The basic formulation of this coupled problem for binary mixtures has been given by Taylor [3], Wagner [4], Ivantsov [5], Lyubov and Temkin [6], who dealt with this problem under a non-stationary state, while Tiller et al. [7], Smith et al. [8] and Chalmers [9] discussed the problem under stationary conditions. In this work we extend the problem by taking into account the effect of density change occurring during the phase change. The physical nature of coupling of temperature and concentration is represented by the phase diagram of the mixture, which is linearized here for simplicity in such a way as to approximate the behavior of a dilute mixture. The problem is formulated as a one-dimensional one, as for example that of a finite slab insulated both with regard to heat and mass transfer on its surfaces. The special case, treated in detail by means of an exact and an approximate solution, of an in finite slab, admits a similarity solution when the surface temperature is kept constant. Further details would be found in reference [i0].

General Formulation

(a) Governing Equations The temperature T(x,t) and the concentration, C(x,t), in the originally

or weight-fraction of solute,

(t = 0) liquid slab (0< x < h) satisfy the diffu-

sion equation, i.e.,

This work was partially supported by a grant from the Office of Naval Research. Scientific Communication

115

116

T. TSUBAKI and B. A. BOLEY

~-~ [K s ~

(Ts)] = PsCs ~-~ (T s) (la,b)

~-£ EDs ~

(PsCs) ] = ~

f o r the s o l i d

~x

region

(PsCs)

0 < x < s(t),

t

> 0

,

and

~x (Tg)] --~x (pgcgTgv) = pgcg ~

(T) (2a,b)

~-~ ED£ ~x (p£Cg)] - ~x (pgCgv) = ~

(p C )

for the liquid region s(t) < x < h(t), t > 0 where v is a possible velocity of the liquid medium and the symbols K,p,c and D stand respectively for the thermal conductivity, efficient.

the density, the specific heat and the diffusion co-

The subscripts s and £ refer to the solid and the liquid, re-

spectively. (b) Initial Conditions (3)

T~(x,0) = T~o , C%(x,0) = C~o , S(0) = 0

for 0 < x < h(O) , where T%o and C£o are constants such that (for solidifica. tion) T%o > T --

- mC O

~O

(c) Fixed-Boundary

Conditions

Ts(0,t ) = T

T~(x,t) or ~x ET (x,t)] prescribed on x = h(t) so

~-f (PsCs)(0,t)

'

(4)

(p%C~)(x,t) prescribed on x = h(t)

= 0 ,

for t > O, where Tso is a constant such that T s o _< T o -- mCg ° . (d) Interface Conditions Ks ~

ds (Ts) -- Kg ~x (T£) = LPs ~

D

~ (PsCs) - D£ ~x (pgC)

~ s ~

T s = T~ = Tm(t), T~ = T

o

=

(i

ds -- k)PsC ~ ~

-- mC~ , C

s

(5)

= kC

for x = s (t), t > 0, where the last two conditions are obtained from the linearized phase diagram (cf. Fig. I).

The symbols L,k, - m ,

T

and T m

stand for the latent heat of fusion, the distribution coefficient,

o

the slope

of the liquidus line, the melting temperature of the mixture and that of the pure solvent, respectively. m<0.

Note that if k < I then m > 0, while if k > i,

SOLIDIFICATION OF BINARY MIXTURES

117

The Similarity Solution of the Problem

For simplicity we now assume that all physical parameters are constant, but not necessarily the same in the liquid and in the solid, and that the region is semi-infinite,i.e., h(t) = =o

Furthermore, V is defined in the following

way ([1,2,11]), which simply represents a rigid body motion of the liquid caused by the volume change due to the phase change at the interface: v(t)

(Ps . . . . P~

I

)ds ~{

(6)

A solution which satisfies all these conditions can be obtained on the basis of similarity, as was done by Neumann for the classic change-of-phase problem, by employing the so-called semi-inverse method, and by noting that all equations can be satisfied in terms of the single variable X

=--2#Kst

(7a)

In particular

where I is a constant as yet unknown, and where K stands for the thermal diffusivity, i.e., K = K/pc.

Then the solution is:

Ts = Tso +
=

erf erf

(8)

kC (X) ,

t or

c

=C

,~oLI

erfc{[~ + (e7

-

I)X]P 6 }

+

I -- d erfc{[~ + (P7 -- I)%3P4}] + - d erfc (~P4P7) J

where d = I - (I -- k ) ~

%P4P7exp(%2P42P72)erfc(%P4P7 )

The constant ~ is the solutLon of exp( -- ~2) _ P2P5P6exp( -- ~2P62P72 ) P3~f~ ~ = erf k erfc(ke6P7)

(9)

118

T. TSUBAKI and B. A. BOLEY

PI C exp( -- k 2)

L

d The dimensionless

PI

]

erfc(kP6P 7)

•

+ P5P6

(lO)

parameters PI,---,P7 are defined as:

mC =

erf k

exp( -- k2P62P72)

T =

go ' P2 To -- Tso

T

- T go

o

-- T

o

so

'

P3

=

- -

cs

L

(To -- Tso)

(ii)

'~S-

K

PS

,[~4--

Discussion

As shown in the previous section, the solution is affected by the eight nondimensional parameters PI,---,P7 and k.

PI and P2 can be either positive or

negative, while the others are non-negative.

The solution reduces to the

classical one of Neumann [i] if PI = 0, P7 = i;

thus PI is the principal

coupling parameter.

Note that all parameters except P4' P7 and k appear in

Neumann's

Hereafter Neumann's

solution.

change,(i.e.,

solution,

corrected

Cor density

when PI = 0) will be denoted by the subscript n.

noted, however,

It should be

that the correction just referred to (i.e., the effect of P7 )

is very small on k when PI = 0 (cf. insert in Fig. 4a).

The effect of coupling is shown in Figs. 2 and 3a,b. k decreases unity.

(Fig. 2);

As PI approaches unity,

it will in fact be zero for PI = I if P2 is also

The solute build-up at the interface increases as P4 increases

(Figs. 3) because of the lower rate of diffusion in the liquid relative to the rate of heat transfer. valid for the experiments of Fig° 4a and 4b.

The parameters in [12].

for Fig. 3b correspond to those

The effect of k is shown by a comparison

We have PI > 0 for 0 < k < i and PI < 0 for k > i from

the definition of the liquidus slope, and the nunerical results show that, for the values of parameters k > I.

Note, however,

chosen, k < kn for 0 < k < I and k > Xn for

that eq.

(I0) shows that k = X n only if PI = O;

the above relations between ~ and k cannot be always precisely true, although it is likely that they hold whenever P1 is not close to zero.

thus

SOLIDIFICATION OF BINARY MIXTURES

119

We will next employ approximations valid for large and small values of X, i.e., I erfc (~P4P 7 )

~

exp(-

~

2p

kP4e 7

4

2p 2 7 )

(ke4P7>>l) (12)

erfc(kP4PT) ~ i , expk)~2P42P72 ~ ~ i

(XP4P7< I)

the former of which is the asymptotic expansion of the error function. easily verified that d ~ k for %P4P7 >> i ;

It is

thus, for example, the concen-

tration C~(X) at the interface becomes "C~o/d

(exact)

J C%(~) = ~\ C~o/k

()~P4P7 >> i)

(13)

I

~C%o/

[I -- (I -- k)~?T %P4P7 ] (XP4P7 < i)

Fig° 5 shows the error incurred by using the above approximate solutions° Note that the result for ~P4P7 >> I coincides with the steady-state solution [9].

Furthermore it follows from the last of eqs. (8) that the temperatures

at the interface for the Neumann's solution (Tn) , the present solution (T) and the approximate one for large XP4P7(T ') are respectively as follows: mC mC Tn = To -- mC~o , T = To -- d~° , T' = T o -- k~° on x = s (t) The position of these in the phase diagram is shown in Fig. i.

(14) Since it is

evident from eq. (9) that only the possibilities k < d < i and I < d < k exist (which can only arise physically if m > 0 and m < O, respectively), it follows that the relation T

n

> T > T' holds in all cases at the interface.

The conditions of similarity (and its consequences, such as the constancy of concentration in the solid and in both phases at the interface) would disappear if more complicated problems were considered.

This would certainly

be the case if a finite slab were considered, or if the actual space and time dependent velocity field in the liquid were to be included.

120

T. TSUBAKI and B. A. BOLEY

References

i. 2. 3. 4. 5. 6. 7. 8. 9. i0. ii. 12.

HoS. Carslaw and JoCo Jaeger, Conduction of Heat in Solids, 2nd ed., p. 282. Clarendon Press, Oxford (1959) Jo Crank, The Mathematics of Diffusion, 2nd ed., p. 299. Clarendon Press, Oxford (1975) J°Ro Ockendon and WoRo Hodgkins, Moving Boundary Problems in Heat Flow and Diffusion, p. 120. Clarendon Press, Oxford (1975) Co Wagner,Trans. AIME 200, 154 (1954) Golo Ivantsov, Akad. Nauk SSSR Dokl. 81, 179 (1951) BoY° Lyubov and D.E. Temkin, Growth of Crystals, Vol. 3, p. 40. Consultants Bureau, New York (1962) WoA. Tiller, K.A. Jackson, J.W. Rutter and B. Chalmers, Acta Meta. ~, 428 (1953) V.Go Smith, W.Ao Tiller and JoWo Rutter, Can. J. Phys., 33, 723 (1955) Bo Chalmers, Principles of Solidification, p. 133. Wiley (1964) T. Tsubaki, M.So Thesis, Northwestern Univ. (1976) V.S. Arpaci, Conduction Heat Transfer, p. 88. Addison-Wesley (1966) J.A. de Leeuw den Bouter, P°Mo Heertjes and FoC.Ho Jongenelen, J. Crystal Growth 6, 327 (1970)

SOLIDIFICATION

OF BINARY

MIXTURES

121

.0 C 1"5 I k,

"T -T. TIo_TI°

C'° I -1,0"

Ilqutdus iidue '

Content

ration

,

0.5

% k T.

0.5

....

B--

T' 2: . . . .

'-;

T

- -

B"

\

:-,

0.1

9.09X

--'-- 1.0 .... IO.O

.

I0 ~

9.05X10' 8.76×10 -z

0~5

kC,(~,) C,o

C,(~..)

1.0

C,o/k

FIG.I: Linearized Phase Diagram and Various Solidification Temperatures (k < I)

FIG.3a: Effect of P4 on Concentration and Temperature D~stribution.

PI=O.I, P2=0.5,

P3=50, P5=P6=P7=I.0, k=0.5

7.Be

/ . . /

/ . ~

C

___~___ . . . . . . 1.0

~.0

II//i ll~/ri '

--

.....

/ Z ~

. . . .

i . . . . 0.5

o

i

0

't.;~-,

....

~.,

~--;

1.0 m Cio PI = To_T, °

T,;-'r,o

,/

./

f / /

0.5

2~63_

Concentration

pero,u,oC+o.o~ Liquidus

. . . . 0.5

1.0

o

x

FIGo3b: D i s t r i b u t i o n

of Concentration

and

P3=50, Ps=P6=P7=I, 0,

Temperature. Pl=l.01 x 10 -1, P2~1.75 x 10 -1, P3=11.9, P4=14.3, P5=4.67 x i0 -1, P6=1.42, P7=I°20, k=O°0, Tl~o=16.I°C , Tso=--3°0°C,

k=0.5, )~ =9.66 x 10 -2

C~o=4.0%, X;9.54 x 10 - 2 , Xn;2.42 x I0 - I

FIG.2: Effect of PI & P4 on X/~ n • P2=0.5

n

'

122

T. TSUBAKI and B. A. BOLEY

P7= 0.1 0.5 1.2

.PT

1.0

k k.

k k.

ko

O. I

9.69x~0

-~

0.5

9.68x10

-z

1.0

9.65x10

-~

1.5

9,64x10 9.63xi0

-z -=

2.0

I.I

,,- oo.,o/

0.5

~-.I 0.5

1.5

9.69xl 0 .2 9 . 6 8 x I 0 -2 9 . 6 5 x I 0 -2 9 . 6 4 x I 0 -2

.0 i.5

1.0

i

,

i

,

0

i

i

i

,

h

T

I

1.0

0.5

L

I

I

t

I

O. I

0.2

0.3

0.4

0.5

= m Clo

PI

mCi

P,

To_T, °

o

To-T. °

FIGo4a: Effect of PI & P7 on k/i n

FIG.4b: Effect of PI & P7 on k/k n

for k
for k>l.

P5=P6=I.0, k=0.5

P5=P6=I.0, k=l.2

P2=0.5, P3=50, P4=IO,

20 / /

C,(X)

k=0.5

cio - -

Exact

----....

Approximate Approximate

(OC ~. I ) (o(~1)

IO

\

.__ k= 2.0

t.O

2.O

3.O

4.0 a=

FIG. 5 :

kP,P~

Comparison of Exact and Approximate Solutions